Year 6: Multi-step problems, estimation and degrees of accuracy
Part of the Year 6 long multiplication, long division and bodmas chapter. Chapter overview · whole chapter on one page
Multi-step problems, estimation and degrees of accuracy
A multi-step problem needs more than one calculation. Start by reading the question carefully and finding the numbers you need. Draw a bar model if it helps. Then decide the order of your steps, and work them out one at a time.
Always estimate first. An estimate shows you roughly what size the answer should be. Then decide how accurate the answer must be. Some answers must be whole numbers, such as the number of buses needed. Money is usually given to the nearest penny.
Key ideas
- Read the question twice. Note the numbers and the words that tell you the operation.
- Estimate first. Round each number to one significant figure and work out the rough answer.
- Choose the degree of accuracy the question asks for, such as the nearest 1,000 or the nearest penny.
- For a remainder in a problem, decide whether to round up, round down or use a fraction.
- Check the answer is sensible, and write the unit.
Key words. To estimate is to find an answer that is close but not exact. To estimate, round each number to one significant figure: keep the first digit and make the other digits zero. The first digit is the most important one.
Method. Round each number to one significant figure, then multiply the rounded numbers.
- Round each number to one significant figure. Keep the first digit and make the other digits zero. Round up if the next digit is 5 or more.4,873 is about 5,000. 29 is about 30.
- Multiply the rounded numbers. This gives an estimate. It is close to the exact answer, but not the same.5,000 × 30 = 150,000
- Decide whether the estimate is sensible. An estimate that is close to the exact answer is a good check.Exact answer 141,317, and 150,000 is close to it
Check. The exact answer is 141,317, which is close to 150,000. ✓
Watch out. An estimate tells you whether an exact answer is sensible. It does not replace the exact answer when a question asks for it.
Key words. To estimate is to find an answer that is close but not exact. To estimate, round each number to one significant figure: keep the first digit and make the other digits zero. The first digit is the most important one.
Method. Work out the cost of each group, then add the two costs.
- Work out the cost of the adults. Multiply the number of adults by the price for one adult.3 × £14 = £42
- Work out the cost of the children. Multiply the number of children by the price for one child.4 × £8 = £32
- Add the two costs. The total cost is the adults plus the children.£42 + £32 = £74
Check. Estimate: 3 × 14 is about 40, and 4 × 8 is about 30, so about 70. £74 is close. ✓
Watch out. Read carefully: the question asks for the total cost of the family, not the cost of one ticket.
Key words. The degree of accuracy is how closely we round an answer, such as to the nearest 10, the nearest whole number or the nearest penny. Rounding a money answer to the nearest penny means two decimal places, but money is usually written in pounds and pence.
Method. Look at the digit after the place you are rounding to. Change the digits after that place to zeros.
- Look at the digit in the column just below hundred thousands. The rounding digit is the one to the right of the hundred thousands column.The digit is 7
- Decide whether to round up or down. 5 or more rounds up. Less than 5 rounds down.7 is 5 or more, so round up
- Write the rounded number. Replace the digits after the hundred thousands column with zeros.2,376,540 to the nearest 100,000 = 2,400,000
Check. 2,400,000 is the closest number to 2,376,540 that ends in 5 zeros. ✓
Watch out. Round the original number, not a number you have already rounded.
Key words. The degree of accuracy is how closely we round an answer, such as to the nearest 10, the nearest whole number or the nearest penny. Rounding a money answer to the nearest penny means two decimal places, but money is usually written in pounds and pence.
Method. Work in pence, then round the answer to the nearest penny.
- Change £1,000 into pence. There are 100 pence in every pound.£1,000 = 100,000p
- Divide by 3. Share the pence equally between the three charities.100,000 ÷ 3 = 33,333 remainder 1
- Round to the nearest penny. The remainder 1 is one third of a penny more, so the exact amount is 33,333 and one third of a penny. The next digit is less than 5, so round down.33,333 pence, so 33,333p to the nearest penny
- Write the answer in pounds. Divide by 100 to change pence into pounds.33,333p = £333.33
Check. 3 × £333.33 = £999.99, which is close to £1,000. ✓
Watch out. Money is written to the nearest penny with two figures after the point, so £333.33, not £333.3.
Watch out
Do not round the numbers too early in a multi-step problem. Round only at the end, when the question asks for an accurate answer or an estimate. Also, an estimate is not the exact answer: 4,873 × 29 is about 150,000, but the exact answer is 141,317.Exercise 2F Estimation, problems and degrees of accuracy
Non-calculatorKey words. To estimate is to find an answer that is close but not exact. To estimate, round each number to one significant figure: keep the first digit and make the other digits zero. The first digit is the most important one.
Method. Round each number to one significant figure, then multiply the rounded numbers.
- Round each number to one significant figure. Keep the first digit and make the other digits zero. Round up if the next digit is 5 or more.3,946 is about 4,000. 58 is about 60.
- Multiply the rounded numbers. This gives an estimate. It is close to the exact answer, but not the same.4,000 × 60 = 240,000
- Decide whether the estimate is sensible. An estimate that is close to the exact answer is a good check.Exact answer 228,868, and 240,000 is close to it
Check. The exact answer is 228,868, which is close to 240,000. ✓
Watch out. An estimate tells you whether an exact answer is sensible. It does not replace the exact answer when a question asks for it.
Key words. To estimate is to find an answer that is close but not exact. To estimate, round each number to one significant figure: keep the first digit and make the other digits zero. The first digit is the most important one.
Method. Round each number to one significant figure, then multiply the rounded numbers.
- Round each number to one significant figure. Keep the first digit and make the other digits zero. Round up if the next digit is 5 or more.612 is about 600. 49 is about 50.
- Multiply the rounded numbers. This gives an estimate. It is close to the exact answer, but not the same.600 × 50 = 30,000
- Decide whether the estimate is sensible. An estimate that is close to the exact answer is a good check.Exact answer 29,988, and 30,000 is close to it
Check. The exact answer is 29,988, which is close to 30,000. ✓
Watch out. An estimate tells you whether an exact answer is sensible. It does not replace the exact answer when a question asks for it.
Key words. To estimate is to find an answer that is close but not exact. To estimate, round each number to one significant figure: keep the first digit and make the other digits zero. The first digit is the most important one.
Method. Round each number to one significant figure, then multiply the rounded numbers.
- Round each number to one significant figure. Keep the first digit and make the other digits zero. Round up if the next digit is 5 or more.2,875 is about 3,000. 33 is about 30.
- Multiply the rounded numbers. This gives an estimate. It is close to the exact answer, but not the same.3,000 × 30 = 90,000
- Decide whether the estimate is sensible. An estimate that is close to the exact answer is a good check.Exact answer 94,875, and 90,000 is close to it
Check. The exact answer is 94,875, which is close to 90,000. ✓
Watch out. An estimate tells you whether an exact answer is sensible. It does not replace the exact answer when a question asks for it.
Method. Estimate 812 × 29 by rounding, then choose the closest option.
- Round each number. Round to one significant figure.812 is about 800, and 29 is about 30
- Estimate the answer. Multiply the rounded numbers.800 × 30 = 24,000
- Choose the closest option. Only one option is close to the estimate.C (24,000) is the closest option
Check. The exact answer is 23,548, which is closest to 24,000. ✓
Watch out. The options 16,000 and 160,000 are very far from the answer.
Key words. The degree of accuracy is how closely we round an answer, such as to the nearest 10, the nearest whole number or the nearest penny. Rounding a money answer to the nearest penny means two decimal places, but money is usually written in pounds and pence.
Method. Look at the digit after the place you are rounding to. Change the digits after that place to zeros.
- Look at the digit in the column just below thousands. The rounding digit is the one to the right of the thousands column.The digit is 6
- Decide whether to round up or down. 5 or more rounds up. Less than 5 rounds down.6 is 5 or more, so round up
- Write the rounded number. Replace the digits after the thousands column with zeros.48,650 to the nearest 1,000 = 49,000
Check. 49,000 is the closest number to 48,650 that ends in 3 zeros. ✓
Watch out. Round the original number, not a number you have already rounded.
Key words. The degree of accuracy is how closely we round an answer, such as to the nearest 10, the nearest whole number or the nearest penny. Rounding a money answer to the nearest penny means two decimal places, but money is usually written in pounds and pence.
Method. Look at the digit after the place you are rounding to. Change the digits after that place to zeros.
- Look at the digit in the column just below ten thousands. The rounding digit is the one to the right of the ten thousands column.The digit is 5
- Decide whether to round up or down. 5 or more rounds up. Less than 5 rounds down.5 is 5 or more, so round up
- Write the rounded number. Replace the digits after the ten thousands column with zeros.2,375,461 to the nearest 10,000 = 2,380,000
Check. 2,380,000 is the closest number to 2,375,461 that ends in 4 zeros. ✓
Watch out. Round the original number, not a number you have already rounded.
Method. Work out the cost of each type of item in pence, then change to pounds and add.
- Work out the pencils. Multiply the number of pencils by the price in pence.36 × 45p = 1,620p
- Change the pencils to pounds. There are 100 pence in every pound.1,620p = £16.20
- Work out the rulers. Multiply the number of rulers by the price in pence.12 × 125p = 1,500p = £15.00
- Add the two costs. Add the totals in pounds.£16.20 + £15.00 = £31.20
Check. Estimate: 40 × 45p is about £18, and 12 × £1.25 is about £15, so about £33. £31.20 is close. ✓
Watch out. Keep the units the same. Change pence to pounds before you add, or add everything in pence and change at the end.
Method. Divide the total by the group size. Then decide what the remainder means in this problem.
- Divide the total by the size of each group. 1,000 ÷ 52 gives 19 with 12 left over.1,000 ÷ 52 = 19 remainder 12
- Decide what to do with the remainder. The 12 left over still need a bus, so we round up.19 + 1 = 20
- Write the answer with a unit. 20 buses
Check. 52 × 19 = 988, and 988 + 12 = 1,000. ✓
Watch out. The answer depends on the story. Ask: does the remainder still need a group, or is it thrown away?
Key words. The order of operations is the order in which we work out a calculation. Brackets come first, then multiplication and division (from left to right), then addition and subtraction (from left to right).
Method. Brackets first, then multiplication and division from left to right, then addition and subtraction from left to right.
- Work out inside the brackets. Brackets come first. Inside them, multiply and divide before adding and subtracting.6 × 17 + 14 = 102 + 14
- Work out inside the brackets. Brackets come first. Inside them, multiply and divide before adding and subtracting.102 + 14 = 116
- Replace the brackets with their answer. The brackets are done, so the calculation is simpler now.250 − ( 6 × 17 + 14 ) = 250 − 116
- Add and subtract, from left to right. Work from the left. Addition and subtraction are done last.250 − 116 = 134
- Write the answer. 134
Check. Read the calculation again and follow the same order. ✓
Watch out. Multiplication and division come before addition and subtraction, even when the addition is written first.
Method. Work out the total number of sheets, then take away the sheets used.
- Work out the total sheets. Split 24 into 20 and 4, multiply each part by 250, then add.250 × 20 = 5,000 and 250 × 4 = 1,000
- Add the two parts. Add the sheets from each part.5,000 + 1,000 = 6,000 sheets
- Take away the sheets used. The school has fewer sheets left after it uses some.6,000 − 3,000 = 3,000
Check. Estimate: 25 × 250 = 6,250, and 6,000 is close. Add 3,000 back: 3,000 + 3,000 = 6,000. ✓
Watch out. Read the question for the words that tell you what to do: total means add, used means take away.
Method. Divide the total by the rate, then round to the nearest minute.
- Divide the litres by the litres per minute. Each minute the pump puts in 45 litres.
26 45 ) 1200 90 -- 30 270 --- 30 - Look at the remainder as a fraction of a minute. The remainder 30 out of 45 is two thirds of a minute, which is more than half.30 ÷ 45 = 2/3
- Round to the nearest minute. Two thirds is more than a half, so round up.26 + 1 = 27 minutes
Check. 45 × 26 = 1,170, and 1,170 + 30 = 1,200. ✓
Watch out. Give the time in whole minutes when the question says nearest minute.
Method. Estimate with one significant figure, then check the size of Ben's answer.
- Estimate each number. Round 4,873 to 5,000 and 29 to 30.5,000 and 30
- Multiply the estimates. Work out the estimate.5,000 × 30 = 150,000
- Compare with Ben's answer. Ben's answer is ten times too big.150,000 is not close to 1,500,000
Check. The exact answer is 141,317, so the estimate of 150,000 is close. ✓
Watch out. Check the number of zeros when you estimate.
Method. Estimate 6,249 ÷ 31 with rounded numbers, then check with the exact answer.
- Round each number. 6,249 is about 6,000, and 31 is about 30.6,000 and 30
- Work out the estimate. Divide the rounded numbers.6,000 ÷ 30 = 200
- Compare with the exact answer. The exact answer is 201 remainder 18, which is 201 and a bit more.6,249 ÷ 31 = 201 remainder 18
Check. 31 × 201 = 6,231, and 6,231 + 18 = 6,249. ✓
Watch out. An estimate is a good estimate if it is close to the exact answer.
Method. Work in pence. Divide by 9, then round to the nearest penny.
- Change £100 into pence. There are 100 pence in each pound.£100 = 10,000p
- Divide by 9. Share the pence equally between the nine friends.10,000 ÷ 9 = 1,111 remainder 1
- Round to the nearest penny. The remainder 1 out of 9 is 0.11 of a penny, which rounds down.1,111p, so 1,111p
- Write the answer in pounds. 1,111p = £11.11
Check. 9 × £11.11 = £99.99, which is close to £100. ✓
Watch out. The answer must be rounded because one penny cannot be split into nine equal parts exactly.