Everything in aptitude eventually becomes a percentage problem.

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Percentage Formula

$$\text{Percentage}=\frac{\text{Part}}{\text{Whole}}\times100$$

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Percentage to Fraction

Memorize these conversions.

Percentage	Fraction
10%	$\frac{1}{10}$
20%	$\frac{1}{5}$
25%	$\frac{1}{4}$
40%	$\frac{2}{5}$
50%	$\frac{1}{2}$
60%	$\frac{3}{5}$
75%	$\frac{3}{4}$
80%	$\frac{4}{5}$
100%	$1$

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Fraction to Percentage

Fraction	Percentage
$\frac{1}{2}$	50%
$\frac{1}{3}$	33.33%
$\frac{1}{4}$	25%
$\frac{1}{5}$	20%
$\frac{1}{8}$	12.5%
$\frac{2}{3}$	66.67%
$\frac{3}{4}$	75%

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Percentage Increase

$$\text{Percentage Increase} = \frac{\text{Increase}}{\text{Original}}\times100$$

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Percentage Decrease

$$\text{Percentage Decrease} = \frac{\text{Decrease}}{\text{Original}}\times100$$

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Important Shortcut

If a value increases by (x%), multiply by:

10%  → ×1.1
20%  → ×1.2
50%  → ×1.5
100% → ×2

Similarly, for decreases:

10% decrease → ×0.9
20% decrease → ×0.8
50% decrease → ×0.5

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2. RATIO & PROPORTION

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Ratio

A ratio compares two quantities.

a : b

This means:

$$\frac{a}{b}$$

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Example

2 : 3

Total parts:

$$2+3=5$$

If ₹500 is divided in the ratio:

2 : 3

First share:

$$500\times\frac{2}{5}=200$$

Second share:

$$500\times\frac{3}{5}=300$$

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Proportion

If

a : b = c : d

then

$$ad=bc$$

This is called cross multiplication.

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Direct Proportion

As one quantity increases, the other also increases.

Examples:

• More workers → More work
• More speed → More distance (for fixed time)

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Inverse Proportion

As one quantity increases, the other decreases.

Examples:

• More workers → Less time
• More speed → Less time (for fixed distance)

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Ratio Shortcut

For ratio:

2 : 3

Difference:

1 part

Total:

5 parts

Useful in partnership and age problems.

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3. AVERAGES

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Formula

$$\text{Average} = \frac{\text{Sum of Observations}} {\text{Number of Observations}}$$

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Example

Numbers:

10, 20, 30

Average

$$\frac{10+20+30}{3} = \frac{60}{3} = 20$$

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Important Shortcut

If all values increase by 5,

the average also increases by 5.

If all values decrease by 10,

the average also decreases by 10.

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Weighted Average

Used in mixtures and combined averages.

Formula:

$$\frac{a_1w_1+a_2w_2}{w_1+w_2}$$

where:

• (a_1,a_2) = averages
• (w_1,w_2) = weights

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4. PROFIT & LOSS

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Terms

CP = Cost Price
SP = Selling Price
MP = Marked Price

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Profit

$$\text{Profit}=SP-CP$$

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Loss

$$\text{Loss}=CP-SP$$

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Profit Percentage

$$\text{Profit \%} = \frac{\text{Profit}}{\text{CP}} \times 100$$

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Loss Percentage

$$\text{Loss \%} = \frac{\text{Loss}}{\text{CP}} \times 100$$

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Important Shortcut

For profit of 25%:

CP = 100
SP = 125

For loss of 20%:

CP = 100
SP = 80

Use the 100-method frequently.

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5. SIMPLE INTEREST

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Formula

$$SI=\frac{PRT}{100}$$

where:

P = Principal
R = Rate (% per annum)
T = Time (years)

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Amount

$$A=P+SI$$

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6. COMPOUND INTEREST

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Amount Formula

$$A = P\left(1+\frac{R}{100}\right)^n$$

where:

• (P) = Principal
• (R) = Rate
• (n) = Number of years

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Compound Interest

$$CI=A-P$$

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Important Shortcut

For 2 years:

$$CI-SI ===== \frac{PR^2}{10000}$$

This is frequently asked in placement exams.

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7. SUCCESSIVE DISCOUNT

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Formula

If two discounts are:

a%
and
b%

Equivalent discount:

$$a+b-\frac{ab}{100}$$

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Example

Discounts:

20%
then
10%

Equivalent discount:

$$20+10-\frac{20\times10}{100}$$ $$=20+10-2$$ $$=28$$

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8. ALLIGATION & MIXTURES

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Rule

Cheaper

Mean

Costlier

Then the ratio becomes:

Costlier − Mean

Mean − Cheaper

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Example

20

25

30

Ratio:

$$30-25 : 25-20$$ $$5:5$$ $$1:1$$

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Most Important Formula

$$\text{Higher} - \text{Mean} : \text{Mean} - \text{Lower}$$

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9. AGES

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Golden Rule

Age differences never change.

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Example

Present ages:

Father = 40

Son = 10

Difference:

$$40-10=30$$

After 20 years:

60 and 30

Difference:

$$60-30=30$$

The difference remains constant.

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Common Equations

Present age:

x

After 5 years:

x + 5

Before 8 years:

x - 8

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Ratio Questions

Convert ratios into variables.

Example:

Father : Son

5 : 2

Assume:

Father = 5x

Son = 2x

Then form equations and solve.

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