Procedure

Turn a worded condition into a linear equation in one unknown

Algebra • Equations • Linear Equations

Statement

To solve a worded problem about a single unknown quantity, give it a letter

n

, write every quantity the problem describes as an expression in

n

, translate the one stated relation between those quantities into an equation, and solve it by clearing denominators and collecting like terms.

How to use it

1Let $n$ be the quantity the question actually asks for, so no back-conversion is needed at the end.
2' $k$ more than' becomes $+ k$ , 'times' becomes multiplication, ' $k$ times as many' becomes $\times k$ ; write each phrase down before combining them.
3Multiply through by the common denominator early — a single equation in $n$ with no fractions is much harder to slip on.

Worked example

A number is tripled and the result exceeds one-fifth of the number by

42

. What is the number?

Step-by-step

1
$3 n - \frac{n}{5} = 42$
Why: Name the number $n$ and write the two described quantities and the relation between them.
2
$\frac{15 n - n}{5} = 42 \Rightarrow \frac{14 n}{5} = 42$
Why: Put the left-hand side over the common denominator $5$ .
3
$14 n = 210 \Rightarrow n = 15$

Answer

15

Watch out

Check the translation against the wording before solving: 'exceeds by' and 'is less than by' differ only in which side carries the surplus, and reversing them changes the sign of the answer.

Also known as

word problem to equation, linear equation in one variable, form and solve an equation, transposition method

Appears in past papers

Statement

Worked example

Appears in past papers

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