Year 6: Estimating, checking and solving percentage problems
Part of the Year 6 percentages chapter. Chapter overview · whole chapter on one page
Estimating, checking and solving percentage problems
Good mathematicians check their answers. Before you calculate, estimate: change the numbers to easy ones, such as 50% for 48% or 200 for 199. Your estimate tells you roughly what size the answer should be.
Some questions give you a percentage of the amount you want to find. For example, if 30% of a number is 45, then 1% is 45 ÷ 30 = 1.5, and the whole (100%) is 1.5 × 100 = 150. Look carefully at which amount is the whole.
Key ideas
- Estimate first by changing the numbers to easy ones: 48% is about 50%.
- To find the whole from a percentage: find 1%, then multiply by 100.
- Solve a long problem in steps. Write down each answer before you use it.
- Check the answer: does it make sense, and is it the size you estimated?
Key words. Of means multiply. So 15% of 360 means 15% × 360. To find a percentage of an amount, write the percentage as a fraction out of 100 and multiply.
Method. Replace the numbers with easier ones, work out the estimate, then check it.
- Use easier numbers. 48% is about 50%, and 199 is about 200.48% ≈ 50%, 199 ≈ 200
- Work out 50% of 200. 50% of 200 is easy to find.50% of 200 = 100
- Write the estimate. The answer is about this size.100
- Check with the exact value. The exact answer is 95.52, which is close to the estimate.48% of 199 = 95.52
Check. The estimate 100 and the exact answer 95.52 are close. ✓
Watch out. An estimate tells you if an answer is sensible. It does not replace the exact answer.
Key words. Of means multiply. So 15% of 360 means 15% × 360. To find a percentage of an amount, write the percentage as a fraction out of 100 and multiply.
Method. Find the discount first, then take it from the price to find the sale price.
- Find 15% of the total. This is the part that is given away.15% of 240 = 36
- Take that part away from the total. What is left is the rest.240 − 36 = 204
- Write the answer. 204
Check. 36 + 204 = 240. ✓
Watch out. Two steps: first find the part, then take it away. Do not stop after the first step.
Key words. The part is the number we are interested in. The whole is the total. To write a part as a percentage of the whole, divide the part by the whole and then multiply by 100.
Method. Find 1% of the whole first, then multiply by 100 to find the whole.
- Write what we know. 30% of the whole amount is 45.30% → 45
- Find 1%. Divide the part by 30, because 30% is 30 lots of 1%.45 ÷ 30 = 1.5
- Find 100%. The whole is 100%, so multiply the 1% amount by 100.1.5 × 100 = 150
- Write the answer. 150
Check. 30% of 150 is 45, which matches the part of 45 in the question. ✓
Watch out. Do not find 30% of 45. The 45 is the part, and we need the whole.
Watch out
Do not find 30% of 45. The 45 is the part, not the whole. When the whole is missing, find 1% first.
Exam tip
Exam tip. Write the amount you know, and the percentage you know, before you start. Then ask: which is the whole?
Exercise 6F Estimating and solving percentage problems
Non-calculatorKey words. Of means multiply. So 15% of 360 means 15% × 360. To find a percentage of an amount, write the percentage as a fraction out of 100 and multiply.
Method. Replace the numbers with easier ones, work out the estimate, then check it.
- Use easier numbers. 52% is about 50%, and 398 is about 400.52% ≈ 50%, 398 ≈ 400
- Work out 50% of 400. 50% of 400 is easy to find.50% of 400 = 200
- Write the estimate. The answer is about this size.200
- Check with the exact value. The exact answer is 206.96, which is close to the estimate.52% of 398 = 206.96
Check. The estimate 200 and the exact answer 206.96 are close. ✓
Watch out. An estimate tells you if an answer is sensible. It does not replace the exact answer.
Key words. Of means multiply. So 15% of 360 means 15% × 360. To find a percentage of an amount, write the percentage as a fraction out of 100 and multiply.
Method. Use 25% of 48 as an estimate for 24% of 49.
- Use easier numbers. 24% is about 25%, and 49 is about 48.24% ≈ 25%, 49 ≈ 48
- Work out 25% of 48. 25% is one quarter, so divide by 4.48 ÷ 4 = 12
- Write the estimate. The answer is about this size.12
- Check with the exact value. The exact answer is 11.76, which is close to 12.24% of 49 = 11.76
Check. 12 and 11.76 are close. ✓
Watch out. The estimate is close, but it is not exact. Use it to check, not to finish the question.
Key words. Of means multiply. So 15% of 360 means 15% × 360. To find a percentage of an amount, write the percentage as a fraction out of 100 and multiply.
Method. Work out 40% of 150, then compare it with 65.
- Find 40% of 150. 1% of 150 is 1.5, so 40% is 40 lots of 1.5.40% of 150 = 60
- Compare with 65. 60 is less than 65.60 < 65
- Write the answer. Less (40% of 150 is 60).
Check. 60 is 5 less than 65. ✓
Watch out. Find the amount first. Then compare it with the number in the question.
Key words. Of means multiply. So 15% of 360 means 15% × 360. To find a percentage of an amount, write the percentage as a fraction out of 100 and multiply.
Method. Find 25% of the price, then take it from the price.
- Find 25% of the total. This is the part that is given away.25% of 64 = 16
- Take that part away from the total. What is left is the rest.64 − 16 = 48
- Write the answer. 48
Check. 16 + 48 = 64. ✓
Watch out. Two steps: first find the part, then take it away. Do not stop after the first step.
Key words. The part is the number we are interested in. The whole is the total. To write a part as a percentage of the whole, divide the part by the whole and then multiply by 100.
Method. Find how many walk, then the rest, then 40% of the rest.
- Find the pupils who walk. 25% is a quarter, so divide by 4.240 ÷ 4 = 60 walk
- Find the pupils who do not walk. Take the walkers from the whole school.240 - 60 = 180
- Find 40% of the 180. 1% of 180 is 1.8, so 40% is 40 lots of 1.8.40% of 180 = 72
- Write the answer. 72
Check. 60 + 180 = 240, and 72 is less than 180. ✓
Watch out. The 40% is of the pupils who do not walk, not of the whole school.
Key words. The part is the number we are interested in. The whole is the total. To write a part as a percentage of the whole, divide the part by the whole and then multiply by 100.
Method. Find 1% of the whole first, then multiply by 100 to find the whole.
- Write what we know. 30% of the whole amount is 45.30% → 45
- Find 1%. Divide the part by 30, because 30% is 30 lots of 1%.45 ÷ 30 = 1.5
- Find 100%. The whole is 100%, so multiply the 1% amount by 100.1.5 × 100 = 150
- Write the answer. 150
Check. 30% of 150 is 45, which matches the part of 45 in the question. ✓
Watch out. Do not find 30% of 45. The 45 is the part, and we need the whole.
Key words. The part is the number we are interested in. The whole is the total. To write a part as a percentage of the whole, divide the part by the whole and then multiply by 100.
Method. Find 1% of the whole first, then multiply by 100 to find the whole.
- Write what we know. 15% of the whole amount is 12.15% → 12
- Find 1%. Divide the part by 15, because 15% is 15 lots of 1%.12 ÷ 15 = 0.8
- Find 100%. The whole is 100%, so multiply the 1% amount by 100.0.8 × 100 = 80
- Write the answer. 80
Check. 15% of 80 is 12, which matches the part of 12 in the question. ✓
Watch out. Do not find 30% of 45. The 45 is the part, and we need the whole.
Key words. Of means multiply. So 15% of 360 means 15% × 360. To find a percentage of an amount, write the percentage as a fraction out of 100 and multiply.
Method. Find how much Ben has saved, then take it from the price.
- Find 60% of £220. 10% is £22, so 60% is 6 lots of £22.6 × 22 = 132
- Find how much more he needs. Take what he has saved from the price.220 - 132 = 88
- Write the answer. 88
Check. 132 + 88 = 220. ✓
Watch out. Find how much he has saved first. The question asks how much more he needs.
Key words. Of means multiply. So 15% of 360 means 15% × 360. To find a percentage of an amount, write the percentage as a fraction out of 100 and multiply.
Method. Find 10% first, then add the 5% and 1% parts that make up 4%.
- Split 4% into 10%, 5% and 1% parts. Ten, five and one make the parts we can find easily.4% = 4 lots of 1%
- Find 1% of 350, then 4 lots. Divide by 100 for 1%.1% of 350 = 3.5; 4 × 3.5 = 14
- Add the parts. Add the answers for each part to find the whole percentage.14 = 14
Check. Fraction check: 4/100 × 350 = 14. ✓
Watch out. Check the parts: 10% is a tenth, so divide by 10. 1% is divide by 100.
Key words. Of means multiply. So 15% of 360 means 15% × 360. To find a percentage of an amount, write the percentage as a fraction out of 100 and multiply.
Method. Find 35% of 60 by building it from 10%, then check Tom’s answer.
- Find 10% of 60. 10% is one tenth, so divide by 10.60 ÷ 10 = 6
- Find 30% of 60. 30% is three lots of 10%.3 × 6 = 18
- Find 5% of 60. Half of 10% is 5%.6 ÷ 2 = 3
- Add to find 35%. 35% is 30% plus 5%.18 + 3 = 21
- Write the answer. Tom is not right. Tom found 30%, not 35%.No, 35% of 60 is 21.
Check. 35% of 60 = 21. ✓
Watch out. 18 is 30% of 60. Check every part of the percentage, not just the tens.
Key words. Of means multiply. So 15% of 360 means 15% × 360. To find a percentage of an amount, write the percentage as a fraction out of 100 and multiply.
Method. Find each part, then add and compare.
- Find 50% of 80. Half of 80.80 ÷ 2 = 40
- Find 10% of 80. 10% is one tenth.80 ÷ 10 = 8
- Add the two parts. Add the 50% and the 10% to find 60%.40 + 8 = 48
- Find 60% of 80 to compare. 60% is 6 lots of 10%.6 × 8 = 48
- Write the answer. The two answers are the same.Yes. 48 = 48.
Check. 50% + 10% = 60%, so the two ways give the same answer. ✓
Watch out. Percentages can be added when they are of the same amount.
Key words. Per cent means out of 100. The sign % means per cent, so 35% means 35 out of 100. It can be written as the fraction 35/100 or the decimal 0.35.
Method. Try a number, then decide.
- Try 10. Choose an easy positive number.10
- Find 60% of 10. Half of 10 is 5. 60% is more than 50%.60% of 10 = 6
- Compare with half. 6 is more than 5, which is half of 10.6 > 5
- Decide for every positive number. Any positive number gives the same result, because 60% is more than 50%.Always (for any number above 0).
Check. Try 20: 60% of 20 = 12, which is more than 10. ✓
Watch out. Half of a number is 50%. Compare every percentage with 50%.
Key words. The part is the number we are interested in. The whole is the total. To write a part as a percentage of the whole, divide the part by the whole and then multiply by 100.
Method. Work out the sales on each day, one after the other.
- Sold on Monday: 25% of 480. 25% is a quarter, so divide by 4.480 ÷ 4 = 120
- Left after Monday. Take Monday’s sales from the total.480 - 120 = 360
- Sold on Tuesday: a third of 360. A third is 360 ÷ 3.360 ÷ 3 = 120
- Left after Tuesday. Take Tuesday’s sales from what was left.360 - 120 = 240
- Write the answer. 240
Check. 120 + 120 + 240 = 480. ✓
Watch out. Each day is worked out from what is left, not from the original 480.