Translating and Checking Multi-Step Word Problems

Hello. In the previous lesson, you practised extracting the right values from tables, charts, and graphs, then choosing a calculation and giving an answer with its unit. This final Numeracy Exam Sprint lesson brings those habits together for longer word problems.

Multi-step problems can look difficult because the information arrives as a story. Usually, however, the arithmetic is familiar. The real task is to decide what each number represents, put the calculations in a sensible order, and check that the final answer fits the situation.

Plan for about $40$ minutes, including the short video and reading.

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Turn the story into a plan

A word problem may contain numbers, units, comparisons, and extra context. Do not begin calculating halfway through reading it. First read all the way to the question at the end.

This “UPS check” visual gives a useful four-part routine.

For exam questions, make the routine more specific:

Step	What to do
U — Underline	Underline what you must find. Circle the important numbers and their units. Cross out no information yet; first decide whether it matters.
P — Plan	Write a short list of what must be found first, second, and last. A sketch, table, or bar model can help.
S — Solve	Calculate one stage at a time. Label intermediate answers so you know what they mean.
Check	Use estimation, units, and the original information to test whether the answer is sensible.

The word at the end of the question often tells you the final quantity:

• How much does each person pay? Your final unit should be money per person.
• How many are left? Your final answer should be a remaining number of items or people.
• What is the total distance? Your final unit should be distance.
• How long altogether? Your final unit should be time.

A calculation can be correct but still answer the wrong question. For example, if a question asks for the cost after a discount, stopping at the original total is incomplete.

Operation words are clues, not commands

Words such as “total,” “remaining,” “each,” and “difference” can help, but they do not solve the problem for you.

For instance, each has two common meanings:

• “There are $8$ pens in each pack” means multiply the number of packs by $8$.
• “Share $40$ pens equally between $5$ tables” means divide $40$ by $5$.

The difference is the unknown. In the first sentence, you know the number in one group and want the total across groups. In the second, you know the total and want the amount in one equal group.

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Make the hidden structure visible

Before using a calculator or doing written arithmetic, turn the wording into a brief plan. You do not need to write a long paragraph. A few labelled lines are enough.

Consider this problem:

A school buys $5$ packs of $12$ pens for an event. Ten pens are used at the registration desk. The remaining pens are shared equally among $5$ classroom tables. How many pens does each table receive?

First, underline the target: pens per table.

Next, organise the facts:

• $5$ packs
• $12$ pens per pack
• $10$ pens used
• $5$ tables

Now write a plan:

1. Find the number of pens bought.
2. Find the number remaining after the registration desk uses some.
3. Share the remaining pens equally between the tables.

The calculations follow that plan:

$$5\times 12=60$$

There are $60$ pens at the start.

$$60-10=50$$

There are $50$ pens remaining.

$$50÷5=10$$

Each classroom table receives $10$ pens.

Notice how the units guide the work:

$$\text{packs}\times \frac{\text{pens}}{\text{pack}}=\text{pens}$$

Then you subtract pens from pens, and finally divide pens by tables. If you ever find yourself trying to add “packs” and “pens,” pause: the calculation is probably not yet set up correctly.

Bar models can clarify relationships

A bar model is a simple diagram made of rectangles. It does not do the arithmetic for you; it shows the relationship between a whole, its parts, and equal groups. It is especially useful when a problem involves “times as many,” sharing, or several linked steps.

You do not need to draw a perfect bar model in an exam. A rough sketch of equal boxes or a labelled list can prevent a major error. Use one when the words “twice,” “three times,” “share equally,” or “remaining” make the relationship hard to hold in your head.

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A complete multi-step example

Now combine skills from earlier lessons: multiplication, percentages, subtraction, and division.

A cinema group buys $18$ tickets at $\pounds 7.50$ each. The group receives a $20\%$ discount on the ticket total. The discounted cost is shared equally between $3$ families. How much does each family pay?

1. Identify the final target

The question asks for the amount each family pays. The final answer should be in pounds per family.

2. Plan the stages

The order matters:

1. Find the full cost of all tickets.
2. Find the discount amount.
3. Subtract the discount from the full cost.
4. Divide the discounted cost between $3$ families.

Writing the plan first protects you from a common mistake: finding $20\%$ of one ticket price instead of $20\%$ of the ticket total.

3. Carry out each calculation

First, calculate the original ticket total:

$$18\times 7.50=135$$

The original total is $\pounds 135$.

Next, calculate $20\%$ of $\pounds 135$:

$$0.20\times 135=27$$

The discount is $\pounds 27$.

Now subtract the discount:

$$135-27=108$$

The discounted group cost is $\pounds 108$.

Finally, share this amount among $3$ families:

$$108÷3=36$$

Each family pays $\pounds 36$.

4. Check the answer

A good check uses the context, not just a second calculation.

• Without the discount, each family would pay:

$$135÷3=45$$

• A $20\%$ discount should make each family’s cost lower than $\pounds 45$.
• The discount on the whole group is $\pounds 27$, so the discount per family is:

$$27÷3=9$$

• Therefore each family should pay:

$$45-9=36$$

This matches the answer $\pounds 36$, so it is reasonable.

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Checking: catch mistakes before they cost marks

Checking is not an optional extra. It is your best chance to catch a wrong operation, a missed step, or an answer with the wrong unit.

Use these five checks.

1. Estimate the size first

Round numbers roughly in your head.

In the cinema example, $18$ tickets at about $\pounds 8$ each cost about $\pounds 144$. After a $20\%$ discount, the group should pay somewhat less than $\pounds 144$, perhaps a little over $\pounds 100$. Dividing among $3$ families should give a little over $\pounds 30$ each.

So $\pounds 36$ is believable. An answer like $\pounds 3.60$ or $\pounds 360$ would need immediate investigation.

2. Check the direction of change

Ask whether the situation should make a number bigger or smaller.

• Adding an extra charge should increase the total.
• A discount should decrease the price.
• Taking away used items should decrease the number remaining.
• Sharing the same total among more people should make the amount per person smaller.

For example, if you calculated a discounted price that was higher than the original price, you may have added the discount instead of subtracting it.

3. Work backwards where possible

Use the inverse operation to reconstruct an earlier value.

For the pen example:

$$10\times 5=50$$

So $10$ pens per table gives $50$ shared pens. Adding the $10$ used at registration gives:

$$50+10=60$$

And $60$ pens equals $5$ packs of $12$:

$$5\times 12=60$$

Every part matches the original story.

4. Check units and the exact wording

Your final sentence should answer what was asked:

• money: $\pounds$
• distance: miles, kilometres, metres
• mass: grams or kilograms
• time: minutes or hours
• items or people: whole numbers, where appropriate

If the question asks, “How much more?” include the comparison in your answer. If it asks, “How much each?”, make clear that the amount is per person, pack, day, or group.

5. Check whether a whole-number answer is required

Some contexts allow decimals:

• money, when written to the nearest penny;
• distances;
• weights;
• average values.

Other contexts usually need whole numbers:

• people;
• buses;
• pencils;
• tables.

If a calculation says $3.4$ buses are needed, you cannot use $0.4$ of a bus. You would normally need $4$ buses. Always return to the real-world meaning before rounding.

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An exam-ready answer format

For multi-step questions, organise your working so a marker can see your reasoning even if one arithmetic slip occurs. A strong layout looks like this:

$$\text{first calculation}=\text{intermediate result}$$ $$\text{next calculation}=\text{intermediate result}$$ $$\text{final calculation}=\text{answer}$$

Then write one direct concluding sentence.

For example:

$$18\times 7.50=135$$ $$0.20\times 135=27$$ $$(135-27)÷3=36$$

Each family pays $\pounds 36$.

You may sometimes be able to combine steps into one line, but separate lines are safer when you are learning or working under pressure. They also make checking much easier.

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Key takeaways

A multi-step word problem becomes manageable when you separate its stages.

1. Read the full problem and underline what you must find.
2. Record important numbers with their units.
3. Write a short plan before calculating.
4. Solve one stage at a time and label intermediate answers.
5. Check by estimating, working backwards, checking units, and considering whether the result fits real life.
6. End with a complete sentence that answers the exact question.

You have now completed the Numeracy Exam Sprint. The next module moves to English exam responses, beginning with how to spot command words and evidence requirements in a question.