IMAT Logical Reasoning · Numerical Reasoning

Turning a Story into Arithmetic

Name the unknown, turn each sentence into a relation, solve, and check the answer against the story; when that is slow, work backwards or test the options.

All Numerical Reasoning notes

Why this matters

Setting up an equation from a story is the commonest skill in this chapter: ages, shared money, tickets and coins in the Cambridge papers, and in 2026 a chain of two relations to rearrange.

Concept 1 of 3: Setting up and solving an equation from a word problem

A word problem is an equation written in sentences. Give the unknown a letter, then each fact in the story becomes one line of algebra. Once it is written down, the solving is routine; the real work is the translation. Always check your answer in the original story, not in your equation, because a slip in translation gives an equation that is solved correctly and still wrong.

Definition

The method in four steps:

Name the unknown with a letter, and write what it stands for (" $d$ = daughter's age now").
Translate each sentence: "is" becomes $=$ , "3 times as many" becomes $3\times$ , "5 less than $x$ " becomes $x - 5$ , "in 4 years" adds 4 to every age.
Solve: one unknown needs one equation; two unknowns need two independent facts.
Answer the question asked and check it in the story. The letter you solved for is often not the quantity the question wants.

Worked example

A mother is three times as old as her daughter. In 12 years she will be twice as old as her daughter. How old are they now?

Practice this conceptself-check · 3 quick reps

Solving for the wrong quantity

If the question asks for the teacher tickets and your letter stands for the student tickets, the number you find is a correct answer to a different question. IMAT puts that number among the options. Re-read the last line of the question before choosing.

"5 less than x" is x − 5

The order in the sentence is the reverse of the order in the algebra: "5 less than

x

" is

x - 5

, not

5 - x

. In the same way, "A has 3 more than B" is

A = B + 3

Concept 2 of 3: Linked relations: substituting one into another

When two relations share a letter, you can replace that letter in one relation by what the other says it equals. The shared letter disappears, and you are left with a direct link between the other two. The same idea solves two equations with two unknowns: make one letter the subject, then put it into the other equation.

Definition

Substitution: if $a = 4b$ and $b = 5c$ , replace $b$ in the first relation to get $a = 4 \times 5c = 20c$ .

Multipliers along a chain multiply, they do not add.
To invert a relation, divide both sides: $a = 20c$ gives $c = a/20$ .
Two unknowns: two facts such as a sum and a difference. Make one letter the subject of one equation and substitute it into the other.

Chaining two relations

a = kb \ \text{and}\ b = mc \;\Rightarrow\; a = kmc, \quad c = \frac{a}{km}

$k, m$ the multipliers in each relation
$a, b, c$ the linked quantities; $b$ is the shared letter that disappears

Worked example

Given

p = 4q

and

q = \dfrac{r}{6}

, write

p

in terms of

r

, and then

r

in terms of

p

Practice this conceptself-check · 3 quick reps

Chained multipliers multiply

a = 4b

and

b = 5c

, then

a = 20c

, not

9c

. Adding the two multipliers is the most common wrong option. Test any answer with a small number before you choose it.

Concept 3 of 3: Working backwards and testing the options

When a story tells you the end result of a series of steps, start at the end and undo each step in reverse order. Each operation has an opposite: undo adding with subtracting, doubling with halving. When the equation is awkward, remember that IMAT gives you five answers: putting each one into the story is a legitimate method, and usually only one survives.

Definition

Two methods that avoid heavy algebra:

Working backwards: begin with the final value and undo the steps in reverse order. "Half of what is left, plus 2" is undone by adding 2 and then doubling.
Testing the options: put each option into the story and keep the one that fits every condition. Start with an option in the middle of the range; if it is too big, the answer is smaller.
Either way, finish by running the answer forwards through the story once.

Worked example

A farmer sells half of his eggs plus 2 more to the first customer. He then sells half of the remaining eggs plus 2 more to the second customer. He has 5 eggs left. How many did he start with?

Practice this conceptself-check · 3 quick reps

Undo the steps in reverse order

If the story says "take half, then subtract 2", going backwards you add 2 first and double second. Doubling first and then adding 2 gives a different, wrong starting value, and that value is often one of the options.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Linked relations: substituting one into another
Chaining two relations
$a = kb \ \text{and}\ b = mc \;\Rightarrow\; a = kmc, \quad c = \frac{a}{km}$

Watch out for (4)

Solving for the wrong quantity
→ Setting up and solving an equation from a word problem
"5 less than x" is x − 5
→ Setting up and solving an equation from a word problem
Chained multipliers multiply
→ Linked relations: substituting one into another
Undo the steps in reverse order
→ Working backwards and testing the options

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