Multi-Step Worded Problems

A multi-step problem is a question in words that needs two or more calculations, one after another. The answer to one step is used in the next step.

For example, Sam buys 6 tickets at £9 each and pays with £60. One step finds the cost of the tickets, 6 \times \pounds 9 = \pounds 54. The next step finds the change, \pounds 60 - \pounds 54 = \pounds 6

In the exam you are given a short story about money, time, lengths, masses or numbers of things. You are asked for a total, the change, how many are left, how many boxes or trips are needed, or which option is cheaper. These questions are usually worth 2 to 5 marks, with marks for each correct step, so write every calculation down.

• Read the whole question, then read the last line again: it says what you must find.
• Pick out the numbers you need, with their units.
• Plan the steps in words, such as "find the cost of the tickets, then take it away from £60".
• Work out each step and write what it finds.
• Check that your answer answers the question and is a sensible size.

Words in the question help you choose each calculation. Total and altogether mean add. Left, change and difference mean subtract. Each and per usually mean multiply. Share equally and how many fit mean divide.

Profit is the money taken from sales, subtract all the costs. So a stall that takes £80 and spent £35 on stock makes \pounds 80 - \pounds 35 = \pounds 45 profit.

Write a few words before each calculation to say what it finds: Cost of tickets = 6 \times \pounds 9 = \pounds 54. The label tells you, and the examiner, what each number means, so you use the right number in the next step.

Before you add or subtract amounts, put them all in the same unit. A question may mix metres and centimetres, kilograms and grams, or pounds and pence.

• 1 metre is 100 centimetres.
• 1 kilogram is 1000 grams.
• 1 litre is 1000 millilitres.
• £1 is 100 pence.
• 1 hour is 60 minutes.

Change to the unit the question wants the answer in. If the answer is to be in grams, 3.2 kg becomes 3.2 \times 1000 = 3200 g.

Some amounts are used once, and some are for each person, item, hour or trip. A delivery charge of £5 is added once, but a price of £5 each is multiplied by how many there are. "There and back" means the distance twice.

Count gaps carefully. A meeting has 4 talks, one after another, each 40 minutes long. There is a 10 minute break between one talk and the next. There are only 3 breaks, because there is no break after the last talk.

Total time = 4 \times 40 + 3 \times 10 = 160 + 30 = 190 minutes.

Many problems end with a division that leaves a remainder. For example, 75 \div 8 = 9 remainder 3, because 9 \times 8 = 72 and 75 - 72 = 3. What you do next depends on the story.

Round up when everything must be packed, carried or bought. 75 books go into boxes that hold 8 books each. 9 boxes hold only 72 books, so 10 boxes are needed.

Round down when you count complete things you can make, fill or afford. 75 cm of string is cut into 8 cm pieces. That gives 9 whole pieces, because the 3 cm left over is too short for another piece. In the same way, £20 buys only 6 pens at £3 each, because 7 pens would cost £21.

If the question asks how many are left over, the answer is the remainder itself: 3 cm of string.

Some questions ask which option is cheaper, or whether there is enough of something. Work out a figure for each option, in the same units, then compare them.

Finish with a sentence that gives your decision and the figures behind it, such as "Shop B is cheaper, because £12.60 is less than £13.20". Figures with no decision, or a decision with no figures, will not get full marks.

If you are asked how much cheaper, subtract the smaller cost from the larger one.

Sometimes you are given the total and asked for one missing price. Undo the story: take away the parts you know, then share what is left between the items that cost the same.

For example, 4 pens and a £3 ruler cost £11 altogether. The pens cost \pounds 11 - \pounds 3 = \pounds 8, so one pen costs \pounds 8 \div 4 = \pounds 2

Check by putting your answer back into the story: 4 \times \pounds 2 + \pounds 3 = \pounds 11

If you are told the change instead of the total, find what was spent first. Paying £20 and getting £4.50 change means \pounds 20 - \pounds 4.50 = \pounds 15.50 was spent.

• Giving an answer from part of the way through. Read the last line again: if it asks for the change, what is left or the difference, the total is not the answer.
• Missing a detail in the question. Count every person and every item, and check whether each amount is paid once or for each one.
• Mixing units. Change every amount to the same unit before you add or subtract.
• Rounding a remainder the wrong way. Round up when everything must be packed or carried; round down when you count complete things you can make, fill or afford.
• Giving figures but no decision. When the question asks which, or whether, finish with a sentence that gives your decision and the figures.
• Writing only the final answer. Write every calculation with a label, so that you can earn marks for each step.