GCSE Maths: Number problems and checking answers (Foundation)
Part of the GCSE Maths number basics chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Number problems and checking answers
Real problems are written in words. A good way to tackle them is: Read the question twice and underline the numbers and the units; Plan which operations are needed and in which order (a bar model or a quick sketch helps); Calculate one step at a time and write every step down; Check that the answer is sensible and answers the question that was asked, with the right units.
Key words. Total, altogether, sum mean add. Difference, change, left, how many more mean subtract. Each, per, lots of, times usually mean multiply, and share, between, per (in a rate), how many fit mean divide.
Remainders. If a division leaves a remainder, think about what it means. For 250 eggs in boxes of 6 you need 42 full boxes and one more for the last 4 eggs: 43 boxes (round up). But if you ask how many full boxes you can fill you say 42 (round down).
Best value. To compare two sizes of the same product, work out the price of the same amount of each. A 500 g pack at £1.35 costs 27p per 100 g and a 750 g pack at £1.95 costs 26p per 100 g, so the larger pack is better value.
Checking. Use the inverse operation (add to check a subtraction, multiply to check a division) or estimate by rounding each number to one significant figure. For 4872 ÷ 24 you can say 5000 ÷ 25 = 200, so an answer of 20 or 2000 would be wrong. Estimation is covered fully in Chapter 4. In a “show that” question you must show every step and write a final sentence that states what you have shown.
Key ideas
- Read, plan, calculate, check. Write every step.
- Use the same units throughout. Say the unit in the answer.
- Remainder: round up if everything must fit in, down if only full groups count.
- Best value: compare the price of the same quantity.
- Check with an inverse operation or an estimate.
Method. Work out the cost of the adult tickets and the child tickets separately, add them, then take the total from the amount paid.
- Work out the cost of the adult tickets. 2 adults each pay £12.50, so multiply.2 × 12.50 = 25.00
- Work out the cost of the child tickets. 3 children each pay £7.25.3 × 7.25 = 21.75
- Add to find the total cost. Line up the decimal points.25.00 + 21.75 = 46.75
- Subtract the total from the amount paid to find the change. Change = amount paid − cost.50.00 − 46.75 = 3.25
Check. Add the change to the cost: 46.75 + 3.25 = 50.00. ✓
Watch out. Write every answer in pounds with two decimal places.
Method. Divide, then think about the remainder. If everything must be fitted in, round the answer up. If only full groups count, round down.
- Decide the calculation. We are finding how many groups of 6 there are in 250. That is a division.250 ÷ 6
- Divide. Use short or long division. The remainder is what is left over.250 ÷ 6 = 41 remainder 4
(check: 6 × 41 = 246, and 250 − 246 = 4) - Decide what to do with the remainder. There are 4 eggs left over. They still need somewhere to go, so one more box is needed. We round up even though 4 is a small number.41 + 1 = 42
- Write the answer in words. 42 boxes
Check. 42 boxes holds up to 252 eggs (enough for 250), but 41 boxes holds only 246, which is not enough. ✓
Watch out. The remainder is not the answer. Ask yourself what the remainder means in the question: round up to fit everyone in, or down to count full groups only.
Method. Find the cost of the same quantity for each item (a unit price) and compare.
- Decide how to compare. The two items are different sizes, so we cannot just compare the prices. We find the cost of the same amount of each, for example the cost of 100 g.price per 100 g
- Work out the price of one 100 g for each. Change the price to pence (£1 = 100p), divide by the number of 100 gs in the pack.The 500 g pack: 135p ÷ 5 = 27p per 100 g
The 750 g pack: 195p ÷ 7.5 = 26p per 100 g - Compare. The better value is the one with the smaller price per 100 g.26p is the smallest
- Write the answer. The 750 g pack is the better value.
Check. Compare using a different amount, for example the cost of 1 g of each; the cheaper price per gram always wins. ✓
Watch out. Do not decide by the price alone. Always compare the price of the same amount.
Watch out
Answer the question that was asked. If a question asks for the change, the total cost is only a middle step. Put the units in your answer (£, kg, minutes) and check that the answer is sensible.Exercise 1K Number problems
Non-calculatorMethod. Multiply to find each cost, then add.
- Adult tickets. 2 × £12.50.2 × 12.50 = 25.00
- Child tickets. 3 × £7.25.3 × 7.25 = 21.75
- Add. 25.00 + 21.75 = 46.75
Check. 46.75 − 21.75 = 25.00. ✓
Method. Add the notes to find what they paid, then subtract the cost.
- Total paid. Two £20 notes and a £10 note.20 + 20 + 10 = 50
- Subtract the cost. Change = amount paid − cost.50.00 − 46.75 = 3.25
Check. 46.75 + 3.25 = 50.00. ✓
Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.
Method. Dividing by a decimal: multiply both numbers by 10, 100... until the divisor is a whole number. Then divide. Long division: work from the left, dividing 85 into each number in turn, carrying the remainder to the next digit.
- Make the number you are dividing by a whole number. Multiplying both numbers by 10 does not change the answer (it is like multiplying the top and bottom of a fraction by the same number). It makes the divisor a whole number, which is much easier.255 ÷ 8.5 = 2550 ÷ 85
(255 × 10 = 2550 and 8.5 × 10 = 85) - Write out the 85 times table up to 85 × 9. Having the multiples ready makes it quick to see how many times 85 goes into each number.85 × 1 = 85
85 × 2 = 170
85 × 3 = 255
85 × 4 = 340
85 × 5 = 425
85 × 6 = 510
85 × 7 = 595
85 × 8 = 680
85 × 9 = 765 - Set out the division. Write 2550 under the "bus stop" and 85 outside it. The answer will be written on top, one digit above each digit we divide.85 ) 2550
- Start with the first 3 digits, 255. 85 does not go into the first digit, so we take digits from the left until we have a number that 85 can go into.85 goes into 255 3 times: 85 × 3 = 255
Subtract: 255 − 255 = 0 - Bring down the next digit, 0. The remainder was 0, so the new number to divide is just that digit.85 does not go into 0, so write 0 in the answer.
- Write the answer. Read the digits along the top.30 85) 2550 255 --- 0
2550 ÷ 85 = 30
Check. Multiply back: 30 × 85 = 2550. ✓
Watch out. If 85 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.
Method. Find how many packs are needed (round up), then multiply by the price of one pack.
- Divide to find the number of packs. 150 cases in packs of 12.150 ÷ 12 = 12 remainder 6
- Round up. 12 packs give only 144 cases, which is not enough. She needs 6 more, so a 13th pack.12 + 1 = 13 packs
- Multiply by the price of a pack. 13 × 1.35 = £17.55
Check. 13 × 12 = 156 cases, which is at least 150, and 12 × 12 = 144, which is not. ✓
Method. Find the cost of the same quantity for each item (a unit price) and compare.
- Decide how to compare. The two items are different sizes, so we cannot just compare the prices. We find the cost of the same amount of each, for example the cost of 100 g.price per 100 g
- Work out the price of one 100 g for each. Change the price to pence (£1 = 100p), divide by the number of 100 gs in the pack.The 250 g jar: 180p ÷ 2.5 = 72p per 100 g
The 400 g jar: 260p ÷ 4 = 65p per 100 g - Compare. The better value is the one with the smaller price per 100 g.65p is the smallest
- Write the answer. The 400 g jar is the better value.
Check. Compare using a different amount, for example the cost of 1 g of each; the cheaper price per gram always wins. ✓
Watch out. Do not decide by the price alone. Always compare the price of the same amount.
Method. Subtract the readings to find the units used, then multiply by the price per unit (in pounds).
- Find the units used. The difference between the two readings.46309 − 45821 = 488
- Change 24.5p into pounds. 100p = £1, so divide by 100.24.5 ÷ 100 = £0.245
- Multiply. 488 × 0.245 = £119.56
Check. Estimate: 500 × 0.25 = 125, close to 119.56. ✓
Method. Multiply the price of one by the number bought.
- Write the calculation. 28 equal-price items.28 × 7.95
- Multiply. 795 × 28 = 795 × 20 + 795 × 8, then put the decimal point back.15900 + 6360 = 22260, so 28 × 7.95 = 222.60
- Write the answer. £222.60
Check. Estimate: 28 × 8 = 224, close to 222.60. ✓
Method. Work out how many litres are used, then multiply by the price per litre.
- Find the litres used. Divide the distance by the distance per litre.360 ÷ 12 = 30 litres
- Multiply by the price per litre. 30 × 1.45 = £43.50
Check. Estimate: 30 × 1.5 = 45, close to 43.50. ✓
Method. Add the hours, then the minutes.
- Add the hours. 19:45 + 2 hours = 21:45
- Add the minutes. 45 + 35 = 80 minutes = 1 hour 20 minutes, so we move on to the next hour.21:45 + 35 minutes = 22:20
- Write the answer. 22:20
Check. 22:20 take away 2 h 35 min = 19:45. ✓
Method. Change 1.5 kg to grams, divide, and round down.
- Change to grams. 1 kg = 1000 g.1.5 × 1000 = 1500 g
- Divide. 1500 ÷ 350 = 4 remainder 100
- Round down. The 100 g left over is not enough for another batch.4 batches
Check. 4 × 350 = 1400, 5 × 350 = 1750 > 1500. ✓
Method. Find the cost of the pens, then subtract from £10.
- Cost of 6 pens. 6 × £1.20.6 × 1.20 = 7.20
- Change. Take the cost from the amount paid.10.00 − 7.20 = 2.80
Check. 2.80 + 7.20 = 10.00. ✓
Method. Work out the adults and the children separately, then add.
- Adults. 2 × £2.80.2 × 2.80 = 5.60
- Children. 3 × £1.40.3 × 1.40 = 4.20
- Add. 5.60 + 4.20 = 9.80
Check. 9.80 − 4.20 = 5.60. ✓
Method. Subtract the stickers he gives away, then add the stickers he buys.
- Subtract. 245 − 78.245 − 78 = 167
- Add. 167 + 120 = 287
Check. 287 − 120 + 78 = 245. ✓
Method. Check the order of operations.
- Find what Jon did. He added 3 + 2 first.4 × (3 + 2) = 20
- Say what is wrong. Multiplication has priority over addition, so 4 × 3 comes first.4 × 3 = 12
- Work out the correct answer. 12 + 2 = 14
Check. 14 − 2 = 12 = 4 × 3. ✓
Method. Round the numbers and divide.
- Round. 4872 ≈ 5000 and 24 ≈ 25
- Divide. 5000 ÷ 25 = 200
- Compare. 20.3 is about ten times smaller than 200.20.3 cannot be right: the answer is about 200.
Check. 203 × 24 = 4872. ✓
Method. Divide 24 by 5 and look at the remainder.
- Divide. 24 ÷ 5 = 4 remainder 4
- Think about the remainder. To share equally with nothing left over, we need the number of sweets to be a multiple of 5. The next multiple of 5 above 24 is 25.25 − 24 = 1 more sweet
Check. 25 ÷ 5 = 5, with no remainder. ✓