Writing Equations from Word Problems

How to translate word problems into algebraic equations and solve them step by step.

The Strategy

1
Read and identify the unknown
Decide what you are looking for. Assign it a letter, e.g. $x$ .
2
Translate words into maths
Look for keywords: more than (+), less than (−), times (×), divided by (÷), is / equals (=).
3
Write the equation
Build the equation from the translated parts.
4
Solve and check
Solve algebraically, then substitute back to verify.

Keyword Guide

Word/phrase	Operation
sum, more than, increased by, added to	$+$
difference, less than, decreased by	$-$
product, times, of	$\times$
quotient, divided by, per	$\div$
is, equals, gives, results in	$=$
twice, double	$\times 2$
half of	$\div 2$

A number increased by 7 equals 15. Find the number.

Let $x$ = the number.

$x + 7 = 15$

$x = 15 - 7 = 8$

Three times a number minus 4 is 11. Find the number.

$3x - 4 = 11$

$3x = 15$

$x = 5$

Ana is 6 years older than her brother. Their ages add up to 28. How old are they?

Let brother's age $= x$ . Ana's age $= x + 6$ .

$x + (x + 6) = 28$

$2x + 6 = 28$

$2x = 22 \Rightarrow x = 11$

Brother is 11, Ana is 17.

Real-World Applications

Real world: Train tickets

An adult train ticket costs €3 more than a child ticket. Two adults and one child pay €19 total. Let the child ticket cost $x$ : $2(x + 3) + x = 19 \implies 3x + 6 = 19 \implies 3x = 13 \implies x \approx 4.33\text{ EUR}$ .

Real world: Mobile data add-ons

A plan includes 2 GB for €10, with extra data at €4 per GB. A customer's bill is €26. How much extra data did they use? $4g + 10 = 26 \implies 4g = 16 \implies g = 4$ extra GB.

Real world: Sports match scoring

A basketball team scored twice as many points in the second half as in the first. The full-game total was 72 points. Let $p$ = first-half points: $p + 2p = 72 \implies 3p = 72 \implies p = 24$ in the first half and 48 in the second.

Tip

Where do you see this? Comparing phone or internet tariffs, splitting costs between friends, calculating distances from speed and time, quantities in recipes or shopping.

Practice Problems

A number divided by 4 is 9. What is the number?

Show answer

x / 4 = 9 → x = 36.

Two numbers differ by 5 and their sum is 23. Find both numbers.

Show answer

Let the numbers be x and x + 5. x + x + 5 = 23 → 2x = 18 → x = 9. Numbers are 9 and 14.

A cinema ticket costs €3 more than a bus ticket. Together they cost €11. Find both prices.

Show answer

Bus = x, Cinema = x + 3. x + x + 3 = 11 → 2x = 8 → x = 4. Bus €4, Cinema €7.