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GCSE Maths · EdexcelChapter 1 · Number basics
GCSE Maths · Number Basics · Higher · Lesson 1.9

GCSE Maths: Number problems and checking answers (Higher)

Part of the GCSE Maths number basics chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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1.9

Number problems and checking answers

Edexcel 1MA1: N2, N3, N13
Grade 2-6

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.

2 adults3 childrentotal cost = ?
A bar model for the cost of a family ticket: the two parts are added to give the total.

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.
Worked example 1.9.1
Adult tickets cost £12.50 and child tickets cost £7.25. A family buys 2 adult tickets and 3 child tickets and pays with a £50 note. Work out the change.

Method. Work out the cost of the adult tickets and the child tickets separately, add them, then take the total from the amount paid.

  1. Work out the cost of the adult tickets. 2 adults each pay £12.50, so multiply.
    2 × 12.50 = 25.00
  2. Work out the cost of the child tickets. 3 children each pay £7.25.
    3 × 7.25 = 21.75
  3. Add to find the total cost. Line up the decimal points.
    25.00 + 21.75 = 46.75
  4. 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.

Worked example 1.9.2
Eggs are packed in boxes of 6. A farm has 250 eggs. How many boxes are needed to pack them all?

Method. Divide, then think about the remainder. If everything must be fitted in, round the answer up. If only full groups count, round down.

  1. Decide the calculation. We are finding how many groups of 6 there are in 250. That is a division.
    250 ÷ 6
  2. 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)
  3. 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
  4. 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.

Worked example 1.9.3
A 500 g pack of rice costs £1.35. A 750 g pack of the same rice costs £1.95. Which pack is better value?

Method. Find the cost of the same quantity for each item (a unit price) and compare.

  1. 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
  2. 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
  3. Compare. The better value is the one with the smaller price per 100 g.
    26p is the smallest
  4. 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.

Worked example 1.9.4
A gym membership costs £34.50 per month or £360 for a year. Show that paying for a year saves £54.

Method. Work out the cost of 12 months at the monthly price, then compare it with the yearly price.

  1. Work out the cost of paying monthly for a year. There are 12 months in a year.
    34.50 × 12 = 414.00
  2. Compare with the yearly price. Subtract the cheaper amount from the dearer.
    414.00 − 360 = 54.00
  3. State the conclusion. We have shown the saving is £54.
    So paying yearly saves £54. ✓

Check. 360 + 54 = 414 = 34.50 × 12. ✓

Watch out. In a “show that” question, always finish with a sentence that states what you have shown.

Worked example 1.9.5
Without a calculator, use an estimate to check whether 4872 ÷ 24 = 203 is sensible, then check it exactly.

Method. Round each number to one significant figure, divide, and compare with the given answer.

  1. Round each number. Rounding to one significant figure gives numbers that are easy to divide.
    4872 is about 5000 and 24 is about 25 (using a friendly number)
  2. Divide the rounded numbers.
    5000 ÷ 25 = 200
  3. Compare. The estimate is 200, so the real answer should be close to 200.
    203 is close to 200, so 203 is sensible; 20 or 2000 would not be.
  4. Check exactly. Multiply back: 203 × 24.
    203 × 24 = 4872 and 4872 ÷ 24 = 203

Check. 4872 = 4872. ✓

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-calculator
1
PSAdult tickets for a zoo cost £12.50 and child tickets cost £7.25. A family buys 2 adult tickets and 3 child tickets.
(a)Work out the total cost.
(b)They pay with two £20 notes and a £10 note. Work out the change.
2
PSDan saves £8.50 each week. How many weeks will it take him to save £255?
3
PSCupcake cases are sold in packs of 12. Mia needs 150 cases. Each pack costs £1.35. How much does she pay?
4
PSA 250 g jar of jam costs £1.80. A 400 g jar of the same jam costs £2.60. Which jar is better value? Show your working.
5
PSAn electricity meter reading goes from 45 821 to 46 309 units. Each unit costs 24.5p. Work out the total cost in pounds.
6
PSA school buys 28 calculators at £7.95 each. How much does it spend?
7
PSA car travels 360 km and uses 1 litre of fuel for every 12 km. Fuel costs £1.45 per litre. Work out the cost of the fuel.
8
PSA film starts at 19:45 and lasts 2 hours 35 minutes. At what time does it finish?
9
PSA recipe needs 350 g of flour. A bag has 1.5 kg of flour. How many whole batches can be made from one bag?
10
RJon says that 4 × 3 + 2 = 20 because “4 × 5 = 20”. Explain his mistake and give the correct answer.
11
RUse an estimate to show that 4872 ÷ 24 = 20.3 must be wrong.
12
PSA room is 5.4 m long and 3.5 m wide. Carpet costs £12.50 per square metre. Work out the cost of carpeting the whole floor.
13
PSAn office uses 5 reams of paper in 4 weeks. Each ream has 500 sheets. How many sheets does the office use in 52 weeks?
14
ROption A: 3 tickets at £8.40 each plus a booking fee of £2.50 for the whole order. Option B: 3 tickets at £9.20 each with no fee. Show that Option B is cheaper, and find by how much.
15
PSAt a school fair, 250 raffle tickets are sold for £1.50 each. The prizes cost £48.75. Work out the profit.
16
RZoe has 24 sweets. She shares them equally between 5 friends and keeps the remainder. What is the smallest number of extra sweets she needs so that nothing is left over to keep? Explain.

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