GCSE Maths · Decimals, Rounding and Estimation · Higher · Lesson 4.1

GCSE Maths: Place value and ordering decimals (Higher)

Part of the GCSE Maths decimals, rounding and estimation chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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4.1

Place value and ordering decimals

Edexcel 1MA1: N1, N2
Grade 1-3

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.

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.

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.

3.47 has 3 units, 4 tenths and 7 hundredths.

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.

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...
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
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.

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
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.
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.

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
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
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
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.

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
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.
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.

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
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.
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

FWrite down the value of the highlighted digit in each number.

(a)4.7

(b)0.072

(d)6.157

(e)7.04

(f)0.917

Done

FWrite down the missing symbol, <, > or =, between each pair of numbers.

(a)0.7 ? 0.07

(b)0.45 ? 0.5

(d)12.50 ? 12.5

(e)0.612 ? 0.62

(f)5.08 ? 5.008

Done

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

Done

FWrite the number that has 5 tens, 3 units, 4 tenths and 7 thousandths.

Done

FWork out

(a)3.8 × 10

(b)0.46 × 100

(d)64 ÷ 10

(e)9.1 ÷ 100

(f)0.7 ÷ 10

Done

FWrite down the decimal shown by each arrow on this number line, which goes from 2 to 3.

(a)arrow P

(b)arrow Q

Done

FFill in the missing number.

(a)5.2 × ? = 520

(b)0.083 × ? = 8.3

Done

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.)

Done

FWrite down a decimal that lies between 2.3 and 2.4.

Done

PSA box of 100 pencils costs £24. Each pencil costs £24 ÷ 100. How much is that for one pencil?

Done

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.)

Done

RAli says: “0.45 is bigger than 0.5 because 45 is bigger than 5.” Is Ali right? Give a reason.

Done

RJess says: “To multiply a number by 10 you just add a zero on the end.” Show with an example that Jess is wrong.

Done

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?

Done