How to turn a word problem into equations

A student can be comfortable rearranging an equation and still find it difficult to write one from a description. The missing step is often deciding what each quantity represents and how the quantities relate.

I like to separate that decision from the calculation. Two short examples show what this looks like.

Define the quantities before choosing a method

Consider this problem:

Sarah has twice as many apples as John. Together they have 30 apples. How many apples does each have?

Start by naming the two unknowns:

• Let $j$ be the number of apples John has.
• Let $s$ be the number of apples Sarah has.

“Sarah has twice as many” tells us that $s = 2j$. The total gives a second equation, $s + j = 30$. Substituting the first into the second:

$$2j + j = 30$$

So $j = 10$ and $s = 20$. Check both parts of the story: $20 = 2 \times 10$ and $20 + 10 = 30$.

The useful habit is to write down what a symbol means. An equation such as $s = 2j$ is easier to judge when you can read it back in words.

Read relationships, not just keywords

Words such as “twice” and “total” can help, but a keyword alone does not determine an equation. Order matters: “five less than $x$” means $x - 5$, while “$x$ less than five” means $5 - x$.

Before calculating, ask:

1. What am I trying to find?
2. What information connects to that unknown?
3. What assumptions does the question require?

For a motion problem, a quick diagram is often more useful than underlining every number.

Worked example: two trains

One train travels 120 km in 1 hour 30 minutes. Another travels 90 km in 1 hour. They leave stations 315 km apart at the same time, travelling towards each other at these constant speeds. When do they meet?

First, put the times in the same units. One hour 30 minutes is $1.5$ hours. The speeds are therefore:

$$v_1 = \frac{120}{1.5} = 80\text{ km/h}$$ $$v_2 = \frac{90}{1} = 90\text{ km/h}$$

Let $t$ be the time from departure until they meet, measured in hours. The first train covers $80t$ kilometres and the second covers $90t$ kilometres.

Now use the diagram or the story: travelling towards each other, their two distances must add to the full separation:

$$80t + 90t = 315$$

Hence:

$$t = \frac{315}{170} = \frac{63}{34}\text{ hours}$$

This is approximately $1.85$ hours, or 1 hour 51 minutes. It is not 1 hour 85 minutes: the decimal part is a fraction of an hour, so multiply it by 60 to convert to minutes.

Check before rounding

Using the exact time, the distances are:

$$d_1 = 80 \times \frac{63}{34} = \frac{2520}{17}\text{ km}$$ $$d_2 = 90 \times \frac{63}{34} = \frac{2835}{17}\text{ km}$$

Adding them gives exactly $315$ km. Rounding the time first would introduce a small discrepancy into this check.

There is also a quick estimate: together the trains close the gap at $170$ km/h. After two hours they would have covered $340$ km, so a meeting time just under two hours is reasonable.

Try a variation

What changes if the second train leaves 15 minutes later?

The first train has travelled $20$ km before the second starts. At that point the remaining separation is $295$ km. The trains then close that gap together at $170$ km/h. Decide whether your final answer measures time from the first departure or the second, and state it clearly.

That is the part I want students to practise: adapting the model when the story changes, rather than recognising a question and repeating a memorised calculation.

For individual support with setting up equations and explaining solutions, see my A-Level Maths tuition. The same approach is useful when Physics problems ask you to translate a physical situation into a mathematical model.