GCSE Maths Decimals, Rounding and Estimation (Higher): check-up and exam-style questions

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.
   0.5 = 0.500 0.45 = 0.450 0.405 = 0.405 0.054 = 0.054
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.
   0.500 = 500 thousandths
   0.450 = 450 thousandths
   0.405 = 405 thousandths
   0.054 = 54 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).
   54, 405, 450, 500
4. Write the original decimals in that order. The answer must use the numbers from the question, not the ones with extra zeros.
   0.054, 0.405, 0.45, 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.

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.
   4.70 + 12.85 -----
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: 7 + 8 = 15, write 5 and carry 1
   Put the decimal point straight down, directly under the other decimal points.
   units: 4 + 2 + 1 (carried) = 7
   tens: 0 + 1 = 1
3. Read the answer. Write the digits, with the decimal point where you put it.
   4.7 + 12.85 = 17.55

Check. Use the opposite operation: 17.55 − 12.85 = 4.7. ✓
Estimate: 5 + 10 = 15, close to 17.55. ✓

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.

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.
   10.00 − 3.64 -----
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 4, so exchange 1 from the next column on the left: 10 − 4 = 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 − 6 = 3
   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 − 3 = 6
   tens: 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.
   10 − 3.64 = 6.36

Check. Use the opposite operation: 6.36 + 3.64 = 10. ✓
Estimate: 10 − 4 = 6, close to 6.36. ✓

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.

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.
   0.6 has 1 decimal place and 0.07 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.
   6 × 7
3. Multiply 6 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 42
   = 42
4. 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.
   42 with 3 decimal places = 0.042

Check. Divide back: 0.042 ÷ 0.07 = 0.6. ✓

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.

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.
   7.2 × 10 = 72
   0.4 × 10 = 4
   So 7.2 ÷ 0.4 = 72 ÷ 4
2. Divide 72 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
   32 ÷ 4 = 8 exactly
3. Read the quotient (the answer on top). Write the digits you found, with the decimal point in the position shown.
   72 ÷ 4 = 18

Check. Multiply back: 18 × 0.4 = 7.2. ✓

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

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.
   0.35: last digit in the hundredths column, so denominator = 100
2. Write the digits after the decimal point over the denominator. The digits after the decimal point make the numerator.
   0.35 = 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

Check. Change back: 7 ÷ 20 = 0.35. ✓

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

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.
   4.0753 (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 5: 5 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.
   7 + 1 = 8
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.
   4.0753 = 4.08 (2 decimal places)

Check. The difference between 4.0753 and 4.08 is 0.0047. That is less than or equal to half of 0.01 (which is 0.005), so 4.08 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.

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 0.04067 is 4, in the hundredths 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 thousandths column.
   0.04067 (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.
   0 + 1 = 1
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.
   0.04067 = 0.041 (2 significant figures)

Check. The difference between 0.04067 and 0.041 is 0.00033. That is less than or equal to half of 0.001 (which is 0.0005), so 0.041 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.

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.
   4.9 → 5 (the next digit is 9, so round up)
   31.2 → 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 153 (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.

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 = 2.7 − 0.05 = 2.65
   upper bound = 2.7 + 0.05 = 2.75
4. Write the error interval with inequality signs. The lower bound 2.65 rounds up to 2.7, so x can equal it: use ≤. The upper bound 2.75 would round up to 2.8, so x must be less than it: use <.
   2.65 ≤ x < 2.75

Check. Test the ends: 2.65 rounds to 2.7 ✓; a number just below the top, such as 2.749, rounds to 2.7 ✓; and 2.75 itself rounds up to 2.8, 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.

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.
   36.50 − 8.74 -----
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 4, so exchange 1 from the next column on the left: 10 − 4 = 6
   tenths: the 5 gave 1 to the column on its right, so it is now 4: 4 is smaller than 7, so exchange 1 from the next column on the left: 14 − 7 = 7
   Put the decimal point straight down, directly under the other decimal points.
   units: the 6 gave 1 to the column on its right, so it is now 5: 5 is smaller than 8, so exchange 1 from the next column on the left: 15 − 8 = 7
   tens: the 3 gave 1 to the column on its right, so it is now 2: 2 − 0 = 2
3. Read the answer. Write the digits, with the decimal point where you put it.
   36.5 − 8.74 = 27.76

Check. Use the opposite operation: 27.76 + 8.74 = 36.5. ✓
Estimate: 40 − 9 = 31, close to 27.76. ✓

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.

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.
   2.8 has 1 decimal place and 0.35 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.
   28 × 35
3. Multiply 28 by each digit of 35. 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.
   28 × 5
   8 × 5 = 40, write 0, carry 4
   2 × 5 + 4 = 14, write 14
   = 140
   
   28 × 3: the 3 is in the tens column, so it really means 30, and we work out 28 × 3 then put 1 zero on the end
   8 × 3 = 24, write 4, carry 2
   2 × 3 + 2 = 8
   84 with 1 zero on the end = 840
4. Add the partial answers. The full product is the sum of the lines above.
   140 + 840 = 980
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.
   980 with 3 decimal places = 0.98

Check. Estimate by rounding to 1 significant figure: 3 × 0.4 = 1.2, which is close to 0.98. ✓

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.

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.
   15.12 × 10 = 151.2
   0.7 × 10 = 7
   So 15.12 ÷ 0.7 = 151.2 ÷ 7
2. Divide 151.2 by 7 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.
   1 ÷ 7 = 0 remainder 1
   15 ÷ 7 = 2 remainder 1
   11 ÷ 7 = 1 remainder 4
   Put the decimal point in the answer straight above the decimal point in the question.
   42 ÷ 7 = 6 exactly
3. Read the quotient (the answer on top). Write the digits you found, with the decimal point in the position shown.
   151.2 ÷ 7 = 21.6

Check. Multiply back: 21.6 × 0.7 = 15.12. ✓

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

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.
   0.45: last digit in the hundredths column, so denominator = 100
2. Write the digits after the decimal point over the denominator. The digits after the decimal point make the numerator.
   0.45 = 45100
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 45: 1, 3, 5, 9, 15, 45
   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.
   45 ÷ 5 = 9 and 100 ÷ 5 = 20
   45100 = 920

Check. Change back: 9 ÷ 20 = 0.45. ✓

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

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.
   78 = 7 ÷ 8 = 0.875
   0.65 is already a decimal
   35 = 3 ÷ 5 = 0.6
   0.7 is already a decimal
2. Give every decimal the same number of decimal places. Add zeros on the end until each has 3 decimal places.
   78 = 0.875
   0.65 = 0.650
   35 = 0.600
   0.7 = 0.700
3. Put them in order, smallest first. Compare column by column from the left.
   0.600, 0.650, 0.700, 0.875
4. Write the original numbers in this order. The answer should use the numbers from the question.
   35,  0.65,  0.7,  78

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.

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.
   4.87 → 5 (the next digit is 8, so round up)
   31.2 → 30 (the next digit is 1, so round down)
   0.52 → 0.5 (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.
   5 × 300.5
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 × 300.5
   = 1500.5
   = 300
4. State the estimate.
   The estimate is 300.

Check. A calculator gives 292 (3 s.f.) for the exact calculation. That is close to the estimate of 300. ✓

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

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.

Method. Find the cost of the pens, then take it away from £20.

1. Write the calculation. Pens cost £1.35 each. Priya buys 12 and pays with a £20 note.
   1.35 × 12
2. Work out the cost of 12 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. Subtract the cost from £20. Line up the decimal points, fill the gaps with zeros, and subtract column by column from the right.
   20.00 − 16.20 ----- 3.80
4. Write the answer as money. Amounts of money are written with two decimal places.
   £3.80

Check. £16.20 + £3.80 = £20. ✓

Watch out. Take care when subtracting from a whole number: write 20 as 20.00.

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 = 8.5 − 0.05 = 8.45
   upper bound = 8.5 + 0.05 = 8.55
4. Write the error interval with inequality signs. The lower bound 8.45 rounds up to 8.5, so L can equal it: use ≤. The upper bound 8.55 would round up to 8.6, so L must be less than it: use <.
   8.45 ≤ L < 8.55

Check. Test the ends: 8.45 rounds to 8.5 ✓; a number just below the top, such as 8.549, rounds to 8.5 ✓; and 8.55 itself rounds up to 8.6, 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.

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.
   6.7 ≤ x < 6.7 + 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.
   6.7 + 0.1 = 6.8
4. Write the error interval. The lower end 6.7 is possible (the digits after it could all be zero), so use ≤. The upper end 6.8 is not possible, so use <.
   6.7 ≤ x < 6.8

Check. Truncate 6.7 to the same place: 6.7 ✓. Truncate a value just below the top, 6.799: 6.7 ✓. Truncating 6.8 gives 6.8, 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.

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 = 6.4 cm (to 1 decimal place): unit = 0.1, half a unit = 0.05
   lower bound = 6.4 − 0.05 = 6.35, upper bound = 6.4 + 0.05 = 6.45
   
   width = 3.2 cm (to 1 decimal place): unit = 0.1, half a unit = 0.05
   lower bound = 3.2 − 0.05 = 3.15, upper bound = 3.2 + 0.05 = 3.25
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.
   6.45 × 3.25 = 20.9625 cm²
4. Calculate the lower bound of the area. Put the other bounds into the calculation.
   6.35 × 3.15 = 20.0025 cm²

Check. Check the bounds make sense: the area found from the original values is 20.48 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.

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 = 250 km (to the nearest ten): unit = 10, half a unit = 5
   lower bound = 250 − 5 = 245, upper bound = 250 + 5 = 255
   
   time = 3.2 hours (to 1 decimal place): unit = 0.1, half a unit = 0.05
   lower bound = 3.2 − 0.05 = 3.15, upper bound = 3.2 + 0.05 = 3.25
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.
   255 ÷ 3.15 = 80.95 km/h (rounded)
4. Calculate the lower bound of the speed. Put the other bounds into the calculation.
   245 ÷ 3.25 = 75.38 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 80, upper bound 80 (the same)
   to 2 s.f.: lower bound 75, upper bound 81 (different)
6. Write the answer to the most accurate level on which they agree. Both bounds round to 80 to 1 s.f., but to 2 s.f. they are different. So 80 is the answer we can be certain of.
   80 km/h (to 1 s.f.)

Check. Check the bounds make sense: the speed found from the original values is 78.125 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.

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 length 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.
   area = 24 cm² (to the nearest whole number): unit = 1, half a unit = 0.5
   lower bound = 24 − 0.5 = 23.5, upper bound = 24 + 0.5 = 24.5
   
   width = 4 cm (to the nearest whole number): unit = 1, half a unit = 0.5
   lower bound = 4 − 0.5 = 3.5, upper bound = 4 + 0.5 = 4.5
2. Decide which bounds to use. To get the biggest possible length, make it as big as it can be: use whichever bound of each measurement makes the result biggest. For the smallest length, do the opposite.
   area: a bigger area makes the length bigger, so for the upper bound of the length use the upper bound of area, and for the lower bound of the length use the lower bound.
   width: a bigger width makes the length smaller, so for the upper bound of the length use the lower bound of width, and for the lower bound of the length use the upper bound.
3. Calculate the upper bound of the length. Put the bounds you chose into the calculation. Use a calculator if the numbers are not easy.
   24.5 ÷ 3.5 = 7 cm

Check. Check the bounds make sense: the length found from the original values is 6 cm, 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.

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.
   9.8 → 10 (the next digit is 8, so round up)
   41 → 40 (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.
   √(10 × 40)
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.
   √(10 × 40)
   = √400
   = 20
4. State the estimate.
   The estimate is 20.

Check. A calculator gives 20 (3 s.f.) for the exact calculation. That is close to the estimate of 20. ✓

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