AFCAT Numerical Ability · Percentage

Percentage Basics: Of, What Percent, and the Whole

A percentage is a fraction out of 100: find x% of a number, say what percent one number is of another, and get the whole back from a part.

All Percentage notes

Why this matters

Only one question on this page is HARD, and two AFCAT past-year questions sit here, from 2016 and 2017. Both are one equation: a difference of two percentages of the same number. Marks are lost on units (minutes against days, grams against kilograms) and on dividing by the wrong base.

Concept 1 of 5: x% of a number, and what percent one number is of another

Per cent means per hundred. 35% of 260 is 35 hundredths of 260. Asking what percent 18 is of 90 is asking how many hundredths of 90 make 18.

Definition

$x\%$ of $y$ is $\dfrac{x}{100} \times y$ .
$a$ as a percentage of $b$ is $\dfrac{a}{b} \times 100$ . The number after 'of' is the base and goes underneath.
Put both quantities in the same unit first: hours into minutes, kilograms into grams.
A percentage can pass 100: 40 is 800% of 5.
$x\%$ of $y$ equals $y\%$ of $x$ , so 8% of 25 is the same as 25% of 8, which is 2.

Percent of, and as a percent

x\% \text{ of } y = \dfrac{xy}{100}, \qquad \dfrac{a}{b} \times 100 = \text{a as a percentage of b}

Worked example

What percent of one and a half hours is 18 minutes?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Percentage · Basic Percentage Calculations

42 minutes is what per cent of 2 days?

Convert the units first

15 minutes of an hour is 25%, but dividing 15 by 1 gives 1,500%. Bring both quantities to minutes (or grams, or rupees) before dividing.

The base follows 'of'

'What percent of 80 is 20' divides by 80 and gives 25%. Dividing 80 by 20 answers a different question.

Drill 16 more on x% of a number, and what percent one number is of another

Concept 2 of 5: Fractions and their percentages

Most AFCAT percentages are friendly fractions in disguise. 12.5% is one eighth and 16⅔% is one sixth, so '12.5% of 640' is just 640 divided by 8.

Definition

A fraction becomes a percentage when multiplied by 100, and a percentage becomes a fraction when divided by 100. Learn the unit fractions below; any other fraction is a multiple of one of them, so three sixteenths is $3 \times 6.25\% = 18.75\%$ .

Fraction	Percentage	Use it like this
$\tfrac 12$	50%	50% of 86 is 43
$\tfrac 13$	$33\tfrac 13\%$	$\tfrac 23 = 66\tfrac 23\%$
$\tfrac 14$	25%	$\tfrac 34 = 75\%$
$\tfrac 15$	20%	$\tfrac 25 = 40\%$
$\tfrac 16$	$16\tfrac 23\%$	$\tfrac 56 = 83\tfrac 13\%$
$\tfrac 17$	$14\tfrac 27\%$	$\tfrac 67 = 85\tfrac 57\%$
$\tfrac 18$	12.5%	$\tfrac 38 = 37.5\%$ , $\tfrac 78 = 87.5\%$
$\tfrac 19$	$11\tfrac 19\%$	$\tfrac 29 = 22\tfrac 29\%$
$\tfrac 1{11}$	$9\tfrac 1{11}\%$	$\tfrac 2{11} = 18\tfrac 2{11}\%$
$\tfrac 1{12}$	$8\tfrac 13\%$	$8\tfrac 13\%$ of 600 is 50
$\tfrac 1{16}$	6.25%	$\tfrac 3{16} = 18.75\%$ Sixteenths turn up as 6.25%, 18.75% and the like: divide by 6.25 to find how many sixteenths.
$\tfrac 54$	125%	a percentage over 100 is a fraction over 1

Decimal to percentage: move the point two places right, so 0.045 is 4.5%.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Percentage · Basic Percentage Calculations

Express

\tfrac{5}{16}

as percentage.

One ninth is not 11%

\tfrac 19

11\tfrac 19\%

, and options often sit 11% beside it. Keep the fraction part; an option written as 11.11% is the same value.

Drill 18 more on fractions and their percentages

Concept 3 of 5: A percentage of a percentage

Taking 20% of 15% of a number is taking 0.2 of 0.15 of it, which is 0.03 of it. Multiply the decimals once, then use the result.

Definition

$a\%$ of $b\%$ of $N$ is $\dfrac{a}{100} \times \dfrac{b}{100} \times N$ .
If that is given and $N$ is asked, divide by the product of the two decimals.
Chains work the same way: if $x$ is 50% of $y$ and $y$ is 70% of $z$ , then $x = 0.5 \times 0.7\,z = 0.35z$ . Take $z = 100$ to read the answer as a percentage.
'x% of y is 7x' cancels the $x$ : $\tfrac{y}{100} = 7$ , so $y = 700$ .

Percent of a percent

a\% \text{ of } b\% \text{ of } N = \dfrac{ab}{10000}\,N

Worked example

Find 20% of 15% of 2,000.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Percentage · Basic Percentage Calculations

30

% of

40

% of a number is

96

, find the number.

Multiply, do not add

20% of 15% is 3%, not 35%. Each 'of' multiplies.

Drill 6 more on a percentage of a percentage

Concept 4 of 5: Finding the whole from a part

If 85% of a load is 1,700 kg, then 1% is 20 kg and the whole is 2,000 kg. Divide the part by its decimal, or go through 1%.

Definition

If $p\%$ of $N$ is $A$ , then $N = \dfrac{A}{p/100} = \dfrac{100A}{p}$ .
When the question then asks another percentage of the same number, find the number first, or scale directly: if 40% is 64, then 75% is $64 \times \tfrac{75}{40} = 120$ .
A loss or a gap is a part too. After 15% is lost, 85% is left; a tank 20% empty is 80% full.
'Half and 60% more' is $0.5 \times 1.6 = 0.8$ of the whole.

The whole from a part

N = \dfrac{100 \times \text{part}}{p}

Worked example

After 15% of a rice consignment is lost in transit, 1,700 kg arrives. How much was sent?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 4 · Percentage · Basic Percentage Calculations

If after 24% of wastage the net output of a coal-mine is 68400 quintals, then the total output of the coal-mine in quintals is

Use the percentage that is left

If 15% is wasted and 1,700 kg remains, divide by 0.85. Dividing by 0.15 finds the waste's whole, a number that means nothing here.

Drill 13 more on finding the whole from a part

Concept 5 of 5: Two percentages of the same number

70% of a number and 55% of the same number differ by 15% of it. Subtract the percentages first; the gap is a single percentage of the number.

Definition

$a\%$ of $N$ exceeds $b\%$ of $N$ by $D$ means $(a - b)\%$ of $N$ is $D$ .
'Adding 42 to 40% of a number gives the number' says the other 60% is 42: $0.4x + 42 = x$ .
A fraction and a percentage mix freely: four fifths is 80%.
Then answer what is asked: the number, or another percentage of it.

Difference of two percentages

(a - b)\% \text{ of } N = D \;\Rightarrow\; N = \dfrac{100D}{a - b}

Worked example

70% of a number exceeds 55% of it by 36. Find 40% of the number.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2016 · AFCAT (I) 2016 — Numerical Ability · 03-210Easy

Example 5 · Percentage · Basic Percentage Calculations

The difference between 63% of a number and 45% of the same number is 342. What is 78% of that number?

Answer the last line

These questions often find the number and then ask for another percentage of it. Stopping at the number picks the distractor that is always among the options.

Drill 8 more on two percentages of the same number

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 (4)

x% of a number, and what percent one number is of another
Percent of, and as a percent
$x\% \text{ of } y = \dfrac{xy}{100}, \qquad \dfrac{a}{b} \times 100 = \text{a as a percentage of b}$
A percentage of a percentage
Percent of a percent
$a\% \text{ of } b\% \text{ of } N = \dfrac{ab}{10000}\,N$
Finding the whole from a part
The whole from a part
$N = \dfrac{100 \times \text{part}}{p}$
Two percentages of the same number
Difference of two percentages
$(a - b)\% \text{ of } N = D \;\Rightarrow\; N = \dfrac{100D}{a - b}$

Reference tables (1)

Fractions and their percentages12 rows

Fraction	Percentage	Use it like this
$\tfrac 12$	50%	50% of 86 is 43
$\tfrac 13$	$33\tfrac 13\%$	$\tfrac 23 = 66\tfrac 23\%$
$\tfrac 14$	25%	$\tfrac 34 = 75\%$
$\tfrac 15$	20%	$\tfrac 25 = 40\%$
$\tfrac 16$	$16\tfrac 23\%$	$\tfrac 56 = 83\tfrac 13\%$
$\tfrac 17$	$14\tfrac 27\%$	$\tfrac 67 = 85\tfrac 57\%$
$\tfrac 18$	12.5%	$\tfrac 38 = 37.5\%$ , $\tfrac 78 = 87.5\%$
$\tfrac 19$	$11\tfrac 19\%$	$\tfrac 29 = 22\tfrac 29\%$
$\tfrac 1{11}$	$9\tfrac 1{11}\%$	$\tfrac 2{11} = 18\tfrac 2{11}\%$
$\tfrac 1{12}$	$8\tfrac 13\%$	$8\tfrac 13\%$ of 600 is 50
$\tfrac 1{16}$	6.25%	$\tfrac 3{16} = 18.75\%$ Sixteenths turn up as 6.25%, 18.75% and the like: divide by 6.25 to find how many sixteenths.
$\tfrac 54$	125%	a percentage over 100 is a fraction over 1

Decimal to percentage: move the point two places right, so 0.045 is 4.5%.

Watch out for (6)

Convert the units first
→ x% of a number, and what percent one number is of another
The base follows 'of'
→ x% of a number, and what percent one number is of another
One ninth is not 11%
→ Fractions and their percentages
Multiply, do not add
→ A percentage of a percentage
Use the percentage that is left
→ Finding the whole from a part
Answer the last line
→ Two percentages of the same number

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