AFCAT Numerical Ability · Decimal and Fractions

Decimals and Percentages: Converting, Percent Of, and Percentage Change

A percentage is a number of hundredths, so moving between decimals and percentages is a shift of two places, and every percentage question is a multiplication or a division by 100.

All Decimal and Fractions notes

Why this matters

The page asks four things: convert, take a percentage of a number, say what percentage one number is of another, and find a percentage change. Each is one line. The marks go on a point moved one place instead of two, and on dividing a change by the new value instead of the original.

Concept 1 of 4: Decimals to percentages and back

Per cent means per hundred. A decimal is turned into hundredths by multiplying by 100, which moves the point two places right; a percentage goes back by moving it two places left.

Definition

Decimal to percentage: multiply by 100 ( $0.06 = 6\%$ ).
Percentage to decimal: divide by 100 ( $7.5\% = 0.075$ ).
A fraction: make it a decimal first, or multiply by 100 directly ( $\tfrac38 = 37.5\%$ ).
Values above 1 are over 100%: $1.25 = 125\%$ .

Decimal and percentage

x = (100x)\%, \qquad p\% = \dfrac{p}{100}

Worked example

Write

0.06

and

1.25

as percentages, and

7.5\%

as a decimal.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Decimal and Fractions · Decimals and Percentages

Convert

0.33

to a percentage:

Two places, not one

0.05 = 5\%

, not

50\%

0.5\%

. The options are usually the right answer with the point moved one place too far or too little.

Going back divides

8\% = 0.08

. Writing

0.8

moves the point only one place.

Drill 15 more on decimals to percentages and back

Concept 2 of 4: A percentage of a number

p\%

of a number is

\tfrac{p}{100}

of it. Ten per cent is the number with its point moved one place left, and most percentages can be built from 10%, 5% and 1%.

Definition

$p\%$ of $N$ is $\dfrac{p}{100} \times N$ .
Quick pieces: 10% moves the point one place left, 5% is half of that, 1% moves it two places.
Fraction equivalents help: $25\% = \tfrac14$ , $12.5\% = \tfrac18$ , $50\% = \tfrac12$ .

Percent of a number

p\% \text{ of } N = \dfrac{p}{100} \times N

Worked example

Find

35\%

of 260, and

12.5\%

of 480.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Decimal and Fractions · Decimals and Percentages

What is

15\%

200

Percent of is not percent off

40\%

of 70 is 28; 70 reduced by

40\%

is 42. Read whether the question wants the part or what is left.

Drill 9 more on a percentage of a number

Concept 3 of 4: What percentage one number is of another

To say what percentage

a

is of

b

, write

a

as a fraction of

b

and multiply by 100. The number after 'of' is always the whole, so it goes underneath.

Definition

$a$ as a percentage of $b$ is $\dfrac ab \times 100\%$ .
The number after 'of' is the whole, the denominator.
Simplify the fraction first when it helps: $\tfrac{18}{72} = \tfrac14 = 25\%$ .
The answer can pass 100% when the part is bigger than the whole.

Part as a percentage of the whole

\dfrac{\text{part}}{\text{whole}} \times 100\%

Worked example

18 is what percentage of 72? And what percentage of 64 is 40?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Decimal and Fractions · Decimals and Percentages

45

is what percentage of

200

The whole goes underneath

'15 is what percentage of 60' is

\tfrac{15}{60} = 25\%

. Turning it round,

\tfrac{60}{15}

, gives

400\%

Drill 7 more on what percentage one number is of another

Concept 4 of 4: Percentage increase and decrease

A change is always measured against where you started. Find the change, divide by the original value, and multiply by 100. To apply a change, multiply by

1 + \tfrac{p}{100}

1 - \tfrac{p}{100}

Definition

Percentage change $= \dfrac{\text{new} - \text{original}}{\text{original}} \times 100\%$ .
Increase by $p\%$ : multiply by $1 + \tfrac{p}{100}$ (up 15% is $\times 1.15$ ).
Decrease by $p\%$ : multiply by $1 - \tfrac{p}{100}$ (down 15% is $\times 0.85$ ).

Percentage change

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

Worked example

A price rises from 240 to 300. Find the percentage rise. Then reduce 300 by

15\%

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 4 · Decimal and Fractions · Decimals and Percentages

A price goes from

400

320

. What is the percentage decrease?

Divide by the original, not the new value

50 to 75 is a

50\%

rise (25 on 50), but 75 back to 50 is a

33\tfrac13\%

fall (25 on 75). The same gap gives a different percentage.

Up and down by the same percentage does not cancel

100 up

10\%

is 110, and 110 down

10\%

is 99, not 100.

Drill 7 more on percentage increase and decrease

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)

Decimals to percentages and back
Decimal and percentage
$x = (100x)\%, \qquad p\% = \dfrac{p}{100}$
A percentage of a number
Percent of a number
$p\% \text{ of } N = \dfrac{p}{100} \times N$
What percentage one number is of another
Part as a percentage of the whole
$\dfrac{\text{part}}{\text{whole}} \times 100\%$
Percentage increase and decrease
Percentage change
$\text{change}\% = \dfrac{\text{new} - \text{original}}{\text{original}} \times 100$

Watch out for (6)

Two places, not one
→ Decimals to percentages and back
Going back divides
→ Decimals to percentages and back
Percent of is not percent off
→ A percentage of a number
The whole goes underneath
→ What percentage one number is of another
Divide by the original, not the new value
→ Percentage increase and decrease
Up and down by the same percentage does not cancel
→ Percentage increase and decrease

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