IMAT Mathematics · Numbers, Powers and Logarithms

Absolute Value, Percentages and Ratios

The absolute value is the distance from zero; a percentage change is a multiplier; a ratio splits a quantity into equal parts.

All Numbers, Powers and Logarithms notes

Why this matters

The 2014 paper asked for an original price from a reduced one, and the 2023 paper asked for a ratio that changes after both parts grow. Absolute value has not been asked on its own, but it appears inside an inequality in 2024 (see Equations and Inequalities).

Concept 1 of 3: Absolute value as distance from zero

The absolute value strips the sign: it measures how far a number is from 0 on the number line. That is why

|a - b|

is the distance between

a

and

b

, and why an equation like

|A| = 7

has two solutions, one on each side.

Definition

$|x| = x$ if $x \ge 0$ and $|x| = -x$ if $x < 0$ . So $|x| \ge 0$ always.
$|a - b|$ is the distance between $a$ and $b$ on the number line.
$\sqrt{x^2} = |x|$ , not $x$ .
$|ab| = |a|\,|b|$ , but in general $|a + b| \le |a| + |b|$ .
$|A| = k$ with $k > 0$ means $A = k$ or $A = -k$ ; with $k < 0$ it has no solution.

Absolute value

|x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}

Worked example

(a) Solve

|2x - 3| = 7

. (b) Solve

|x + 4| = -2

. (c) Evaluate

|-3| - |5 - 8|

Practice this conceptself-check · 4 quick reps

Minus x is not always negative

|x| = -x

is true for every

x \le 0

: if

x = -5

, then

-x = 5

. An option claiming

|x| = -x

is impossible is wrong.

Concept 2 of 3: Percentages: change, reverse percentage and successive changes

Increasing by 20% is the same as multiplying by 1.20, and decreasing by 20% is multiplying by 0.80. Thinking in multipliers makes reverse questions a single division and chains of changes a single product.

Definition

$p\%$ of a quantity is $\frac{p}{100}$ times it.
A rise of $p\%$ multiplies by $1 + \frac{p}{100}$ ; a fall multiplies by $1 - \frac{p}{100}$ .
Reverse percentage: to find the original, divide the new value by the multiplier.
Successive changes multiply: $+20\%$ then $-25\%$ is $1.20 \times 0.75 = 0.90$ , a 10% fall overall.
Percentage change $= \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$ . Word problems with percentages are covered in the Logic chapter.

Percentage change as a multiplier

\text{new} = \text{original} \times \left(1 \pm \frac{p}{100}\right)

$p$ the percentage rise (+) or fall (minus)

Worked example

(a) After a 25% increase a jacket costs €75. What did it cost before? (b) A price rises by 20% and then falls by 25%. What is the overall change?

Practice this conceptself-check · 4 quick reps

A rise and a fall of the same percentage do not cancel

Up 20% then down 20% gives

1.2 \times 0.8 = 0.96

: a 4% loss overall. The fall is a percentage of the larger, already-raised value.

Concept 3 of 3: Ratios and sharing in a given ratio

A ratio

a : b

says the whole is made of

a + b

equal parts. Call the size of one part

k

; then the amounts are

ak

and

bk

. Most ratio questions become one linear equation in

k

Definition

A ratio is unchanged when both parts are multiplied or divided by the same number: $45 : 60 = 3 : 4$ .
Sharing a total $T$ in the ratio $a : b$ : the parts are $\frac{a}{a+b}T$ and $\frac{b}{a+b}T$ .
Changing ratios: write the amounts as $ak$ and $bk$ , apply the change, and set up an equation with the new ratio.
Two quantities are in direct proportion when their ratio stays fixed (double one, double the other).

Sharing in the ratio a : b

\text{parts} = \frac{a}{a+b}\,T \quad\text{and}\quad \frac{b}{a+b}\,T

$T$ the total being shared

Worked example

(a) Share €84 in the ratio 3 : 4. (b) In a class the ratio of boys to girls is 5 : 7, and there are 12 more girls than boys. How many students are there?

Practice this conceptself-check · 4 quick reps

Adding the same amount to both parts changes the ratio

Adding 1 to each part of

2 : 3

gives

3 : 4

, a different ratio. Only multiplying or dividing both parts by the same number keeps a ratio.

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 (3)

Absolute value as distance from zero
Absolute value
$|x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}$
Percentages: change, reverse percentage and successive changes
Percentage change as a multiplier
$\text{new} = \text{original} \times \left(1 \pm \frac{p}{100}\right)$
Ratios and sharing in a given ratio
Sharing in the ratio a : b
$\text{parts} = \frac{a}{a+b}\,T \quad\text{and}\quad \frac{b}{a+b}\,T$

Watch out for (3)

Minus x is not always negative
→ Absolute value as distance from zero
A rise and a fall of the same percentage do not cancel
→ Percentages: change, reverse percentage and successive changes
Adding the same amount to both parts changes the ratio
→ Ratios and sharing in a given ratio

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