AFCAT Numerical Ability · Percentage

Percentage Change, and Comparing Two Quantities

Measure a change on the starting value, recover the value before a change, and compare two quantities by how much one is more or less than the other.

All Percentage notes

Why this matters

Three AFCAT past-year questions are on this page, from 2015 and 2018, and all three compare two quantities rather than track one. The marks go on the base: 'A is 60% more than B' and 'B is 60% less than A' are different statements, and the options always carry both readings.

Concept 1 of 5: Percentage change measured on the old value

A price that goes from ₹480 to ₹600 rises by ₹120. As a percentage it is 120 out of the ₹480 it started from, so 25%.

Definition

Percentage change $= \dfrac{\text{new} - \text{old}}{\text{old}} \times 100$ . A negative answer is a decrease.
The old value is always the base, whether the change is a rise or a fall.
So a rise and the fall that undoes it have different percentages: 480 to 600 is +25%, and 600 back to 480 is −20%.
An improvement in rank from 60 to 51 is a fall of 9 on 60, which is 15%.

Percentage change

\%\text{ change} = \dfrac{\text{new} - \text{old}}{\text{old}} \times 100

Worked example

A price rises from ₹480 to ₹600. Find the percentage rise, and the percentage fall if it later goes back to ₹480.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Percentage · Percentage Change and Comparison

A district falls from

2,50,000

2,28,800

. What is the percentage decrease?

Divide by the starting value

From 40,000 to 37,400 is 2,600 on 40,000, which is 6.5%. Dividing by 37,400 gives about 6.95%, a common distractor.

Drill 11 more on percentage change measured on the old value

Concept 2 of 5: The value before a percentage change

A 20% rise multiplies a number by 1.2. To get back the number before the rise, divide by 1.2. Taking 20% off the new number does not undo it.

Definition

After a rise of $r\%$ : new $=$ old $\times \left(1 + \tfrac{r}{100}\right)$ , so old $=$ new $\div \left(1 + \tfrac{r}{100}\right)$ .
After a fall of $r\%$ : old $=$ new $\div \left(1 - \tfrac{r}{100}\right)$ .
Then answer what the question asks of the original: the number itself, or another percentage of it.
Given both 'up by $x\%$ gives $A$ ' and 'down by $x\%$ gives $B$ ': adding the two equations gives the number, $N = \tfrac{A + B}{2}$ .

Undoing a change

\text{old} = \dfrac{\text{new}}{1 \pm \tfrac{r}{100}}

Worked example

After a 20% rise, a salary is ₹27,000. What was it before the rise?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Percentage · Percentage Change and Comparison

A number when increased by

12

% becomes

448

. What is

25

% of the original number?

Do not take the percentage off the new value

₹27,000 after a 20% rise came from ₹22,500. Taking 20% off ₹27,000 gives ₹21,600, which is wrong, because the 20% was of the old salary.

Drill 10 more on the value before a percentage change

Concept 3 of 5: More than and less than: the base changes

If A is 60% more than B, take B as 100, so A is 160. B is 60 less than A, but as a share of A's 160 that is 37.5%, not 60%.

Definition

'A is $r\%$ more than B' uses B as the base. 'B is ?% less than A' uses A as the base.
If A is $r\%$ more than B, then B is $\dfrac{r}{100 + r} \times 100\%$ less than A.
If A is $r\%$ less than B, then B is $\dfrac{r}{100 - r} \times 100\%$ more than A.
The fastest check is to put the base at 100 and read the other number.

Swapping the base

\text{more by } r\% \;\to\; \text{less by } \dfrac{100r}{100 + r}\%, \qquad \text{less by } r\% \;\to\; \text{more by } \dfrac{100r}{100 - r}\%

Worked example

A's income is 60% more than B's. By what percent is B's income less than A's?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2015 · AFCAT (II) 2015 — Numerical Ability · 03-211Easy

Example 3 · Percentage · Percentage Change and Comparison

A

's salary is 50% more than

B

's. How much per cent is

B

's salary less than

A

's?

The two percentages are never equal

If A is 60% more than B, B is 37.5% less than A. The same gap is measured on a different base, so the percentages differ.

Drill 4 more on more than and less than: the base changes

Concept 4 of 5: Two quantities measured against a third

When both numbers are described against a third one, set the third at 100. Both numbers then appear as plain values and can be compared directly.

Definition

'Two numbers are 30% and 50% more than a third': with the third at 100 they are 130 and 150.
'Less than a third by 35% and 61%' makes them 65 and 39.
A chain (A against B, B against C) works the same way: put the last one at 100 and walk back along the chain.
Then compare as in the last concept, on whichever base the question names.

Set the reference at 100

\text{third} = 100 \;\Rightarrow\; \text{first} = 100 \pm a, \quad \text{second} = 100 \pm b

Worked example

Two numbers are 30% and 50% more than a third number. What percent of the second number is the first?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 4 · Percentage · Percentage Change and Comparison

The income of

A

is 20% higher than that of

B

. The income of

B

is 25% less than that of

C

. What per cent less is

A

's income than

C

's income?

Percentages along a chain multiply

30% less than 60% more is

0.7 \times 1.6 = 1.12

, so 12% more. Subtracting the percentages gives 30%, which is wrong.

Drill 7 more on two quantities measured against a third

Concept 5 of 5: When x% of one equals y% of another

40% of A being equal to 25% of B means B must be the bigger number. The equation

0.4A = 0.25B

is a ratio in disguise:

A : B = 25 : 40

Definition

$x\%$ of $A = y\%$ of $B$ gives $A : B = y : x$ (the percentages cross over).
B as a percentage of A is then $\dfrac{x}{y} \times 100$ .
With a sum or a difference of A and B, split it in that ratio.
If the percentages are of sums and differences, $a\%$ of $(P + Q) = b\%$ of $(P - Q)$ , expand and collect P and Q.

Equal percentages

\dfrac{x}{100}A = \dfrac{y}{100}B \;\Rightarrow\; A : B = y : x

Worked example

40% of A equals 25% of B. Find A : B, and B as a percentage of A.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2018 · AFCAT (II) 2018 — Numerical Ability · 03-206Moderate

Example 5 · Percentage · Percentage Change and Comparison

The difference of two numbers is 1660. If

6\frac{1}{2}

% of one number is

8\frac{1}{2}

% of the other number, the smaller number is

The ratio crosses over

40% of A = 25% of B gives

A : B = 25 : 40

, not

40 : 25

. The number with the smaller percentage is the larger number.

Drill 11 more on when x% of one equals y% of another

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 (5)

Percentage change measured on the old value
Percentage change
$\%\text{ change} = \dfrac{\text{new} - \text{old}}{\text{old}} \times 100$
The value before a percentage change
Undoing a change
$\text{old} = \dfrac{\text{new}}{1 \pm \tfrac{r}{100}}$
More than and less than: the base changes
Swapping the base
$\text{more by } r\% \;\to\; \text{less by } \dfrac{100r}{100 + r}\%, \qquad \text{less by } r\% \;\to\; \text{more by } \dfrac{100r}{100 - r}\%$
Two quantities measured against a third
Set the reference at 100
$\text{third} = 100 \;\Rightarrow\; \text{first} = 100 \pm a, \quad \text{second} = 100 \pm b$
When x% of one equals y% of another
Equal percentages
$\dfrac{x}{100}A = \dfrac{y}{100}B \;\Rightarrow\; A : B = y : x$

Watch out for (5)

Divide by the starting value
→ Percentage change measured on the old value
Do not take the percentage off the new value
→ The value before a percentage change
The two percentages are never equal
→ More than and less than: the base changes
Percentages along a chain multiply
→ Two quantities measured against a third
The ratio crosses over
→ When x% of one equals y% of another

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