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GCSE Maths · EdexcelChapter 4 · Decimals, rounding and estimation
Chapter 4 · Higher tier

GCSE Maths Decimals, Rounding and Estimation (Higher): worked examples and questions with answers

By the end of this chapter you will be able to

  • Use place value to read, write, compare and order decimals
  • Add, subtract, multiply and divide decimals using formal written methods
  • Change terminating decimals into fractions and fractions into decimals
  • Round to a number of decimal places or significant figures, and truncate
  • Estimate by rounding to 1 significant figure and use estimates to check answers
  • Write error intervals for rounded and truncated numbers and use limits of accuracy
  • (Higher) Find upper and lower bounds of calculations

You should already know

  • Place value and the four operations with whole numbers (Chapter 1)
  • Multiplying and dividing by 10, 100 and 1000
  • Simplifying fractions (Chapter 3)
  • Rounding whole numbers to the nearest 10 or 100

Where this leads

  • Percentages (Chapter 5)
  • Standard form (Chapter 6)
  • Substituting into formulae (Chapter 8)
  • Compound measures (Chapter 16)
  • Accuracy of answers in trigonometry (Chapter 20)

Edexcel GCSE (9–1) Mathematics 1MA1, Higher tier. Content references: N2, N10, N14, N15, N16.

Printable worksheets. Every lesson in this chapter has a free worksheet and an answer sheet to download and print.

Exercise P  Do you remember?

Non-calculator
1
FIn the number 36 480, write down the value of the digit 6.
2
FWork out 345 × 10.
3
FRound 4673 to the nearest 100.
4
FWork out 36 × 12.
5
FWrite the fraction 1220 in its simplest form.
4.1

Place value and ordering decimals

Edexcel 1MA1: N1, N2
Grade 1-3

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Place value and ordering decimalsComing soon

A decimal is a number that has a decimal point. The point separates the whole-number part from the part that is smaller than 1. When you see £3.75 you already use decimals: 3 whole pounds, then 75 hundredths of a pound (75 pence).

Every digit in a number has a place value, which is how much that digit is worth because of the column it is in. Moving one column to the left makes a digit worth 10 times more. Moving one column to the right makes a digit worth 10 times less. This carries on past the decimal point: after the units column come tenths, then hundredths, then thousandths.

hundreds3tens0units5tenths6hundredths2thousandths7
The number 305.627: 3 hundreds, 0 tens, 5 units, 6 tenths, 2 hundredths and 7 thousandths. The dot is the decimal point.

So in 305.627 the digit 6 is worth 6 tenths, which is 0.6, and the digit 7 is worth 7 thousandths, which is 0.007. The zero in the tens column is a placeholder: it is there so that the 3 stays in the hundreds column. Without it the number would be 35.627, which is a very different number.

A zero on the end of a decimal does not change its value. 0.5, 0.50 and 0.500 are all the same number: five tenths. We can add zeros on the end whenever we need to.

A hundred square split into 100 small squares. 37 of them are shaded, so 0.37 of the square is shaded: 3 tenths and 7 hundredths.

Comparing and ordering decimals

To compare two decimals, write them with the same number of decimal places by adding zeros on the end. Then compare the digits column by column, starting from the left. The first column where the digits differ decides which number is bigger. For example 0.5 and 0.45: write 0.5 as 0.50. Now 0.50 and 0.45 have the same number of places. The tenths digits are 5 and 4, and 5 is bigger, so 0.5 is bigger than 0.45. Do not be fooled by the number of digits: a longer decimal is not always bigger.

We use these symbols: < means “is less than”, > means “is greater than” and = means “is equal to”. The sign always opens towards the bigger number: 0.45 < 0.5.

00.51AB
A number line from 0 to 1 split into 10 equal parts. Each part is 0.1, so A is 0.3 and B is 0.8.

Multiplying and dividing by 10, 100 and 1000

Because each column is worth 10 times the one to its right, multiplying by 10 moves every digit one column to the left. Multiplying by 100 moves every digit two columns to the left, and by 1000 three columns. Dividing moves the digits to the right instead. The decimal point itself stays where it is; it is the digits that move. If a column becomes empty, fill it with a 0.

units3tenths4hundredths7
3.47 has 3 units, 4 tenths and 7 hundredths.
hundreds3tens4units7
3.47 × 100 = 347: each digit has moved 2 columns to the left, so the 3 is now in the hundreds column.

Key ideas

  • Each column is worth 10 times the column on its right: … hundreds, tens, units, tenths, hundredths, thousandths…
  • A 0 on the end of a decimal does not change its value: 2.3 = 2.30 = 2.300.
  • To compare or order decimals, give them the same number of decimal places (add zeros on the end), then compare column by column from the left.
  • × 10, 100, 1000 moves the digits left 1, 2, 3 places. ÷ 10, 100, 1000 moves them right. The decimal point does not move.
Worked example 4.1.1
In the number 52.384, write down the value of the digit 8.

Key words. Place value is the value of a digit because of where it is in the number. Tenths, hundredths, thousandths are the columns to the right of the decimal point.

Method. The value of a digit depends on the column it is in. Find the column, then multiply the digit by the column value.

  1. Write the number in a place value chart. Each column is worth ten times the column to its right. The columns to the right of the decimal point are tenths, hundredths, thousandths...
    tens5units2tenths3hundredths8thousandths4
  2. Find the column of the digit and read its name. The digit 8 is in the hundredths column, so each unit in this column is worth 0.01.
    8 is in the hundredths column
  3. Work out its value. Multiply the digit by the value of its column.
    8 × 1100 = 8 ÷ 100 = 0.08

Check. Put the value back into the chart: 0.08 has a 8 in the hundredths column and zeros elsewhere. ✓

Watch out. The value of the digit depends on its column: the 5 in 0.05 is worth 5 hundredths (0.05), not 5.

Worked example 4.1.2
Write the correct sign, < or >, between 0.6 and 0.58.

Method. To compare decimals, line up the decimal points, add zeros so both have the same number of places, then compare column by column from the left.

  1. Give both numbers the same number of decimal places by adding zeros on the end. Adding a zero at the end of a decimal does not change its value (0.5 = 0.50), but it lets us compare the columns one at a time.
    0.60 0.58
  2. Compare the digits column by column, starting from the left (the biggest column). The first column where the digits are different decides which number is bigger. Earlier columns that match do not matter.
    The first difference is in the tenths column: 6 and 5.
  3. Decide. 6 is bigger than 5, so the first number is bigger.
    0.6 > 0.58

Check. Write both as whole numbers of hundredths: 60 and 58, and 60 is bigger than 58. ✓

Watch out. A decimal with more digits is not always bigger. 0.5 is bigger than 0.45 because 5 tenths beats 4 tenths.

Worked example 4.1.3
Write these numbers in order, smallest first:   0.35,   0.305,   0.5,   0.053

Method. To put decimals in order, give them all the same number of decimal places (add zeros), then compare them like whole numbers.

  1. Write every number with the same number of decimal places. Add zeros on the end until every number has 3 decimal places. This does not change their values and makes the comparison fair.
    0.35 = 0.350 0.305 = 0.305 0.5 = 0.500 0.053 = 0.053
  2. Now compare them as whole numbers of thousandths. Every number is now a count of thousandths, so the bigger the whole number, the bigger the decimal.
    0.350 = 350 thousandths
    0.305 = 305 thousandths
    0.500 = 500 thousandths
    0.053 = 53 thousandths
  3. Put the whole numbers in order, smallest first. Write the numbers from the smallest to the largest (or the other way round if the question asks).
    53, 305, 350, 500
  4. Write the original decimals in that order. The answer must use the numbers from the question, not the ones with extra zeros.
    0.053, 0.305, 0.35, 0.5

Check. Look at the answer: each number is bigger than the one before it. ✓

Watch out. Do not decide by the number of digits. 0.7 is bigger than 0.65 even though 0.65 looks longer.

Worked example 4.1.4
Work out 5.62 × 1000.

Method. Multiplying by 10, 100, 1000... makes the digits move to the left; dividing makes them move to the right. One place for each zero.

  1. Decide how many places the digits move. 1000 has 3 zeros, so every digit moves 3 places. Each place is a factor of 10.
    × 1000: digits move 3 places to the left
  2. Move the digits (the decimal point stays where it is). When we multiply, the number gets bigger, so the digits move left. Fill any gaps with zeros.
    units5tenths6hundredths2
  3. Write the answer.
    5.62 × 1000 = 5620

Check. Do the opposite operation: 5620 ÷ 1000 = 5.62. ✓

Watch out. The decimal point does not move; the digits do. Check that the answer is bigger when you multiply and smaller when you divide.

Worked example 4.1.5
Work out 8.4 ÷ 100.

Method. Multiplying by 10, 100, 1000... makes the digits move to the left; dividing makes them move to the right. One place for each zero.

  1. Decide how many places the digits move. 100 has 2 zeros, so every digit moves 2 places. Each place is a factor of 10.
    ÷ 100: digits move 2 places to the right
  2. Move the digits (the decimal point stays where it is). When we divide, the number gets smaller, so the digits move right. Fill any gaps with zeros. Do not forget a 0 before the decimal point if there are no units.
    units8tenths4
  3. Write the answer.
    8.4 ÷ 100 = 0.084

Check. Do the opposite operation: 0.084 × 100 = 8.4. ✓

Watch out. The decimal point does not move; the digits do. Check that the answer is bigger when you multiply and smaller when you divide.

Watch out

A decimal with more digits is not bigger. 0.5 is bigger than 0.45, because 5 tenths is more than 4 tenths. Always add zeros (0.50 and 0.45) before you compare.

Exercise 4A  Place value and ordering decimals

Non-calculator
1
FWrite down the value of the highlighted digit in each number.
(a)4.7
(b)0.072
(c)17.3
(d)6.157
(e)7.04
(f)0.917
2
FWrite down the missing symbol, <, > or =, between each pair of numbers.
(a)0.7   ?   0.07
(b)0.45   ?   0.5
(c)3.09   ?   3.9
(d)12.50   ?   12.5
(e)0.612   ?   0.62
(f)5.08   ?   5.008
3
FWrite each set of numbers in order, starting with the smallest.
(a)0.4, 0.04, 0.44
(b)2.5, 2.05, 2.55, 2.505
(c)0.81, 0.8, 0.081, 0.18
4
FWrite the number that has 5 tens, 3 units, 4 tenths and 7 thousandths.
5
FWork out
(a)3.8 × 10
(b)0.46 × 100
(c)27.5 × 1000
(d)64 ÷ 10
(e)9.1 ÷ 100
(f)0.7 ÷ 10
6
FWrite down the decimal shown by each arrow on this number line, which goes from 2 to 3.
22.12.22.32.42.52.62.72.82.93PQ
(a)arrow P
(b)arrow Q
7
FFill in the missing number.
(a)5.2 × ? = 520
(b)0.083 × ? = 8.3
(c)4500 ÷ ? = 4.5
8
PSFour swimmers have these times, in seconds, in a 50 m race:   28.4,   28.04,   28.44,   28.39. Write down the fastest time. (The smallest time is the fastest.)
9
FWrite down a decimal that lies between 2.3 and 2.4.
10
PSA box of 100 pencils costs £24. Each pencil costs £24 ÷ 100. How much is that for one pencil?
11
PSA paper clip is 3.2 cm long. Work out the length of a chain of 10 000 paper clips, in metres. (1 m = 100 cm.)
12
RAli says: “0.45 is bigger than 0.5 because 45 is bigger than 5.” Is Ali right? Give a reason.
13
RJess says: “To multiply a number by 10 you just add a zero on the end.” Show with an example that Jess is wrong.
14
RFiona has three number cards, 1, 5 and 9, and a decimal point card. She uses all four cards to make a number. The decimal point cannot go at the start or at the end. Write down all the different numbers she can make, in order, starting with the smallest. How many are there?
4.2

Adding and subtracting decimals

Edexcel 1MA1: N2
Grade 1-3

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Adding and subtracting decimalsComing soon

You can only add or subtract things that are the same kind. Hundreds are added to hundreds, tens to tens, and so on. This is also true for decimals: tenths are added to tenths and hundredths to hundredths. That is why the decimal points must be lined up in a vertical line: it puts every column of the same value under each other.

The method is called a formal written method (also called column addition and column subtraction):

  1. Write the numbers one under the other with the decimal points in a vertical line.
  2. If one number has fewer decimal places, fill the gaps with zeros. This does not change its value.
  3. Work from the right (the smallest column) to the left, adding or subtracting one column at a time.
  4. Put the decimal point in the answer straight down, in line with the others.

A whole number has a decimal point that you cannot see, after the units digit: 7 is the same as 7.0 or 7.00. When the question has a whole number, write the point and the zeros yourself.

2.41.752.4 + 1.75 = 4.15
A bar model: the whole bar is the total, split into the two amounts being added.

Layout for 2.4 + 1.75 (a zero is added so both numbers have 2 decimal places):

2.40 + 1.75 ---- 4.15

Adding. If a column adds to 10 or more, write the units digit and carry the ten to the next column on the left. Subtracting. If the top digit is smaller than the bottom digit, you cannot take away. Exchange one from the next column on the left: that one becomes ten in the column you are working on. Always subtract the smaller number from the larger number. After you finish, check by doing the opposite: add to check a subtraction, or subtract to check an addition.

Key ideas

  • Write the numbers with the decimal points in a vertical line.
  • Fill empty places with zeros, e.g. 5 = 5.00.
  • Work from the right, one column at a time. Carry when adding, exchange when subtracting.
  • The decimal point in the answer goes straight down.
  • Check with the opposite operation.
Worked example 4.2.1
Work out 45.3 + 28.75.

Key words. A column holds digits of the same place value. Carrying moves ten of one column into one of the next column to the left. Exchanging (borrowing) moves one of the next column on the left into ten of this column.

Method. To add decimals, write the numbers in columns with the decimal points in a vertical line, then add column by column.

  1. Write the numbers in columns, with the decimal points directly under each other. Fill the gaps with zeros. Each digit must sit in the right column (units under units, tenths under tenths...). A zero on the end of a decimal does not change its value, so writing 4.5 as 4.50 is allowed.
    45.30 + 28.75 -----
  2. Add column by column, starting at the right (the smallest column). When a column adds to 10 or more, write the units digit and carry the 10 to the next column to the left.
    hundredths: 0 + 5 = 5
    tenths: 3 + 7 = 10, write 0 and carry 1
    Put the decimal point straight down, directly under the other decimal points.
    units: 5 + 8 + 1 (carried) = 14, write 4 and carry 1
    tens: 4 + 2 + 1 (carried) = 7
  3. Read the answer. Write the digits, with the decimal point where you put it.
    45.3 + 28.75 = 74.05

Check. Use the opposite operation: 74.05 − 28.75 = 45.3. ✓
Estimate: 50 + 30 = 80, close to 74.05. ✓

Watch out. Line up the decimal points, not the right-hand ends of the numbers. In 3.5 + 0.27 the 5 (tenths) goes above the 2 (tenths), not above the 7.

Worked example 4.2.2
Work out 9.04 − 3.576.

Key words. A column holds digits of the same place value. Carrying moves ten of one column into one of the next column to the left. Exchanging (borrowing) moves one of the next column on the left into ten of this column.

Method. To subtract decimals, write the numbers in columns with the decimal points in a vertical line, then subtract column by column.

  1. Write the numbers in columns, with the decimal points directly under each other. Fill the gaps with zeros. Each digit must sit in the right column (units under units, tenths under tenths...). A zero on the end of a decimal does not change its value, so writing 4.5 as 4.50 is allowed.
    9.040 − 3.576 -----
  2. Subtract column by column, starting at the right (the smallest column). When the top digit is smaller than the bottom digit, exchange 1 from the next column on the left. That 1 is worth 10 in this column, so add 10 to the top digit.
    thousandths: 0 is smaller than 6, so exchange 1 from the next column on the left: 10 − 6 = 4
    hundredths: the 4 gave 1 to the column on its right, so it is now 3: 3 is smaller than 7, so exchange 1 from the next column on the left: 13 − 7 = 6
    tenths: the top digit is 0, but the column to its right needs 1 from it. So it takes 1 from the next column on the left (making 10) and then gives 1 away, leaving 9. 9 − 5 = 4
    Put the decimal point straight down, directly under the other decimal points.
    units: the 9 gave 1 to the column on its right, so it is now 8: 8 − 3 = 5
  3. Read the answer. Write the digits, with the decimal point where you put it.
    9.04 − 3.576 = 5.464

Check. Use the opposite operation: 5.464 + 3.576 = 9.04. ✓
Estimate: 9 − 4 = 5, close to 5.464. ✓

Watch out. Line up the decimal points, and fill gaps with zeros first: 5 − 2.37 needs 5.00 − 2.37. Subtract the smaller number from the bigger one.

Worked example 4.2.3
Work out 100 − 7.95.

Key words. A column holds digits of the same place value. Carrying moves ten of one column into one of the next column to the left. Exchanging (borrowing) moves one of the next column on the left into ten of this column.

Method. To subtract decimals, write the numbers in columns with the decimal points in a vertical line, then subtract column by column.

  1. Write the numbers in columns, with the decimal points directly under each other. Fill the gaps with zeros. Each digit must sit in the right column (units under units, tenths under tenths...). A zero on the end of a decimal does not change its value, so writing 4.5 as 4.50 is allowed.
    100.00 − 7.95 ------
  2. Subtract column by column, starting at the right (the smallest column). When the top digit is smaller than the bottom digit, exchange 1 from the next column on the left. That 1 is worth 10 in this column, so add 10 to the top digit.
    hundredths: 0 is smaller than 5, so exchange 1 from the next column on the left: 10 − 5 = 5
    tenths: the top digit is 0, but the column to its right needs 1 from it. So it takes 1 from the next column on the left (making 10) and then gives 1 away, leaving 9. 9 − 9 = 0
    Put the decimal point straight down, directly under the other decimal points.
    units: the top digit is 0, but the column to its right needs 1 from it. So it takes 1 from the next column on the left (making 10) and then gives 1 away, leaving 9. 9 − 7 = 2
    tens: the top digit is 0, but the column to its right needs 1 from it. So it takes 1 from the next column on the left (making 10) and then gives 1 away, leaving 9. 9 − 0 = 9
    hundreds: the 1 gave 1 to the column on its right, so it is now 0: 0 − 0 = 0
  3. Read the answer. Write the digits, with the decimal point where you put it.
    100 − 7.95 = 92.05

Check. Use the opposite operation: 92.05 + 7.95 = 100. ✓
Estimate: 100 − 8 = 92, close to 92.05. ✓

Watch out. Line up the decimal points, and fill gaps with zeros first: 5 − 2.37 needs 5.00 − 2.37. Subtract the smaller number from the bigger one.

Worked example 4.2.4
Rasha buys a book for £4.85 and a magazine for £12.50. She pays with a £20 note. How much change does she get?

Method. Add the prices to find the total cost, then subtract the total from the money paid.

  1. Write down what we know. A book costs £4.85 and a magazine costs £12.50. Rasha pays with a £20 note.
    £4.85, £12.50 and £20
  2. Add the two prices. Line up the decimal points, fill the gaps with zeros, and add column by column from the right.
    4.85 + 12.50 ----- 17.35
  3. Subtract the total from £20 to find the change. Line up the decimal points, fill the gaps with zeros, and subtract column by column from the right.
    20.00 − 17.35 ----- 2.65
  4. Write the answer as money. Amounts of money are written with two decimal places.
    £2.65

Check. Add the change to the total: £17.35 + £2.65 = £20. ✓

Watch out. Write the £ sign in the answer, and give amounts of money with two decimal places (£2.65, not £2.6).

Watch out

Line up the decimal points, not the ends of the numbers. In 3.5 + 0.27 the 5 (tenths) goes above the 2 (tenths). If you line up the right-hand ends you will add tenths to hundredths and get the wrong answer.

Exercise 4B  Adding and subtracting decimals

Non-calculator
1
FWork out
(a)3.4 + 2.5
(b)0.6 + 0.27
(c)12.7 + 8.5
(d)4.56 + 3.78
(e)0.907 + 0.85
(f)25 + 3.64
2
FWork out
(a)7.9 − 3.4
(b)6.5 − 2.8
(c)12.4 − 5.67
(d)10 − 3.45
(e)8.003 − 2.9
(f)0.5 − 0.347
3
PSTom buys a drink for £1.85 and a sandwich for £3.60. He pays with a £10 note. How much change should he get?
4
PSThree parcels have masses 2.35 kg, 4.8 kg and 0.975 kg. A courier bag can hold at most 10 kg. How much more mass can be added to the bag after the three parcels?
5
PSA ribbon is 3 m long. A piece 1.85 m long is cut off. How much ribbon is left?
6
FFind the missing number.
(a)4.6 + ? = 9.1
(b)? + 2.75 = 6
(c)12 − ? = 7.4
7
RSara works out 3.5 + 0.27 and gets 0.62. Explain her mistake and work out the correct answer.
8
PSFour runners in a relay team run 11.43 s, 10.98 s, 11.07 s and 11.52 s. Work out the total time of the team in seconds.
9
PSThe rectangle has length 5.4 cm and width 3.65 cm. Work out its perimeter.
5.4 cm3.65 cm
10
PSAt midnight the temperature was −3.4°C. By morning it had risen by 5.7°C. What was the temperature in the morning?
11
RShow that 2.4 + 3.7 + 1.9 = 8.
12
RTwo decimals have a total of 1 and a difference of 0.3. Work out the two decimals.
13
RTwo decimals x and y satisfy x + y = 5.28 and x − y = 1.46. Work out x and y.
4.3

Multiplying decimals

Edexcel 1MA1: N2
Grade 2-4

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Multiplying decimalsComing soon

What is 0.3 × 0.4? Think of a big square with side 1 whose area is 1. Cut it into 10 rows and 10 columns: that makes 100 small squares, and each one is worth one hundredth (0.01). A rectangle that is 0.3 high and 0.4 wide covers 3 rows and 4 columns, which is 3 × 4 = 12 small squares. Twelve hundredths is 0.12. So 0.3 × 0.4 = 0.12.

0.30.411
The shaded rectangle is 0.3 by 0.4. It covers 12 of the 100 small squares, so 0.3 × 0.4 = 0.12.

This shows why the method works. Tenths times tenths make hundredths: 0.3 = 310 and 0.4 = 410, and 310 × 410 = 12100 = 0.12 (multiply tops, multiply bottoms, as in Chapter 3). So we can multiply the digits as if they were whole numbers (3 × 4 = 12), and then the decimal points tell us where to put the decimal point in the answer.

The method

  1. Count the decimal places in each number and add them up. (The total is the number of decimal places the answer will have.)
  2. Ignore the decimal points and multiply the numbers as whole numbers. Use long multiplication if you need to.
  3. Put the decimal point back so that the answer has the total number of decimal places. If there are not enough digits, add zeros at the front.
  4. Check with an estimate: round each number to 1 significant figure and multiply (see section 4.7).

Notice that you do not line up the decimal points when you multiply. That rule is only for adding and subtracting.

Using a fact you already know

If you know 23 × 45 = 1035, you can write down 2.3 × 4.5 without any new multiplying. 2.3 is 23 divided by 10, and 4.5 is 45 divided by 10, so the answer is 1035 divided by 10 and divided by 10 again: 10.35. The digits stay the same; only the position of the decimal point changes.

Multiplying by a number less than 1

Multiplying by a number between 0 and 1 gives a smaller answer. For example 8 × 0.5 = 4: this is half of 8. Do not expect multiplication to always make things bigger.

Key ideas

  • Count the total number of decimal places in the question. The answer has that many.
  • Multiply the digits as whole numbers, then put the decimal point back.
  • Do not line up the decimal points when multiplying.
  • Use a known fact: if you multiply a number by 10, the answer is 10 times bigger; if you divide a number by 10, the answer is 10 times smaller.
  • Multiplying by a number between 0 and 1 makes the answer smaller.
Worked example 4.3.1
Work out 0.3 × 0.4.

Key words. A decimal place is a digit after the decimal point. The product is the answer to a multiplication.

Method. Multiply decimals like whole numbers, then put the decimal point back: the answer has as many decimal places as the two numbers together.

  1. Ignore the decimal points for now and count the decimal places. Multiplying decimals is the same as multiplying whole numbers, then fixing the decimal point at the end. First count how many digits come after the decimal point in each number.
    0.3 has 1 decimal place and 0.4 has 1 decimal place.
    Total decimal places = 1 + 1 = 2
  2. Write the numbers without the decimal points and multiply them as whole numbers. Removing the points turns the question into a whole-number multiplication.
    3 × 4
  3. Multiply 3 by 4, one digit at a time, from the right. Multiply each digit by the multiplier; when the answer is 10 or more, write the units digit and carry the tens.
    3 × 4 = 12, write 12
    = 12
  4. Put the decimal point in the answer: count 2 places from the right. The number of decimal places in the answer equals the total number of decimal places in the question. Add zeros at the front if there are not enough digits.
    12 with 2 decimal places = 0.12

Check. Divide back: 0.12 ÷ 0.4 = 0.3. ✓

Watch out. Do not line up the decimal points when multiplying. Count the total decimal places instead. Also check the size of the answer with an estimate.

Worked example 4.3.2
Work out 4.36 × 7.

Key words. A decimal place is a digit after the decimal point. The product is the answer to a multiplication.

Method. Multiply decimals like whole numbers, then put the decimal point back: the answer has as many decimal places as the two numbers together.

  1. Ignore the decimal points for now and count the decimal places. Multiplying decimals is the same as multiplying whole numbers, then fixing the decimal point at the end. First count how many digits come after the decimal point in each number.
    4.36 has 2 decimal places and 7 has 0 decimal places.
    Total decimal places = 2 + 0 = 2
  2. Write the numbers without the decimal points and multiply them as whole numbers. Removing the points turns the question into a whole-number multiplication.
    436 × 7
  3. Multiply 436 by 7, one digit at a time, from the right. Multiply each digit by the multiplier; when the answer is 10 or more, write the units digit and carry the tens.
    6 × 7 = 42, write 2, carry 4
    3 × 7 + 4 = 25, write 5, carry 2
    4 × 7 + 2 = 30, write 30
    = 3052
  4. Put the decimal point in the answer: count 2 places from the right. The number of decimal places in the answer equals the total number of decimal places in the question. Add zeros at the front if there are not enough digits.
    3052 with 2 decimal places = 30.52

Check. Estimate by rounding to 1 significant figure: 4 × 7 = 28, which is close to 30.52. ✓

Watch out. Do not line up the decimal points when multiplying. Count the total decimal places instead. Also check the size of the answer with an estimate.

Worked example 4.3.3
Work out 3.6 × 0.24.

Key words. A decimal place is a digit after the decimal point. The product is the answer to a multiplication.

Method. Multiply decimals like whole numbers, then put the decimal point back: the answer has as many decimal places as the two numbers together.

  1. Ignore the decimal points for now and count the decimal places. Multiplying decimals is the same as multiplying whole numbers, then fixing the decimal point at the end. First count how many digits come after the decimal point in each number.
    3.6 has 1 decimal place and 0.24 has 2 decimal places.
    Total decimal places = 1 + 2 = 3
  2. Write the numbers without the decimal points and multiply them as whole numbers. Removing the points turns the question into a whole-number multiplication.
    36 × 24
  3. Multiply 36 by each digit of 24. Long multiplication: multiply by the units digit, then by the tens digit (the digit in the tens column is worth ten times as much, so put a 0 on the end of that line), and so on.
    36 × 4
    6 × 4 = 24, write 4, carry 2
    3 × 4 + 2 = 14, write 14
    = 144

    36 × 2: the 2 is in the tens column, so it really means 20, and we work out 36 × 2 then put 1 zero on the end
    6 × 2 = 12, write 2, carry 1
    3 × 2 + 1 = 7
    72 with 1 zero on the end = 720
  4. Add the partial answers. The full product is the sum of the lines above.
    144 + 720 = 864
  5. Put the decimal point in the answer: count 3 places from the right. The number of decimal places in the answer equals the total number of decimal places in the question. Add zeros at the front if there are not enough digits.
    864 with 3 decimal places = 0.864

Check. Estimate by rounding to 1 significant figure: 4 × 0.2 = 0.8, which is close to 0.864. ✓

Watch out. Do not line up the decimal points when multiplying. Count the total decimal places instead. Also check the size of the answer with an estimate.

Worked example 4.3.4
Given that 23 × 45 = 1035, work out 2.3 × 0.45.

Method. Use the fact you know. Work out how each number has been multiplied or divided by a power of 10, and make the same changes to the known answer.

  1. Compare the first numbers. See what has happened to each number compared to the one in the fact you know.
    2.3 = 23 ÷ 10
  2. Compare the second numbers. Do the same for the other number.
    0.45 = 45 ÷ 100
  3. Make the same changes to the known answer. If a number is made 10 times smaller, the product is 10 times smaller. Start from the known answer 23 × 45 = 1035.
    1035 ÷ 10 ÷ 100 = 1.035

Check. Estimate: 2 × 0.5 = 1, close to 1.035. ✓

Watch out. If you divide one number by 10 and the other by 100, the product is divided by 10 and then by 100: that is 1000 in total.

Worked example 4.3.5
Imran buys 12 pens at £1.35 each. How much does he pay in total?

Method. Cost of 12 pens = price of one pen multiplied by 12.

  1. Write the calculation. Each pen costs £1.35 and Imran buys 12.
    1.35 × 12
  2. Multiply the price of one pen by the number of pens. Ignore the decimal points and multiply as whole numbers. Then put the decimal point back, so the answer has as many decimal places as the two numbers together.
    1.35 × 12: ignore the points, 135 × 12 = 1620
    2 + 0 = 2 decimal places in total, so the answer is 16.2
  3. Write the answer as money. Amounts of money are written with two decimal places.
    £16.20

Check. Estimate: £1 × 12 = £12 and £1.35 is a bit more, so about £16 is sensible. ✓

Watch out. The answer is money, so write it with 2 decimal places (£16.20, not £16.2).

Watch out

The number of decimal places in the answer is the total for both numbers, not the number of decimal places in one of them. 0.2 × 0.3 has 1 + 1 = 2 decimal places, so the answer is 0.06 (not 0.6).

Exercise 4C  Multiplying decimals

Non-calculator
1
FWork out
(a)0.4 × 0.2
(b)0.5 × 0.3
(c)0.6 × 7
(d)1.2 × 3
(e)0.07 × 0.9
(f)2.5 × 0.4
2
FWork out
(a)3.4 × 5
(b)2.15 × 4
(c)6.3 × 0.8
(d)0.25 × 12
(e)1.8 × 1.5
(f)0.45 × 0.06
3
FWork out
(a)2.4 × 1.3
(b)0.36 × 0.15
(c)1.07 × 0.35
(d)5.2 × 0.07
(e)3.14 × 2.5
(f)0.9 × 0.004
4
FUse the fact that 36 × 47 = 1692 to work out
(a)3.6 × 47
(b)0.36 × 4.7
(c)360 × 0.47
(d)0.036 × 0.47
5
PSMaria buys 8 books that each cost £6.75. Work out the total cost.
6
PSGran buys 12 packs of rice. Each pack has a mass of 0.45 kg. Work out the total mass of rice.
7
PSA rectangle has length 6.5 cm and width 3.2 cm. Work out its area.
6.5 cm3.2 cm
8
PSA fence is made from 15 planks laid end to end. Each plank is 2.4 m long. How long is the fence?
9
RTick the correct statement and give a reason: “Multiplying a number by 0.5 always makes it bigger.” True or false?
10
RSam says: “0.2 × 0.3 = 0.6 because 2 × 3 = 6.” Explain what is wrong and give the correct answer.
11
RWithout working it out, is 0.7 × 0.8 bigger than 0.7 or smaller than 0.7? Give a reason, then work out 0.7 × 0.8.
12
RShow that 0.15 × 0.24 = 0.036. Then use this result to write down the value of 1.5 × 2.4 and the value of 150 × 0.024.
13
RFind the missing digit:   3.□ × 0.4 = 1.36
4.4

Dividing decimals

Edexcel 1MA1: N2
Grade 2-4

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Dividing decimalsComing soon

Division is the opposite of multiplication, and there are two situations to learn: dividing a decimal by a whole number, and dividing by a decimal.

Dividing a decimal by a whole number

Use short division (the “bus stop” method) exactly as you do with whole numbers, but keep the decimal point in the answer directly above the decimal point in the question. Work from the left. At each digit, divide, write the answer above, and carry any remainder to the next digit. If you reach the end of the number and there is still a remainder, add zeros after the decimal point and carry on. Zeros after the decimal point do not change the value, so 7.5 = 7.50 = 7.500. For example 7.5 ÷ 4 = 1.875.

Dividing by a decimal

It is hard to divide by a decimal, so we change the question into an easier one with the same answer. Multiply both numbers by 10, 100... until the number you are dividing by is a whole number. This works because you can multiply the top and bottom of a fraction by the same number without changing its value (Chapter 3). For example 6 ÷ 0.3 asks “how many 0.3s fit into 6?” This is the same as asking how many 3s fit into 60, which is 60 ÷ 3 = 20.

Dividing by a number less than 1 makes the answer bigger. For example 5 ÷ 0.5 = 10, because there are 10 halves in 5.

Using known facts

If you know 96 ÷ 8 = 12, you can work out 9.6 ÷ 8 = 1.2, 0.96 ÷ 8 = 0.12 and 960 ÷ 8 = 120. The digits stay the same, only the place value changes. Be careful when the divisor changes: dividing by a smaller number gives a bigger answer.

0.30.30.30.30.31.5 is made of five pieces of 0.3, so 1.5 &divide; 0.3 = 5
How many 0.3s are in 1.5? Count the pieces: 5.

Key ideas

  • Short division: keep the decimal point in the answer above the decimal point in the question.
  • If there is a remainder at the end, add zeros after the decimal point and keep dividing.
  • To divide by a decimal, multiply both numbers by 10, 100... so the divisor is a whole number.
  • Dividing by a number less than 1 gives a bigger answer.
  • Check by multiplying the answer by the divisor.
Worked example 4.4.1
Work out 7.5 ÷ 4.

Key words. The divisor is the number you divide by. The quotient is the answer to a division. The remainder is what is left over.

Method. To divide a decimal by a whole number, use short division and keep the decimal point in line.

  1. Set up the short division. Write the number being divided under the bus stop and the divisor outside it. The decimal point in the answer will sit directly above the decimal point in the question.
    4 ) 7.5
  2. Divide 7.5 by 4 using short division (the bus stop method), working left to right. At each digit: put any remainder in front of the next digit, divide, and write the answer on top. If a remainder is left at the end of the number, add zeros after the decimal point and keep going. Adding zeros after the decimal point does not change the value.
    7 ÷ 4 = 1 remainder 3
    Put the decimal point in the answer straight above the decimal point in the question.
    35 ÷ 4 = 8 remainder 3
    30 ÷ 4 = 7 remainder 2 (an extra zero added after the decimal point)
    20 ÷ 4 = 5 exactly (an extra zero added after the decimal point)
  3. Read the quotient (the answer on top). Write the digits you found, with the decimal point in the position shown.
    7.5 ÷ 4 = 1.875

Check. Multiply back: 1.875 × 4 = 7.5. ✓

Watch out. Keep the decimal point in the answer directly above the decimal point in the question, and do not forget to add zeros if a remainder is left.

Worked example 4.4.2
Work out 8.64 ÷ 3.

Key words. The divisor is the number you divide by. The quotient is the answer to a division. The remainder is what is left over.

Method. To divide a decimal by a whole number, use short division and keep the decimal point in line.

  1. Set up the short division. Write the number being divided under the bus stop and the divisor outside it. The decimal point in the answer will sit directly above the decimal point in the question.
    3 ) 8.64
  2. Divide 8.64 by 3 using short division (the bus stop method), working left to right. At each digit: put any remainder in front of the next digit, divide, and write the answer on top. If a remainder is left at the end of the number, add zeros after the decimal point and keep going. Adding zeros after the decimal point does not change the value.
    8 ÷ 3 = 2 remainder 2
    Put the decimal point in the answer straight above the decimal point in the question.
    26 ÷ 3 = 8 remainder 2
    24 ÷ 3 = 8 exactly
  3. Read the quotient (the answer on top). Write the digits you found, with the decimal point in the position shown.
    8.64 ÷ 3 = 2.88

Check. Multiply back: 2.88 × 3 = 8.64. ✓

Watch out. Keep the decimal point in the answer directly above the decimal point in the question, and do not forget to add zeros if a remainder is left.

Worked example 4.4.3
Work out 2.4 ÷ 0.06.

Key words. The divisor is the number you divide by. The quotient is the answer to a division. The remainder is what is left over.

Method. To divide by a decimal, first multiply both numbers by 10, 100... so that the divisor is a whole number. Then use short division, keeping the decimal point in line.

  1. Make the number you divide by (the divisor) a whole number: multiply both numbers by 100. Multiplying both numbers by the same thing does not change the answer (it works like equivalent fractions), and dividing by a whole number is much easier.
    2.4 × 100 = 240
    0.06 × 100 = 6
    So 2.4 ÷ 0.06 = 240 ÷ 6
  2. Divide 240 by 6 using short division (the bus stop method), working left to right. At each digit: put any remainder in front of the next digit, divide, and write the answer on top. If a remainder is left at the end of the number, add zeros after the decimal point and keep going. Adding zeros after the decimal point does not change the value.
    2 ÷ 6 = 0 remainder 2
    24 ÷ 6 = 4 exactly
    0 ÷ 6 = 0 exactly
  3. Read the quotient (the answer on top). Write the digits you found, with the decimal point in the position shown.
    240 ÷ 6 = 40

Check. Multiply back: 40 × 0.06 = 2.4. ✓

Watch out. You must multiply both numbers by the same power of 10. 6 ÷ 0.3 becomes 60 ÷ 3 (not 6 ÷ 3).

Worked example 4.4.4
Work out 0.36 ÷ 0.8.

Key words. The divisor is the number you divide by. The quotient is the answer to a division. The remainder is what is left over.

Method. To divide by a decimal, first multiply both numbers by 10, 100... so that the divisor is a whole number. Then use short division, keeping the decimal point in line.

  1. Make the number you divide by (the divisor) a whole number: multiply both numbers by 10. Multiplying both numbers by the same thing does not change the answer (it works like equivalent fractions), and dividing by a whole number is much easier.
    0.36 × 10 = 3.6
    0.8 × 10 = 8
    So 0.36 ÷ 0.8 = 3.6 ÷ 8
  2. Divide 3.6 by 8 using short division (the bus stop method), working left to right. At each digit: put any remainder in front of the next digit, divide, and write the answer on top. If a remainder is left at the end of the number, add zeros after the decimal point and keep going. Adding zeros after the decimal point does not change the value.
    3 ÷ 8 = 0 remainder 3
    Put the decimal point in the answer straight above the decimal point in the question.
    36 ÷ 8 = 4 remainder 4
    40 ÷ 8 = 5 exactly (an extra zero added after the decimal point)
  3. Read the quotient (the answer on top). Write the digits you found, with the decimal point in the position shown.
    3.6 ÷ 8 = 0.45

Check. Multiply back: 0.45 × 0.8 = 0.36. ✓

Watch out. You must multiply both numbers by the same power of 10. 6 ÷ 0.3 becomes 60 ÷ 3 (not 6 ÷ 3).

Worked example 4.4.5
Four friends share a bill of £25.80 equally. How much does each friend pay?

Method. Share equally = total ÷ number of people.

  1. Write the calculation. Four friends share £25.80 equally.
    25.80 ÷ 4
  2. Divide the total by the number of friends. Use short division, keeping the decimal point in line.
    2 ÷ 4 = 0 remainder 2
    25 ÷ 4 = 6 remainder 1
    Put the decimal point in the answer straight above the decimal point in the question.
    18 ÷ 4 = 4 remainder 2
    20 ÷ 4 = 5 exactly (an extra zero added after the decimal point)
    25.8 ÷ 4 = 6.45
  3. Write the answer as money. Amounts of money are written with two decimal places.
    £6.45

Check. 4 × £6.45 = £25.80. ✓

Watch out. When the answer is money, write two decimal places: £6.45.

Watch out

When you multiply both numbers to remove the decimal from the divisor, you must do it to both numbers. 6 ÷ 0.3 becomes 60 ÷ 3 (not 6 ÷ 3).

Exercise 4D  Dividing decimals

Non-calculator
1
FWork out
(a)0.8 ÷ 2
(b)6.9 ÷ 3
(c)0.45 ÷ 5
(d)12.6 ÷ 6
(e)4.8 ÷ 8
(f)9.1 ÷ 7
2
FWork out
(a)5.6 ÷ 4
(b)9.45 ÷ 3
(c)0.72 ÷ 9
(d)13.5 ÷ 5
(e)1.2 ÷ 8
(f)2.07 ÷ 9
3
FWork out
(a)6 ÷ 0.3
(b)2.4 ÷ 0.6
(c)0.84 ÷ 0.07
(d)1.5 ÷ 0.05
(e)9 ÷ 0.15
(f)4.2 ÷ 0.35
4
FUse the fact that 96 ÷ 8 = 12 to work out
(a)9.6 ÷ 8
(b)0.96 ÷ 8
(c)960 ÷ 0.8
(d)0.96 ÷ 0.08
5
PSThree friends share a taxi fare of £18.60 equally. How much does each friend pay?
6
PSA roll of ribbon is 7.2 m long. How many pieces of length 0.4 m can be cut from it?
7
PSA baker has 12 kg of flour. She puts 0.25 kg of flour into each bag. How many bags can she fill?
8
PSA car travels 78.4 miles using 8 litres of fuel. How many miles does it travel on 1 litre?
9
PS2.5 kg of apples cost £4.50. 1.5 kg of pears cost £3.15. Which fruit is cheaper per kilogram?
10
FFill in the gap:   ? × 0.4 = 3.2
11
PSWork out   3.6 × 0.5 + 4.2 ÷ 0.6
12
RWithout working it out, is 5 ÷ 0.5 bigger or smaller than 5? Give a reason, then work it out.
13
RBen says: “4.8 ÷ 0.6 = 0.8 because 48 ÷ 6 = 8.” Explain his mistake and give the correct answer.
14
RA decimal is divided by 0.2 and the answer is 35. What was the decimal?
15
PSThe area of a rectangle is 2.24 cm². Its length is 3.5 cm. Work out its width.
4.5

Decimals and fractions

Edexcel 1MA1: N10, N2
Grade 3-5

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Decimals and fractionsComing soon

Decimals and fractions are two ways of writing the same numbers. The columns after the decimal point are fractions: tenths 110, hundredths 1100 and thousandths 11000. So 0.7 means 710, and 0.37 means 37100. A terminating decimal is a decimal that stops (like 0.375). Every terminating decimal can be written as a fraction.

Changing a decimal into a fraction

  1. Look at the last digit and say which column it is in (tenths, hundredths, thousandths…). That gives the denominator: 10, 100, 1000…
  2. Write the digits after the decimal point as the numerator.
  3. Simplify the fraction by dividing the top and bottom by their highest common factor (Chapter 3).
  4. If there is a whole number part (like the 2 in 2.35), keep it in front as a mixed number.
75 of the 100 squares are shaded, so 0.75 = 75100 = 34.

Changing a fraction into a decimal

A fraction is a division: 38 means 3 ÷ 8. So you can divide the top by the bottom using short division (section 4.4), adding zeros after the decimal point until the remainder is 0. If the denominator goes into 10, 100 or 1000, there is a quicker way: make an equivalent fraction with denominator 10, 100 or 1000. For example 720 = 35100 = 0.35.

It helps to remember these:

121434151101838120
0.50.250.750.20.10.1250.3750.05

Not every fraction gives a terminating decimal. 13 = 0.3333… goes on for ever (a recurring decimal). You will meet these in Chapter 3 (Higher). Here we only use fractions that give decimals that stop.

Comparing fractions and decimals

To order a mixture of fractions and decimals, change everything to decimals, then compare as in section 4.1.

Key ideas

  • The last decimal place gives the denominator: tenths → 10, hundredths → 100, thousandths → 1000.
  • Always simplify the fraction (divide top and bottom by the HCF).
  • Fraction → decimal: top ÷ bottom, or make the denominator 10, 100 or 1000.
  • To compare a fraction with a decimal, change both to decimals.
Worked example 4.5.1
Write 0.375 as a fraction in its simplest form.

Key words. A terminating decimal stops after a few digits (like 0.375). A mixed number has a whole number part and a fraction part.

Method. A decimal is a fraction with a denominator of 10, 100, 1000... The last decimal place tells you which. Write the fraction, then simplify.

  1. Say the decimal part in words to find the denominator. The last digit is in the thousandths column, so the denominator is 1000.
    0.375: last digit in the thousandths column, so denominator = 1000
  2. Write the digits after the decimal point over the denominator. The digits after the decimal point make the numerator.
    0.375 = 3751000
  3. List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).
    Factors of 375: 1, 3, 5, 15, 25, 75, 125, 375
    Factors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
    The highest common factor is 125.
  4. Divide the top and the bottom by 125. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.
    375 ÷ 125 = 3 and 1000 ÷ 125 = 8
    3751000 = 38

Check. Change back: 3 ÷ 8 = 0.375. ✓

Watch out. Always simplify. 0.25 = 25/100 is not finished until it is 1/4.

Worked example 4.5.2
Write 2.35 as a mixed number in its simplest form.

Key words. A terminating decimal stops after a few digits (like 0.375). A mixed number has a whole number part and a fraction part.

Method. A decimal is a fraction with a denominator of 10, 100, 1000... The last decimal place tells you which. Write the fraction, then simplify.

  1. Say the decimal part in words to find the denominator. The last digit is in the hundredths column, so the denominator is 100.
    2.35: last digit in the hundredths column, so denominator = 100
  2. Keep the whole number part to one side. Write the decimal part as a fraction. The digits after the decimal point make the numerator.
    2.35 = 2 and 35100
  3. List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).
    Factors of 35: 1, 5, 7, 35
    Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
    The highest common factor is 5.
  4. Divide the top and the bottom by 5. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.
    35 ÷ 5 = 7 and 100 ÷ 5 = 20
    35100 = 720
  5. Write the whole number and the fraction together. This gives a mixed number.
    2720

Check. Change back: 47 ÷ 20 = 2.35. ✓

Watch out. Always simplify. 0.25 = 25/100 is not finished until it is 1/4.

Worked example 4.5.3
Write 720 as a decimal.

Method. Make the denominator 10, 100 or 1000 with an equivalent fraction, then read off the decimal.

  1. Find what to multiply the denominator by to get 100. A fraction with denominator 10, 100 or 1000 is easy to write as a decimal. We use equivalent fractions: multiply the top and bottom by the same number.
    20 × 5 = 100
  2. Multiply the top by the same number. Top and bottom must be multiplied by the same number to keep the fraction the same size.
    720 = 35100
  3. Write as a decimal. The denominator 100 tells us the last digit is in the hundredths column.
    35100 = 0.35

Check. Divide the numerator by the denominator: 7 ÷ 20 = 0.35. ✓

Watch out. Whatever you multiply the bottom by, multiply the top by as well.

Worked example 4.5.4
Write 38 as a decimal.

Method. A fraction means top divided by bottom. Use short division and add zeros after the decimal point until there is no remainder.

  1. A fraction is a division: the top divided by the bottom. The fraction bar means divide, so 38 means 3 ÷ 8.
    3 ÷ 8
  2. Write 3 as 3.0 and use short division by 8. Add a decimal point and zeros after the number (this does not change its value) so that you can keep dividing.
    3 ÷ 8 = 0 remainder 3
    The number has no decimal point written, so imagine one after the last digit (3 is 3.0). Put the decimal point in the answer straight above it.
    30 ÷ 8 = 3 remainder 6 (an extra zero added after the decimal point)
    60 ÷ 8 = 7 remainder 4 (an extra zero added after the decimal point)
    40 ÷ 8 = 5 exactly (an extra zero added after the decimal point)
  3. Read the answer.
    3 ÷ 8 = 0.375

Check. Multiply back: 0.375 × 8 = 3. ✓

Watch out. Divide the top by the bottom, never the other way round.

Worked example 4.5.5
Write these in order, smallest first:   0.6,   58,   0.58,   34

Method. Change everything to decimals, then order them.

  1. Change every fraction into a decimal. Dividing the top by the bottom gives the decimal. Now all the numbers are in the same form and can be compared.
    0.6 is already a decimal
    58 = 5 ÷ 8 = 0.625
    0.58 is already a decimal
    34 = 3 ÷ 4 = 0.75
  2. Give every decimal the same number of decimal places. Add zeros on the end until each has 3 decimal places.
    0.6 = 0.600
    58 = 0.625
    0.58 = 0.580
    34 = 0.750
  3. Put them in order, smallest first. Compare column by column from the left.
    0.580, 0.600, 0.625, 0.750
  4. Write the original numbers in this order. The answer should use the numbers from the question.
    0.58,  0.6,  58,  34

Check. Each number in the answer is bigger than the one before it. ✓

Watch out. You can also change everything to fractions with a common denominator, but decimals are usually quicker when the fractions have denominators like 2, 4, 5, 8 or 10.

Watch out

The denominator comes from the last decimal place, not the first. 0.05 has its last digit in the hundredths column, so 0.05 = 5100 = 120 (not 510).

Exercise 4E  Decimals and fractions

Non-calculator
1
FWrite each decimal as a fraction in its simplest form.
(a)0.7
(b)0.25
(c)0.6
(d)0.35
(e)0.125
(f)0.04
2
FWrite each fraction as a decimal.
(a)12
(b)35
(c)920
(d)1325
(e)1750
(f)45
3
FWrite each fraction as a decimal.
(a)38
(b)58
(c)18
(d)740
4
FWrite each decimal as a mixed number in its simplest form.
(a)3.5
(b)1.25
(c)2.04
(d)6.8
5
FThe diagram is a 10 by 10 grid. Some of the squares are shaded. Write the shaded part as a decimal and as a fraction in its simplest form.
6
FWrite these in order, smallest first.
(a)0.4, 38, 0.35
(b)710, 0.65, 45, 0.7
(c)0.55, 12, 920, 0.5
7
PSWhich is bigger, 38 or 0.4? Show your working.
8
PSA bag of flour has a mass of 1.5 kg. Joe uses 35 of the flour. How many kilograms of flour does he use?
9
RExplain why 0.5 = 510 = 12.
10
RIs 0.3 the same as 3100? Give a reason.
11
RWrite a fraction that is between 0.3 and 0.4. Give your answer in the form ab where b is no more than 20.
12
FWrite 0.0125 as a fraction in its simplest form.
13
RShow that 316 = 0.1875 as a decimal, and write 3.1875 as a mixed number in its simplest form.
4.6

Rounding and truncation

Edexcel 1MA1: N15
Grade 2-5

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Rounding and truncationComing soon

Often an exact number is too long or too exact to be useful. A calculator might say a plank is 2.4638291 m long, but nobody measures that accurately. We round the number to an approximate value that is easier to use. To round, we decide how accurate we need to be, then find the nearest number with that accuracy.

Rounding to a number of decimal places

A decimal place (d.p.) is a digit after the decimal point. To round to a number of decimal places:

  1. Find the last digit you are keeping (for 2 d.p. it is the second digit after the point).
  2. Look at the next digit. This is the decider.
  3. If the decider is 5 or more, round up (add 1 to the last digit you keep). If it is 4 or less, leave the last digit as it is.
  4. Remove all the digits after the last digit you kept.
3.43.453.5halfway3.46rounds downrounds up
3.46 is between 3.4 and 3.5. The halfway point is 3.45. 3.46 is past halfway, so it is nearer 3.5. To 1 decimal place, 3.46 = 3.5.

The decider only needs the digit straight after the last digit you keep. Look at it once: do not round the number twice. If you round up a 9, it becomes 10: write 0 and carry 1 to the next column on the left (for example 3.996 to 2 d.p. is 4.00). Keep the zeros at the end to show how accurate the answer is. “Nearest whole number” means 0 decimal places, and “nearest penny” means 2 decimal places.

Rounding to a number of significant figures

The significant figures (s.f.) of a number are counted from the first non-zero digit. Zeros at the start of a decimal do not count (they only hold the place), but zeros between other digits do count. In 0.003962 the first significant figure is the 3. In 4.06 all three digits are significant. To round to a number of significant figures: find the last significant figure you are keeping, look at the decider, and round as before. For a big number, fill the gaps up to the decimal point with placeholder zeros, so 45 678 to 2 s.f. is 46 000 (not 46).

0.0380.03850.039halfway0.0384rounds downrounds up
0.0384 to 2 significant figures: the first significant figure is the 3 (thousandths), the second is the 8 (ten-thousandths). Rounding to 2 s.f. gives the nearest 0.001: 0.038.

Truncation

To truncate a number is to cut it off after a chosen digit, without looking at the next digit. There is no rounding up. So 3.789 truncated to 1 decimal place is 3.7, but rounded to 1 decimal place it is 3.8. Truncating always gives a number that is less than or equal to the original (for positive numbers).

Rounding in real life

Round only your final answer. If you round in the middle of a calculation, the small errors add up. Give money to the nearest penny (2 d.p.). If a question asks for a “suitable degree of accuracy”, choose the accuracy that matches the numbers in the question, for example 3 s.f.

Key ideas

  • Decider 5 or more: round up. Decider 4 or less: stay the same.
  • Decimal places count digits after the decimal point; significant figures count from the first non-zero digit.
  • Zeros at the start do not count as significant figures. Use placeholder zeros for big numbers: 45 678 → 46 000 (2 s.f.).
  • Truncate = chop off, never round up.
  • Round only the final answer.
Worked example 4.6.1
Round 4.9672 to 2 decimal places.

Key words. A decimal place (d.p.) is a digit after the decimal point. A significant figure (s.f.) is a digit that counts towards accuracy, starting from the first non-zero digit. The decider is the digit just after the one you are keeping.

Method. To round, find the last digit you want to keep, look at the next digit (the decider), and round up if it is 5 or more.

  1. Find the last digit you are keeping: the second digit after the decimal point. 2 decimal places means 2 digits after the decimal point. That digit is in the hundredths column.
    4.9672 (blue = last digit kept)
  2. Look at the next digit to the right: the decider. If the decider is 5 or more, round up. If it is 4 or less, round down (the last digit stays the same).
    The decider is 7: 7 is 5 or more, so round up.
  3. Round up: add 1 to the last digit you are keeping. Rounding up makes the last kept digit one bigger.
    6 + 1 = 7
  4. Write the rounded answer. Remove all the digits after the last digit you kept. Keep any zeros that are at the end, because they show how accurate the answer is.
    4.9672 = 4.97 (2 decimal places)

Check. The difference between 4.9672 and 4.97 is 0.0028. That is less than or equal to half of 0.01 (which is 0.005), so 4.97 is the closest value with that accuracy. ✓

Watch out. Look at only the one digit right after the last digit you keep, and do not round twice.

Worked example 4.6.2
Round 3.996 to 2 decimal places.

Key words. A decimal place (d.p.) is a digit after the decimal point. A significant figure (s.f.) is a digit that counts towards accuracy, starting from the first non-zero digit. The decider is the digit just after the one you are keeping.

Method. To round, find the last digit you want to keep, look at the next digit (the decider), and round up if it is 5 or more.

  1. Find the last digit you are keeping: the second digit after the decimal point. 2 decimal places means 2 digits after the decimal point. That digit is in the hundredths column.
    3.996 (blue = last digit kept)
  2. Look at the next digit to the right: the decider. If the decider is 5 or more, round up. If it is 4 or less, round down (the last digit stays the same).
    The decider is 6: 6 is 5 or more, so round up.
  3. Round up. The kept digit is 9. Adding 1 gives 10, so we write 0 and carry 1 into the next column to the left.
    3.996 → 4.00
  4. Write the rounded answer. Remove all the digits after the last digit you kept. Keep any zeros that are at the end, because they show how accurate the answer is.
    3.996 = 4.00 (2 decimal places)

Check. The difference between 3.996 and 4.00 is 0.004. That is less than or equal to half of 0.01 (which is 0.005), so 4.00 is the closest value with that accuracy. ✓

Watch out. Keep a zero on the end if it is needed: 4.00 has 2 decimal places, and writing it as 4 would show less accuracy.

Worked example 4.6.3
Write 0.003962 to 2 significant figures.

Key words. A decimal place (d.p.) is a digit after the decimal point. A significant figure (s.f.) is a digit that counts towards accuracy, starting from the first non-zero digit. The decider is the digit just after the one you are keeping.

Method. To round, find the last digit you want to keep, look at the next digit (the decider), and round up if it is 5 or more.

  1. Find the first significant figure. Significant figures are counted from the first digit that is not zero. Zeros at the start of a decimal do not count, because they only hold the place.
    The first non-zero digit of 0.003962 is 3, in the thousandths column.
  2. Count 2 significant figures to find the last digit you are keeping. The first significant figure is number 1. The digit number 2 is in the ten-thousandths column.
    0.003962 (blue = last digit kept)
  3. Look at the next digit to the right: the decider. If the decider is 5 or more, round up. If it is 4 or less, round down (the last digit stays the same).
    The decider is 6: 6 is 5 or more, so round up.
  4. Round up. The kept digit is 9. Adding 1 gives 10, so we write 0 and carry 1 into the next column to the left.
    0.003962 → 0.0040
  5. Write the rounded answer. Remove all the digits after the last digit you kept. Keep any zeros that are at the end, because they show how accurate the answer is.
    0.003962 = 0.0040 (2 significant figures)

Check. The difference between 0.003962 and 0.0040 is 0.000038. That is less than or equal to half of 0.0001 (which is 0.00005), so 0.0040 is the closest value with that accuracy. ✓

Watch out. Do not count the zeros at the start of a decimal as significant figures, and put in placeholder zeros for large numbers: 45 678 to 2 s.f. is 46 000, not 46.

Worked example 4.6.4
Write 45 678 to 2 significant figures.

Key words. A decimal place (d.p.) is a digit after the decimal point. A significant figure (s.f.) is a digit that counts towards accuracy, starting from the first non-zero digit. The decider is the digit just after the one you are keeping.

Method. To round, find the last digit you want to keep, look at the next digit (the decider), and round up if it is 5 or more.

  1. Find the first significant figure. Significant figures are counted from the first digit that is not zero. Zeros at the start of a decimal do not count, because they only hold the place.
    The first non-zero digit of 45 678 is 4, in the ten thousands column.
  2. Count 2 significant figures to find the last digit you are keeping. The first significant figure is number 1. The digit number 2 is in the thousands column.
    45678 (blue = last digit kept)
  3. Look at the next digit to the right: the decider. If the decider is 5 or more, round up. If it is 4 or less, round down (the last digit stays the same).
    The decider is 6: 6 is 5 or more, so round up.
  4. Round up: add 1 to the last digit you are keeping. Rounding up makes the last kept digit one bigger.
    5 + 1 = 6
  5. Write the rounded answer. Digits before the decimal point cannot be removed, or the number would become much smaller. Replace them with zeros (placeholder zeros).
    45 678 = 46 000 (2 significant figures)

Check. The difference between 45 678 and 46 000 is 322. That is less than or equal to half of 1000 (which is 500), so 46 000 is the closest value with that accuracy. ✓

Watch out. Do not count the zeros at the start of a decimal as significant figures, and put in placeholder zeros for large numbers: 45 678 to 2 s.f. is 46 000, not 46.

Worked example 4.6.5
Round 4561 to the nearest hundred.

Key words. A decimal place (d.p.) is a digit after the decimal point. A significant figure (s.f.) is a digit that counts towards accuracy, starting from the first non-zero digit. The decider is the digit just after the one you are keeping.

Method. To round, find the last digit you want to keep, look at the next digit (the decider), and round up if it is 5 or more.

  1. Find the digit in the hundreds column. This is the last digit you are keeping. Rounding to the nearest hundred means the answer ends in the hundreds column.
    4561 (blue = last digit kept)
  2. Look at the next digit to the right: the decider. If the decider is 5 or more, round up. If it is 4 or less, round down (the last digit stays the same).
    The decider is 6: 6 is 5 or more, so round up.
  3. Round up: add 1 to the last digit you are keeping. Rounding up makes the last kept digit one bigger.
    5 + 1 = 6
  4. Write the rounded answer. Digits before the decimal point cannot be removed, or the number would become much smaller. Replace them with zeros (placeholder zeros).
    4561 = 4600 (the nearest hundred)

Check. The difference between 4561 and 4600 is 39. That is less than or equal to half of 100 (which is 50), so 4600 is the closest value with that accuracy. ✓

Watch out. Your answer must end with the right number of zeros: 45 678 to the nearest thousand is 46 000, not 46.

Worked example 4.6.6
Truncate 3.789 to 1 decimal place.

Key words. To truncate a number is to cut it off after a given digit, without looking at the next digit.

Method. To truncate, keep the digits up to the place you are told and cut off everything after it, without rounding.

  1. Find the last digit you are keeping. 1 decimal place means 1 digit after the decimal point, in the tenths column.
    3.789 (blue = last digit kept)
  2. Cut off every digit after it. Do not look at the next digit and do not round up. Truncating means chopping the number off at that point, even if the next digit is large.
    3.789: everything after the blue digit is removed
  3. Write the truncated number. Keep the digits up to the last one you kept.
    3.789 truncated = 3.7

Check. A truncated number is never bigger than the original, and the difference is less than one unit of the last column: 0.089 < 0.1. ✓
If we had rounded instead, we would have got 3.8.

Watch out. Truncating is not the same as rounding. 3.79 truncated to 1 decimal place is 3.7, but rounded it is 3.8.

Worked example 4.6.7
Use a calculator to work out 17.8 ÷ 7. Give your answer correct to 3 significant figures.

Method. Work out the value with a calculator, then round the final answer to 3 significant figures.

  1. Use the calculator. Type 17.8 ÷ 7 and read the full display.
    17.8 ÷ 7 = 2.542857143
  2. Find the first significant figure. Significant figures are counted from the first digit that is not zero. Zeros at the start of a decimal do not count, because they only hold the place.
    The first non-zero digit of 2.542857143 is 2, in the units column.
  3. Count 3 significant figures to find the last digit you are keeping. The first significant figure is number 1. The digit number 3 is in the hundredths column.
    2.542857143 (blue = last digit kept)
  4. Look at the next digit to the right: the decider. If the decider is 5 or more, round up. If it is 4 or less, round down (the last digit stays the same).
    The decider is 2: 2 is less than 5, so round down (the kept digit does not change).
  5. Write the rounded answer. Remove all the digits after the last digit you kept. Keep any zeros that are at the end, because they show how accurate the answer is.
    2.542857143 = 2.54 (3 significant figures)

Check. The difference between 2.542857143 and 2.54 is 0.002857143. That is less than or equal to half of 0.01 (which is 0.005), so 2.54 is the closest value with that accuracy. ✓

Watch out. Do not round the digits you type into the calculator. Round only the answer at the end.

Watch out

Round using only the one digit after the last digit you keep. To round 5.449 to 1 d.p., the decider is 4 (so the answer is 5.4). Do not round 5.449 to 5.45 first and then to 5.5: that is rounding twice.

Exercise 4F  Rounding to decimal places and place values

Non-calculator
1
FRound each number to the nearest whole number.
(a)3.6
(b)12.3
(c)7.5
(d)0.49
(e)19.5
(f)99.9
2
FRound each number to 1 decimal place.
(a)4.36
(b)0.847
(c)12.05
(d)9.96
(e)3.449
(f)0.052
3
FRound each number to 2 decimal places.
(a)5.678
(b)0.0942
(c)12.999
(d)3.005
(e)7.1349
(f)0.396
4
FWhich of these numbers are equal to 3.4 when rounded to 1 decimal place?   3.35,   3.449,   3.45,   3.39,   3.351,   3.5
5
PSThe price of an item is £17.865. Round this to the nearest penny.
6
RDev rounds 5.449 to 1 decimal place. He says: “5.449 rounds to 5.45, and 5.45 rounds to 5.5.” Is he correct? Explain.
7
PSA plank has length 2.4638 m. Write its length to the nearest centimetre.
8
RWrite down the smallest number with 2 decimal places that rounds to 5.7 when rounded to 1 decimal place. Then write down the largest such number.
(a)smallest
(b)largest
9
RRound 9.996 to 2 decimal places and explain why the answer has a zero in the units column.
10
RThe number x is 3.5 when rounded to 1 decimal place, and 4 when rounded to the nearest whole number. Write down a possible value of x with 2 decimal places.

Exercise 4G  Rounding to significant figures and truncation

Calculator allowed
1
FRound each number to 1 significant figure.
(a)48
(b)0.0762
(c)5921
(d)0.00349
(e)3.96
(f)95
2
FRound each number to 2 significant figures.
(a)3.482
(b)0.05269
(c)7619
(d)12.96
(e)0.003051
(f)450.7
3
FRound each number to 3 significant figures.
(a)4.5678
(b)0.0123456
(c)56789
(d)2.998
(e)0.80049
(f)1234.5
4
FTruncate each number as shown.
(a)4.78 to 1 decimal place
(b)12.999 to 2 decimal places
(c)0.0567 to 2 significant figures
(d)4567 to 2 significant figures
(e)9.99 to 1 decimal place
(f)0.8 to the nearest whole number
5
FWrite the number 8.746:
(a)rounded to 2 decimal places
(b)truncated to 2 decimal places
6
FUse your calculator to work out 17.8 ÷ 7. Give your answer correct to 3 significant figures.
7
PS£100 is shared equally between 7 people. How much does each person get, to the nearest penny?
8
PSA bottle holds 1.5 litres. It is shared equally between 7 glasses. How much is in each glass? Give your answer in litres to 2 significant figures.
9
PSThe population of a town is 24 683. Write this number (a) to the nearest thousand, (b) to 2 significant figures.
(a)to the nearest thousand
(b)to 2 significant figures
10
FThe value of π is 3.14159265… Write π correct to the accuracy shown.
(a)to 3 significant figures
(b)to 2 decimal places
(c)to 5 significant figures
11
RExplain why 0.0307 is 0.031 to 2 significant figures, not 0.03.
12
RWrite down a number that is 0.1 when rounded to 1 significant figure and 0.11 when rounded to 2 significant figures.
13
RShow that 9.995 and 10.004 both give 10.0 when rounded to 3 significant figures, but give different numbers when truncated to 3 significant figures.
4.7

Estimation and checking answers

Edexcel 1MA1: N14
Grade 3-5

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Estimation and checking answersComing soon

An estimate is an approximate answer found using simple numbers. We use estimates to check that an answer is sensible, and to get a rough idea of the size of a calculation when an exact answer is not needed. This is very useful when you use a calculator: it is easy to press a wrong key or to miss a decimal point, and an estimate will catch it.

How to estimate

  1. Round every number in the calculation to 1 significant figure.
  2. Write the calculation again with the rounded numbers. Keep the same operations.
  3. Work it out (without a calculator), remembering the order of operations: brackets first, then multiply and divide, then add and subtract.

Rounding to 1 significant figure leaves one non-zero digit, so the numbers are easy to work with. For example 4.8 × 31 becomes 5 × 30 = 150. 596 ÷ 2.9 becomes 600 ÷ 3 = 200. The estimate does not have to be exact, it only has to be close. A good estimate is usually within a few percent of the exact answer.

Be careful with these. Never round a number to 0 (0.52 becomes 0.5, not 1 or 0). Round each number separately, before you do the calculation. Do not round the answer at the end: the estimate is what you get from the rounded numbers.

Using an estimate to check an answer

Work out the estimate. If the answer from the calculator is close to the estimate, it is probably correct. If it is about 10 or 100 times too big or too small, the decimal point is probably in the wrong place.

Is the estimate too big or too small?

You can often say whether an estimate is an over-estimate (bigger than the exact value) or an under-estimate (smaller). Think about each number: which way was it rounded, and how does that number change the answer? If you multiply by a number that was rounded up, the answer is pushed up. But if you divide by a number that was rounded up, the answer is pushed down. If the roundings push in different directions you cannot tell.

44.55halfway4.87rounds downrounds up
Estimating: 4.87 is past the halfway mark between 4 and 5, so we round it up to 5. (To 1 significant figure, a number between 1 and 10 is rounded to the nearest whole number.)

Key ideas

  • Round every number to 1 significant figure, then do the easy calculation.
  • Follow BIDMAS: brackets, then × and ÷, then + and −.
  • Never round a number to 0.
  • If the exact answer is far from the estimate (about 10 times too big or small), check the decimal point.
  • Rounding a number up makes the answer bigger if it is multiplied or added, but smaller if it is divided by or taken away.
Worked example 4.7.1
Estimate the value of 4.8 × 31.

Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.

Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.

  1. Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.
    4.8 → 5 (the next digit is 8, so round up)
    31 → 30 (the next digit is 1, so round down)
  2. Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.
    5 × 30
  3. Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.
    5 × 30
    = 150
  4. State the estimate.
    The estimate is 150.

Check. A calculator gives 149 (3 s.f.) for the exact calculation. That is close to the estimate of 150. ✓

Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.

Worked example 4.7.2
Estimate the value of 6.2 + 3.95.1 − 1.2.

Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.

Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.

  1. Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.
    6.2 → 6 (the next digit is 2, so round down)
    3.9 → 4 (the next digit is 9, so round up)
    5.1 → 5 (the next digit is 1, so round down)
    1.2 → 1 (the next digit is 2, so round down)
  2. Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.
    6 + 45 − 1
  3. Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.
    6 + 45 − 1
    = 104
    = 2.5
  4. State the estimate.
    The estimate is 2.5.

Check. A calculator gives 2.59 (3 s.f.) for the exact calculation. That is close to the estimate of 2.5. ✓

Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.

Worked example 4.7.3
Estimate the value of 596 ÷ 2.9.

Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.

Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.

  1. Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.
    596 → 600 (the next digit is 9, so round up)
    2.9 → 3 (the next digit is 9, so round up)
  2. Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.
    600 ÷ 3
  3. Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.
    600 ÷ 3
    = 200
  4. State the estimate.
    The estimate is 200.

Check. A calculator gives 206 (3 s.f.) for the exact calculation. That is close to the estimate of 200. ✓

Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.

Worked example 4.7.4
Estimate the value of 0.42 × 19.6.

Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.

Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.

  1. Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.
    0.42 → 0.4 (the next digit is 2, so round down)
    19.6 → 20 (the next digit is 9, so round up)
  2. Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.
    0.4 × 20
  3. Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.
    0.4 × 20
    = 8
  4. State the estimate.
    The estimate is 8.

Check. A calculator gives 8.23 (3 s.f.) for the exact calculation. That is close to the estimate of 8. ✓

Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.

Worked example 4.7.5
Dilip uses a calculator and gets 4.8 × 31 = 1488. Use an estimate to show that this answer is wrong.

Method. Estimate the answer with numbers rounded to 1 significant figure and compare it with the answer given.

  1. Estimate the answer by rounding each number to 1 significant figure. The estimate tells us roughly how big the answer should be.
    5 × 30 = 150
  2. Compare the estimate with the answer given. If the answer is close to the estimate, it is probably right. If it is about 10 times too big or too small, the decimal point is probably in the wrong place.
    Answer given: 1488. Estimate: 150.
  3. Decide. The two numbers are not close.
    1488 is far from 150, so the answer is not sensible (it is about 10 times the estimate).

Check. The exact answer is 149 (3 s.f.), so the answer given was wrong. ✓

Watch out. An estimate only checks the size of the answer. If it matches the estimate, the answer could still have a small mistake.

Worked example 4.7.6
Work out an estimate for 4.8 × 3.90.44. Is your estimate bigger or smaller than the exact value? Give a reason.

Method. Work out what each rounding does to the answer. If they all push the answer the same way, you know whether the estimate is too big or too small.

  1. Round to 1 significant figure and work out the estimate. This is the normal estimating method.
    5 × 4 ÷ 0.4 = 50
  2. Look at what each rounding does to the answer. Rounding a number up or down changes the answer. Whether the answer gets bigger or smaller depends on where the number is in the calculation: a number being added or multiplied pushes the answer the same way; a number we divide by pushes it the opposite way.
    4.8 is rounded up to 5. A bigger number here makes the answer bigger, so this makes the estimate bigger than the true answer.
    3.9 is rounded up to 4. A bigger number here makes the answer bigger, so this makes the estimate bigger than the true answer.
    0.44 is rounded down to 0.4. A bigger number here makes the answer smaller, so this makes the estimate bigger than the true answer.
  3. Decide. Every rounding pushes the estimate in the same direction, so we can be sure.
    The estimate is an over-estimate: it is bigger than the exact value.

Check. The exact value is 42.5 (3 s.f.) and the estimate is 50; the estimate is bigger. ✓

Watch out. When you divide by a number that was rounded up, the estimate gets smaller. Think about each number separately.

Watch out

When you estimate, round each number to 1 significant figure first. Do not round the final answer, and do not round a small decimal to 0: 0.52 → 0.5, not 1. And remember: 1 significant figure for 0.0472 is 0.05, not 0.

Exercise 4H  Estimation and checking answers

Non-calculator
1
FEstimate the value of each calculation by rounding each number to 1 significant figure.
(a)4.8 × 6.1
(b)9.7 × 3.2
(c)31 × 1.9
(d)0.82 × 48
(e)72 × 0.48
(f)6.3 × 8.9
2
FEstimate the value of each calculation.
(a)596 ÷ 2.9
(b)81.3 ÷ 3.9
(c)4.9 ÷ 0.51
(d)1920 ÷ 39
(e)6.4 ÷ 0.31
3
FEstimate the value of each calculation.
(a)(3.8 + 6.1) × 2.9
(b)4.9 × 210.48
(c)7.9 − 2.10.52
(d)48.2 − 19.7
4
PSA shop sells 38 bags of oranges at £4.95 each. Estimate the total amount of money the shop takes.
5
PSA concert has 312 tickets sold at £19.50 each. Estimate the total money from ticket sales.
6
PSA car uses 6.2 litres of petrol for every 100 km. Estimate the amount of petrol needed for a journey of 487 km. (Use 6.2 × 487100.)
7
PSJo has £50. She wants to buy three items that cost £14.75, £22.99 and £8.49. By rounding each price up to the next pound, show that she has enough money.
8
REach of these answers was found with a calculator. Use an estimate to decide whether each answer is sensible.
(a)4.8 × 31 = 1488
(b)596 ÷ 2.9 = 20.55
(c)7.2 × 0.48 = 3.456
(d)81.3 ÷ 3.9 = 20.846… (to 5 s.f.)
9
RFor each calculation, work out an estimate by rounding to 1 significant figure. Say whether your estimate is an over-estimate or an under-estimate. Give a reason.
(a)4.8 × 3.9 ÷ 0.44
(b)6.7 × 8.6
(c)29 ÷ 0.52
(d)4.2 × 21
10
RZoe says: “To estimate 0.52 × 48, I round 0.52 to 1 significant figure and get 1.” Explain what she has done wrong and work out a correct estimate.
11
RWithout a calculator, decide whether 31.2 × 0.42 is closest to 1.2, 12 or 120. Show your working.
12
PSEstimate the value of √(9.8 × 41).
13
REstimate 0.0081 × 620.019 and explain why the exact answer is not 2.64.
4.8

Error intervals and limits of accuracy

Edexcel 1MA1: N15, N16
Grade 4-6

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Error intervals and limits of accuracyComing soon

When a number has been rounded, we no longer know the original number exactly. All we know is that it was somewhere in a range of values. That range is called an error interval. For example, the length of a leaf is 3.4 cm, correct to 1 decimal place. The real length could be 3.36, 3.4 or even 3.44, because all of these round to 3.4. But it cannot be 3.3 or 3.5.

Error intervals from rounding

A number rounded to a given accuracy could have been up to half a unit smaller or bigger than the rounded value. The unit is the value of the last column: for 1 d.p. the unit is 0.1, so half a unit is 0.05. So 3.4 (to 1 d.p.) lies between 3.4 − 0.05 = 3.35 and 3.4 + 0.05 = 3.45. 3.35 would be rounded up to 3.4, so the lower end is included. 3.45 would be rounded up to 3.5, so the upper end is not included. We write this with inequality signs:

3.35 ≤ x < 3.45

Here x stands for the original number. The sign ≤ means “is less than or equal to” (the end value is allowed). The sign < means “is less than” (the end value is not allowed). On a number line, a solid circle means the end is included and an open circle means it is not.

3.353.453.40given value
The error interval 3.35 ≤ x < 3.45: the solid circle at 3.35 means included, the open circle at 3.45 means not included.

Lower bound and upper bound

The lower bound is the smallest value the number could be (3.35 here), and the upper bound is the value at the top of the range (3.45 here). Together they give the limits of accuracy of a measurement. Although the upper bound is not actually included, in calculations we use it as if it were a possible value. You will see why in section 4.9.

Error intervals from truncation

If a number has been truncated, digits have been chopped off, so the original number is at least the truncated value, and less than one whole unit more. For example 5.3 truncated to 1 d.p. means 5.3 ≤ x < 5.4. The interval is one whole unit wide and starts at the truncated value.

Other accuracies

The same idea works for other roundings. A number that is 450 to the nearest 10 has a unit of 10, so half a unit is 5: 445 ≤ x < 455. A number that is 0.062 to 2 significant figures has its last significant figure in the thousandths column, so the unit is 0.001 and the interval is 0.0615 ≤ x < 0.0625.

Key ideas

  • A rounded number lies within half a unit either side: lower bound = value − half a unit, upper bound = value + half a unit.
  • Error interval: lower bound ≤ x < upper bound. The upper end uses <.
  • Truncated number: value ≤ x < value + one unit.
  • Unit = the value of the last digit: 0.1 for 1 d.p., 0.01 for 2 d.p., 10 for the nearest 10.
  • Solid circle → included. Open circle → not included.
Worked example 4.8.1
A length is 3.4 cm, correct to 1 decimal place. Write down the error interval for the length.

Key words. An error interval shows all the values a number could have been before rounding or truncating. The lower bound is the smallest possible value and the upper bound is the limit at the top.

Method. A number rounded to a given accuracy could have been anywhere within half a unit of the rounded value. The error interval says exactly where.

  1. Find the size of the last place. The number is given to 1 decimal place, so the last digit is in the tenths column. One unit of that column is 0.1.
    unit = 0.1
  2. Halve the unit. A number rounds to the given value if it is within half a unit of it. Anything further away would round to a different value.
    0.1 ÷ 2 = 0.05
  3. Subtract and add half a unit to find the two bounds. The lowest value that rounds to the given number is the lower bound. The upper bound is the point where it would round up to the next value.
    lower bound = 3.4 − 0.05 = 3.35
    upper bound = 3.4 + 0.05 = 3.45
  4. Write the error interval with inequality signs. The lower bound 3.35 rounds up to 3.4, so L can equal it: use ≤. The upper bound 3.45 would round up to 3.5, so L must be less than it: use <.
    3.35 ≤ L < 3.45

Check. Test the ends: 3.35 rounds to 3.4 ✓; a number just below the top, such as 3.449, rounds to 3.4 ✓; and 3.45 itself rounds up to 3.5, so it is not included. ✓

Watch out. The top end uses < (not ≤). A number exactly half way rounds up, so the upper bound itself is not a possible value.

Worked example 4.8.2
The mass of a parcel is 450 g, correct to the nearest 10 g. Write down the error interval for the mass.

Key words. An error interval shows all the values a number could have been before rounding or truncating. The lower bound is the smallest possible value and the upper bound is the limit at the top.

Method. A number rounded to a given accuracy could have been anywhere within half a unit of the rounded value. The error interval says exactly where.

  1. Find the size of the rounding unit. The number was rounded to the nearest ten, so the unit is 10.
    unit = 10
  2. Halve the unit. A number rounds to the given value if it is within half a unit of it. Anything further away would round to a different value.
    10 ÷ 2 = 5
  3. Subtract and add half a unit to find the two bounds. The lowest value that rounds to the given number is the lower bound. The upper bound is the point where it would round up to the next value.
    lower bound = 450 − 5 = 445
    upper bound = 450 + 5 = 455
  4. Write the error interval with inequality signs. The lower bound 445 rounds up to 450, so m can equal it: use ≤. The upper bound 455 would round up to 460, so m must be less than it: use <.
    445 ≤ m < 455

Check. Test the ends: 445 rounds to 450 ✓; a number just below the top, such as 454.9, rounds to 450 ✓; and 455 itself rounds up to 460, so it is not included. ✓

Watch out. The top end uses < (not ≤). A number exactly half way rounds up, so the upper bound itself is not a possible value.

Worked example 4.8.3
A number x is 0.062 correct to 2 significant figures. Write down the error interval for x.

Key words. An error interval shows all the values a number could have been before rounding or truncating. The lower bound is the smallest possible value and the upper bound is the limit at the top.

Method. A number rounded to a given accuracy could have been anywhere within half a unit of the rounded value. The error interval says exactly where.

  1. Find the size of the last place that is shown. The number is given to 2 significant figures. The last significant figure of 0.062 is in the thousandths column. One unit of that column is 0.001.
    unit = 0.001
  2. Halve the unit. A number rounds to the given value if it is within half a unit of it. Anything further away would round to a different value.
    0.001 ÷ 2 = 0.0005
  3. Subtract and add half a unit to find the two bounds. The lowest value that rounds to the given number is the lower bound. The upper bound is the point where it would round up to the next value.
    lower bound = 0.062 − 0.0005 = 0.0615
    upper bound = 0.062 + 0.0005 = 0.0625
  4. Write the error interval with inequality signs. The lower bound 0.0615 rounds up to 0.062, so x can equal it: use ≤. The upper bound 0.0625 would round up to 0.063, so x must be less than it: use <.
    0.0615 ≤ x < 0.0625

Check. Test the ends: 0.0615 rounds to 0.062 ✓; a number just below the top, such as 0.06249, rounds to 0.062 ✓; and 0.0625 itself rounds up to 0.063, so it is not included. ✓

Watch out. The top end uses < (not ≤). A number exactly half way rounds up, so the upper bound itself is not a possible value.

Worked example 4.8.4
The number y is 5.3 when truncated to 1 decimal place. Write down the error interval for y.

Key words. An error interval shows all the values a number could have been before rounding or truncating. The lower bound is the smallest possible value and the upper bound is the limit at the top.

Method. A number truncated to a given accuracy could have been anywhere from the truncated value up to (but not including) one whole unit more.

  1. Find the size of the last place. The number is given to 1 decimal place, so the last digit is in the tenths column. One unit of that column is 0.1.
    unit = 0.1
  2. Think about what truncating does. Truncating cuts the digits off without rounding, so the original number is never smaller than the truncated value, and is less than one whole unit bigger.
    5.3 ≤ y < 5.3 + 0.1
  3. Add one unit to get the upper end. If the number were one unit more it would have a bigger digit in the last column.
    5.3 + 0.1 = 5.4
  4. Write the error interval. The lower end 5.3 is possible (the digits after it could all be zero), so use ≤. The upper end 5.4 is not possible, so use <.
    5.3 ≤ y < 5.4

Check. Truncate 5.3 to the same place: 5.3 ✓. Truncate a value just below the top, 5.399: 5.3 ✓. Truncating 5.4 gives 5.4, which is different. ✓

Watch out. For truncation the interval starts at the given value and goes up by one whole unit, not half a unit.

Worked example 4.8.5
A rope has length 2.5 m, correct to 1 decimal place. Is it possible that the length is 2.55 m?

Method. Compare the number with the lower end and with the upper end of the error interval.

  1. Write the error interval and the number side by side. The number is possible only if it lies inside the interval.
    2.45 ≤ x < 2.55
    number: 2.55
  2. Check the lower end. The number must be at least the lower bound (it can equal it).
    2.55 ≥ 2.45: true
  3. Check the upper end. The number must be less than the upper bound (it cannot equal it).
    2.55 < 2.55: false
  4. Decide. Both checks must be true.
    No, 2.55 is not a possible value.

Check. Picture it on a number line: 2.55 is outside the interval. ✓

Watch out. An open circle (a < sign) means the end value itself is not allowed.

Watch out

The upper end of an error interval uses <, not ≤. A number exactly half-way rounds up, so the upper bound is not a possible value of the original number. Also, the interval is half a unit either side for rounding, but a whole unit for truncation.

Exercise 4I  Error intervals and limits of accuracy

Non-calculator
1
FWrite down the lower bound and the upper bound of each measurement.
(a)7.3 cm (correct to 1 decimal place)
(b)45 kg (correct to the nearest ten)
(c)0.82 (correct to 2 decimal places)
(d)2400 people (correct to the nearest hundred)
(e)3.5 m (correct to 2 significant figures)
(f)6 s (correct to the nearest whole number)
2
FEach number is correct to the accuracy shown. Write down the error interval for each.
(a)3.4 to 1 decimal place
(b)52 to the nearest whole number
(c)0.85 to 2 decimal places
(d)6700 to the nearest hundred
(e)4.5 to 2 significant figures
(f)0.062 to 2 significant figures
3
FEach number has been truncated to the accuracy shown. Write down the error interval for each.
(a)5.3 truncated to 1 decimal place
(b)12 truncated to the nearest whole number
(c)0.47 truncated to 2 decimal places
(d)2400 truncated to the nearest hundred
4
FA number is 3.4 correct to 1 decimal place. So its error interval is 3.35 ≤ x < 3.45. Could the number be
(a)3.4
(b)3.45
(c)3.35
(d)3.3499
(e)3.5
5
PSA bag of sugar is labelled 1.5 kg, correct to 1 decimal place. Could the mass of the sugar be
(a)1.46 kg
(b)1.55 kg
(c)1.449 kg
6
PSA rope is 250 cm long, correct to the nearest 10 cm. A student says a 254 cm rope is possible. Is she right?
7
PSAt a match the attendance was reported as 24 000, correct to 2 significant figures. Write down the error interval for the attendance.
8
PSThe mass of an elephant is 4.5 tonnes, correct to 1 decimal place. What is the largest possible error in this value?
9
RHana says: “x is 3.4 correct to 1 decimal place, so x could be 3.45.” Is she right? Explain.
10
RJoe writes the error interval for 6, correct to the nearest whole number, as 5 ≤ x ≤ 7. Write down two mistakes he has made, and write the correct error interval.
11
RA number x is 8.2 when rounded to 1 decimal place and 8.2 when truncated to 1 decimal place. Write down the error interval for x.
12
RThe error interval for a number is 4.45 ≤ x < 4.55. The number was rounded. Write down the number and the accuracy to which it was rounded.
4.9

Upper and lower bounds in calculations

Edexcel 1MA1: N16
Grade 7-9

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Upper and lower bounds in calculationsComing soon

In section 4.8 you found the lower bound and the upper bound of a single rounded measurement. In real problems we often calculate with measurements, for example finding an area from a length and a width. If both are rounded, the answer also has a range of possible values. The upper bound of the answer is the biggest value it could be (the limit at the top), and the lower bound is the smallest.

The key idea is to choose the bounds that make the answer as big (or as small) as possible. It is not always the upper bound of everything, because some measurements work against the answer. To make a − b as big as possible, take the biggest a and subtract the smallest b. To make a ÷ b as big as possible, divide the biggest a by the smallest b (dividing by a small number gives a big answer).

CalculationUpper bound of the answerLower bound of the answer
a + bUB of a + UB of bLB of a + LB of b
a − bUB of a − LB of bLB of a − UB of b
a × bUB of a × UB of bLB of a × LB of b
a ÷ bUB of a ÷ LB of bLB of a ÷ UB of b

(UB means upper bound and LB means lower bound.) Always check the numbers by asking: does this combination really give the biggest (or smallest) answer?

12.4 cm8.6 cm
A rectangle with length 12.4 cm and width 8.6 cm, each correct to 1 decimal place. The area could be anywhere between 105.5925 cm² and 107.6925 cm².
105.5925107.6925106.6400given value
The lower and upper bounds of the area, and the area found from the values given (106.64 cm²).

Answering to an appropriate degree of accuracy

Once you have the lower bound and upper bound of the answer, you can say how accurately the answer is known. Round both bounds to 1, 2, 3… significant figures. Find the most accurate version on which they agree. For the area above, the bounds 105.5925 and 107.6925 both round to 110 to 2 significant figures, but to 3 significant figures they are 106 and 108, which are different. So we can only say the area is 110 cm² to 2 s.f.

Exact values and truncation

Some values are exact, for example “5 identical bags”. An exact value has no error, so its lower bound and upper bound are both the value itself. If a value is truncated, the lower bound is the value shown and the upper bound is one whole unit more.

Key ideas

  • Sum or product: upper bound with upper bounds, lower bound with lower bounds.
  • Difference a − b: UB = UBa − LBb, LB = LBa − UBb.
  • Quotient a ÷ b: UB = UBa ÷ LBb, LB = LBa ÷ UBb.
  • Exact values have no error: use them as they are.
  • To give an answer to an appropriate accuracy, round the bounds to different numbers of significant figures and find the most accurate level on which they agree.
Worked example 4.9.1
A rectangle has length 12.4 cm and width 8.6 cm, each correct to 1 decimal place. Work out the lower bound and the upper bound of the area. Hence write the area to a suitable degree of accuracy.

Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.

Method. Find the bounds of every measurement, then choose the combination that makes the area as big (upper bound) or as small (lower bound) as possible.

  1. Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.
    length = 12.4 cm (to 1 decimal place): unit = 0.1, half a unit = 0.05
    lower bound = 12.4 − 0.05 = 12.35, upper bound = 12.4 + 0.05 = 12.45

    width = 8.6 cm (to 1 decimal place): unit = 0.1, half a unit = 0.05
    lower bound = 8.6 − 0.05 = 8.55, upper bound = 8.6 + 0.05 = 8.65
  2. Decide which bounds to use. To get the biggest possible area, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest area, do the opposite.
    length: a bigger length makes the area bigger, so for the upper bound of the area use the upper bound of length, and for the lower bound of the area use the lower bound.
    width: a bigger width makes the area bigger, so for the upper bound of the area use the upper bound of width, and for the lower bound of the area use the lower bound.
  3. Calculate the upper bound of the area. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.
    12.45 × 8.65 = 107.6925 cm²
  4. Calculate the lower bound of the area. Put the other bounds into the calculation.
    12.35 × 8.55 = 105.5925 cm²
  5. Compare the two bounds rounded to 1, 2, 3… significant figures. If the lowest and highest possible values both round to the same number, then we can be sure of that number to that accuracy.
    to 1 s.f.: lower bound 100, upper bound 100 (the same)
    to 2 s.f.: lower bound 110, upper bound 110 (the same)
    to 3 s.f.: lower bound 106, upper bound 108 (different)
  6. Write the answer to the most accurate level on which they agree. Both bounds round to 110 to 2 s.f., but to 3 s.f. they are different. So 110 is the answer we can be certain of.
    110 cm² (to 2 s.f.)

Check. Check the bounds make sense: the area found from the original values is 106.64 cm², which lies between the lower bound and the upper bound. ✓

Watch out. Use the upper bound of every measurement for the upper bound of the answer, and the lower bound of every measurement for the lower bound. Do not mix them in a sum or product.

Worked example 4.9.2
a = 8.4 and b = 3.7, each correct to 1 decimal place. Work out the upper bound and the lower bound of a − b.

Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.

Method. Find the bounds of every measurement, then choose the combination that makes the difference as big (upper bound) or as small (lower bound) as possible.

  1. Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.
    a = 8.4 (to 1 decimal place): unit = 0.1, half a unit = 0.05
    lower bound = 8.4 − 0.05 = 8.35, upper bound = 8.4 + 0.05 = 8.45

    b = 3.7 (to 1 decimal place): unit = 0.1, half a unit = 0.05
    lower bound = 3.7 − 0.05 = 3.65, upper bound = 3.7 + 0.05 = 3.75
  2. Decide which bounds to use. To get the biggest possible difference, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest difference, do the opposite.
    a: a bigger a makes the difference bigger, so for the upper bound of the difference use the upper bound of a, and for the lower bound of the difference use the lower bound.
    b: a bigger b makes the difference smaller, so for the upper bound of the difference use the lower bound of b, and for the lower bound of the difference use the upper bound.
  3. Calculate the upper bound of the difference. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.
    8.45 − 3.65 = 4.8
  4. Calculate the lower bound of the difference. Put the other bounds into the calculation.
    8.35 − 3.75 = 4.6

Check. Check the bounds make sense: the difference found from the original values is 4.7, which lies between the lower bound and the upper bound. ✓

Watch out. For a subtraction or a division, the bound you need is not the same one each time: to make a − b biggest you need the upper bound of a but the lower bound of b. The same is true for dividing by b.

Worked example 4.9.3
A cyclist rides 150 km, correct to the nearest kilometre, in 2.4 hours, correct to 1 decimal place. Work out the upper and lower bounds of the average speed. Give the speed to a suitable degree of accuracy.

Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.

Method. Find the bounds of every measurement, then choose the combination that makes the speed as big (upper bound) or as small (lower bound) as possible.

  1. Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.
    distance = 150 km (to the nearest whole number): unit = 1, half a unit = 0.5
    lower bound = 150 − 0.5 = 149.5, upper bound = 150 + 0.5 = 150.5

    time = 2.4 hours (to 1 decimal place): unit = 0.1, half a unit = 0.05
    lower bound = 2.4 − 0.05 = 2.35, upper bound = 2.4 + 0.05 = 2.45
  2. Decide which bounds to use. To get the biggest possible speed, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest speed, do the opposite.
    distance: a bigger distance makes the speed bigger, so for the upper bound of the speed use the upper bound of distance, and for the lower bound of the speed use the lower bound.
    time: a bigger time makes the speed smaller, so for the upper bound of the speed use the lower bound of time, and for the lower bound of the speed use the upper bound.
  3. Calculate the upper bound of the speed. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.
    150.5 ÷ 2.35 = 64.04 km/h (rounded)
  4. Calculate the lower bound of the speed. Put the other bounds into the calculation.
    149.5 ÷ 2.45 = 61.02 km/h (rounded)
  5. Compare the two bounds rounded to 1, 2, 3… significant figures. If the lowest and highest possible values both round to the same number, then we can be sure of that number to that accuracy.
    to 1 s.f.: lower bound 60, upper bound 60 (the same)
    to 2 s.f.: lower bound 61, upper bound 64 (different)
  6. Write the answer to the most accurate level on which they agree. Both bounds round to 60 to 1 s.f., but to 2 s.f. they are different. So 60 is the answer we can be certain of.
    60 km/h (to 1 s.f.)

Check. Check the bounds make sense: the speed found from the original values is 62.5 km/h, which lies between the lower bound and the upper bound. ✓

Watch out. For a subtraction or a division, the bound you need is not the same one each time: to make a − b biggest you need the upper bound of a but the lower bound of b. The same is true for dividing by b.

Worked example 4.9.4
The length of a rectangle is 7.6 cm, truncated to 1 decimal place. Its width is exactly 5 cm. Work out the upper and lower bounds of its perimeter.

Key words. The upper bound and lower bound of a calculation are the biggest and smallest answers that are possible given the accuracy of the numbers used.

Method. Find the bounds of every measurement, then choose the combination that makes the perimeter as big (upper bound) or as small (lower bound) as possible.

  1. Find the lower bound and the upper bound of each measurement. A measurement given to a certain accuracy could be half a unit smaller or half a unit bigger than the value shown. Do this for every number in the question.
    length = 7.6 cm (truncated to 1 decimal place): unit = 0.1
    lower bound = 7.6 (truncating never rounds up, so the value shown is the smallest possible), upper bound = 7.6 + 0.1 = 7.7

    width = 5 cm is an exact value, so it has no error: lower bound = upper bound = 5
  2. Decide which bounds to use. To get the biggest possible perimeter, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest perimeter, do the opposite.
    length: a bigger length makes the perimeter bigger, so for the upper bound of the perimeter use the upper bound of length, and for the lower bound of the perimeter use the lower bound.
  3. Calculate the upper bound of the perimeter. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.
    2 × (7.7 + 5) = 25.4 cm
  4. Calculate the lower bound of the perimeter. Put the other bounds into the calculation.
    2 × (7.6 + 5) = 25.2 cm

Check. Check the bounds make sense: the perimeter found from the original values is 25.2 cm, which lies between the lower bound and the upper bound. ✓

Watch out. Use the upper bound of every measurement for the upper bound of the answer, and the lower bound of every measurement for the lower bound. Do not mix them in a sum or product.

Watch out

For subtraction and division you do not use the upper bounds of both numbers. For the upper bound of a − b you need the lower bound of b. Check each time: which choice makes the answer biggest?

Exercise 4J  Bounds in calculations

Calculator allowed
1
FEach number is correct to the accuracy shown. Work out the bound asked for.
(a)a = 5.6 and b = 3.2 (1 d.p.). Upper bound of a + b.
(b)a = 9.5 and b = 4.2 (1 d.p.). Lower bound of a − b.
(c)a = 3.5 and b = 2.4 (1 d.p.). Upper bound of a × b.
(d)a = 13 and b = 3 (both to the nearest whole number). Upper bound of a ÷ b.
2
FA rectangle has length 7.2 cm and width 4.5 cm, each correct to 1 decimal place. Work out
7.2 cm4.5 cm
(a)the lower bound of the area
(b)the upper bound of the perimeter
3
FBox A has a mass of 40 g and box B has a mass of 60 g. Each mass is correct to the nearest 10 g. Work out the lower bound and the upper bound of the total mass of the two boxes.
4
PSA metal block has mass 240 g, correct to the nearest 10 g, and volume 85 cm³, correct to the nearest cm³. Density = mass ÷ volume. Work out the density, giving your answer to a suitable degree of accuracy. You must show all your working.
5
PSA sprinter runs 100 m, correct to the nearest metre, in 12.4 seconds, correct to 1 decimal place. Calculate her average speed in metres per second. Give your answer to a suitable degree of accuracy.
6
PSx = 4.2, y = 3.1 and z = 0.5, each correct to 1 decimal place. Work out the lower bound and the upper bound of x + yz.
7
RThe area of a rectangle is found from a length of 12.4 cm and a width of 8.6 cm, each correct to 1 decimal place. (a) Show that the area could be more than 107 cm². (b) Is it possible that the area is exactly 108 cm²?
(a)show that the area could be more than 107 cm²
(b)is it possible that the area is exactly 108 cm²?
8
FWrite each result to a suitable degree of accuracy, given its lower bound and upper bound.
(a)lower bound 7.31, upper bound 7.34
(b)lower bound 3.846, upper bound 3.912
(c)lower bound 0.4551, upper bound 0.4612
(d)lower bound 1204, upper bound 1231
9
FA square tile has side 12.5 cm, correct to 1 decimal place. Work out the upper bound of the area of the tile.
10
PSParcel A has mass 2.35 kg, correct to 2 decimal places. Parcel B has mass 4.8 kg, correct to 1 decimal place. The postal limit is 7.2 kg. The total mass of the two parcels is worked out as 2.35 + 4.8 = 7.15 kg. Is it possible that the total mass is more than 7.2 kg? You must show your working.
11
RBen says: “The upper bound of a ÷ b is UB of a divided by UB of b.” Use a = 12 and b = 4, each correct to the nearest whole number, to show that Ben is wrong.
12
RExplain why the lower bound of a − b uses the upper bound of b.
13
RThe area of a rectangle is 24 cm², correct to the nearest cm². Its width is 4 cm, correct to the nearest cm. Work out the upper bound of the length of the rectangle. (Length = area ÷ width.)
14
Fa = 40 and b = 60, each correct to the nearest 10. Work out the lower bound of a × b without a calculator.
15
PSFive identical planks are laid end to end. Each plank has length 2.4 m, truncated to 1 decimal place. Work out the lower bound and the upper bound of the total length of the five planks.
16
Rx = 6.3 and y = 2.1, each correct to 1 decimal place. Show that xy could be less than 3.

Chapter summary

SkillMethod
Place value and orderingColumns are worth 10 times the one on their right. Add zeros so decimals have the same number of places, then compare column by column.
× or ÷ by 10, 100, 1000Move the digits 1, 2, 3 places left (×) or right (÷). The decimal point does not move.
Add and subtract decimalsDecimal points in a vertical line, fill gaps with zeros, work from the right, carry or exchange.
Multiply decimalsCount total decimal places, multiply as whole numbers, put the point back so the answer has that many decimal places.
Divide decimalsMake the divisor a whole number (multiply both numbers by 10, 100…), then short division with the point kept in line.
Decimal ↔ fractionLast decimal place gives the denominator (10, 100, 1000); simplify. Fraction to decimal: top ÷ bottom or make the denominator 10, 100, 1000.
Rounding to d.p. and s.f.Find the last digit to keep, look at the decider (5 or more rounds up). Significant figures start at the first non-zero digit. Use placeholder zeros for big numbers.
TruncationCut off after the chosen digit without rounding.
EstimationRound every number to 1 s.f., follow BIDMAS, compare with the exact answer to check it. Never round a number to 0.
Error intervalsRounding: value ± half a unit, written lower ≤ x < upper. Truncation: value ≤ x < value + one unit.
Bounds in calculations (Higher)Sum and product use matching bounds. For a − b: UBa − LBb. For a ÷ b: UBa ÷ LBb. Then round both bounds to find a suitable accuracy.

Can you…?

  • I can read, write, compare and order decimals using place value.
  • I can multiply and divide by 10, 100 and 1000.
  • I can add and subtract decimals using column methods.
  • I can multiply decimals by counting decimal places.
  • I can divide by whole numbers and by decimals.
  • I can convert between terminating decimals and fractions.
  • I can round to a given number of decimal places or significant figures.
  • I can truncate a number.
  • I can estimate an answer by rounding to 1 significant figure and use it to check a calculation.
  • I can write an error interval for a rounded or truncated number.
  • (Higher) I can find the upper and lower bounds of a calculation and give the answer to a suitable accuracy.

Key terms

Decimal place
A digit after the decimal point.
Place value
The value a digit has because of the column it is in.
Placeholder zero
A zero used to keep the other digits in the correct columns.
Terminating decimal
A decimal that stops after a fixed number of digits, such as 0.375.
Round
To replace a number by a nearby number that is easier to use.
Decider
The digit just after the last digit you keep when rounding.
Significant figure
A digit that counts towards the accuracy of a number, counted from the first non-zero digit.
Truncate
To cut off a number after a chosen digit without rounding.
Estimate
An approximate answer found with simple numbers, usually by rounding to 1 significant figure.
Error interval
The range of values a number could have had before it was rounded or truncated.
Lower bound
The smallest value that rounds to a given number.
Upper bound
The value at the top of the error interval; the least number that would round up to the next value.
Limits of accuracy
The lower and upper bounds of a measurement or calculation.
Inequality
A statement using <, >, ≤ or ≥, such as 3.35 ≤ x < 3.45.

Exercise 4Z  Check-up (non-calculator)

Non-calculator
1
Write these numbers in order, smallest first:   0.5,   0.45,   0.405,   0.054
2
Work out 4.7 + 12.85.
3
Work out 10 − 3.64.
4
Work out 0.6 × 0.07.
5
Work out 7.2 ÷ 0.4.
6
Write 0.35 as a fraction in its simplest form.
7
Round 4.0753 to 2 decimal places.
8
Write 0.04067 correct to 2 significant figures.
9
Work out an estimate for 4.9 × 31.2.
10
The number x is 2.7 correct to 1 decimal place. Write down the error interval for x.

Total 44 marks. Allow about 44 minutes.

Exercise 4X  Exam-style questions

Non-calculator
1
FWork out 36.5 − 8.74.[2 marks]
2
FWork out 2.8 × 0.35.[2 marks]
3
FWork out 15.12 ÷ 0.7.[3 marks]
4
FWrite 0.45 as a fraction in its simplest form.[2 marks]
5
FWrite these numbers in order, starting with the smallest.   78,   0.65,   35,   0.7[3 marks]
6
F(a) Write 12.0672 correct to 2 decimal places.   (b) Write 0.00493 correct to 2 significant figures.[2 marks]
(a)12.0672 to 2 decimal places
(b)0.00493 to 2 significant figures
7
FWork out an estimate for 4.87 × 31.20.52. You must show your working.[3 marks]
8
FGina uses her calculator and says that 4.8 × 31 = 1488. Without working out the exact answer, show that Gina is wrong.[2 marks]
9
FPens cost £1.35 each. Priya buys 12 pens and pays with a £20 note. How much change does she get?[3 marks]
10
FThe length of a path is 8.5 m, correct to 1 decimal place. Write down the error interval for the length.[2 marks]
11
FA bag of sugar is labelled 1.5 kg, correct to 1 decimal place. (a) Write down the lower bound and the upper bound of the mass. (b) Could the mass of the sugar be 1.56 kg? Give a reason.[3 marks]
(a)lower bound and upper bound
(b)1.56 kg
12
FA number is 6.7 when truncated to 1 decimal place. Write down the error interval for the number.[2 marks]
13
FA rectangle has length 6.4 cm and width 3.2 cm, each correct to 1 decimal place. Work out the lower bound and the upper bound of the area of the rectangle.[4 marks]
14
FA train travels 250 km, correct to the nearest 10 km, in 3.2 hours, correct to 1 decimal place. Work out the average speed in km/h. Give your answer to a suitable degree of accuracy. You must show all your working.[5 marks]
15
FA rectangle has an area of 24 cm², correct to the nearest cm², and a width of 4 cm, correct to the nearest cm. Work out the upper bound of the length of the rectangle.[3 marks]
16
FWork out an estimate for √(9.8 × 41).[3 marks]
← Chapter 3: FractionsChapter 5: Percentages →

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