Percentage — AFCAT Numerical Ability Formulas and Traps

Percentage Basics: Of, What Percent, and the Whole

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}

Fractions and their percentages

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%.

Common traps

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.

One ninth is not 11%

\tfrac 19

is

11\tfrac 19\%

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

Multiply, do not add

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

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.

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.

Percentage Change, and Comparing Two Quantities

Learn this subtopic in the notes

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

Common traps

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.

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.

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.

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.

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.

Successive Changes, Price and Consumption

Learn this subtopic in the notes

The net effect of successive changes

Two successive changes

\text{net } \% = a + b + \dfrac{ab}{100}

Undoing a change: the percentage needed to get back

The change that undoes another

\text{after a fall of } r\%: \; \dfrac{100r}{100 - r}\% \text{ rise}, \qquad \text{after a rise of } r\%: \; \dfrac{100r}{100 + r}\% \text{ fall}

When a quantity is a product: area, volume, takings

Change in a product

Q = XY \;\Rightarrow\; \text{factor}_Q = \left(1 + \tfrac{a}{100}\right)\left(1 + \tfrac{b}{100}\right)

Keeping expenditure fixed when the price changes

Consumption cut for a fixed budget

\text{cut} = \dfrac{r}{100 + r} \times 100\%

A cheaper price buys more for the same money

Reduced price from the extra quantity

\text{reduced price} = \dfrac{\tfrac{r}{100} \times M}{\text{extra quantity}}

Common traps

Up and down by the same percentage is a loss

Up 30% then down 30% is

1.3 \times 0.7 = 0.91

, a 9% fall. The fall is taken on the larger number, so it removes more than the rise added.

Do not add the percentages

Up 30% then down 20% is not the difference of the two. The second change acts on the new value, which is why the formula has the extra

\tfrac{ab}{100}

term.

The same percentage does not undo it

A 40% cut followed by a 40% rise ends at

0.6 \times 1.4 = 0.84

, still 16% short. The rise has to be measured on the smaller, cut price.

Area squares the change

A side 10% longer gives an area 21% larger, not 10%. A perimeter, though, grows by the same 10%.

The cut is smaller than the rise

A 60% price rise needs only a 37.5% cut in use, because the cut is taken on the old consumption while the rise was on the old price.

The saving method gives the reduced price

The money saved buys the extra kilograms at the new price. If the question asks for the original price, divide by

1 - \tfrac{r}{100}

at the end.

Per dozen, per kg, per item

Read the unit asked. A reduced price of ₹1.10 an apple is ₹13.20 a dozen, and both are usually among the options.

Population Growth and Depreciation

Learn this subtopic in the notes

Growth and depreciation year after year

Compound growth

P_n = P\left(1 + \dfrac{r}{100}\right)^n

Going back in time

The starting value

P = \dfrac{P_n}{\left(1 \pm \tfrac{r}{100}\right)^n}

A different rate each year

Changing rates

P_{\text{final}} = P\left(1 + \tfrac{r_1}{100}\right)\left(1 + \tfrac{r_2}{100}\right)\left(1 + \tfrac{r_3}{100}\right)

Finding the rate of growth

Rate from two figures

1 + \dfrac{r}{100} = \left(\dfrac{P_n}{P}\right)^{1/n}

The make-up of a population

A share of the whole

\text{total} = \dfrac{\text{size of a part} \times 100}{\text{its percentage}}

Common traps

Compound, not simple

12.5% a year for 3 years is not 37.5% in all. Each year's growth is on the new figure, so 64,000 becomes 91,125, not 88,000.

Divide, do not subtract

For 60,500 after two years of 10% growth, taking 10% off twice gives 49,005. The answer is 50,000: the growth was on the smaller, earlier figure.

The second loss is on the survivors

10% die and then 20% of the rest leave is a loss of 28%, not 30%. The 20% is of the 90% left.

Do not divide the total growth by the years

44% growth in two years is 20% a year, not 22%. The yearly rate is a root, not an average.

Weight each part's change

Urban up 10% and rural down 10% is not 'no change' when the parts are 70% and 30%. The larger part pulls the total its way, here to a 4% rise.

Shares of a Whole: Budgets, Savings and Groups

Learn this subtopic in the notes

Parts of a whole, and the rest

The rest, and the whole

\text{rest} = 100\% - \sum \text{parts}, \qquad \text{whole} = \dfrac{100 \times \text{part}}{p}

Income, spending and savings

Savings from spending

\text{income} = \dfrac{\text{savings} \times 100}{100 - \text{spending}\%}

A share of what is left

Amount left after shares of the rest

\text{left} = S\left(1 - \tfrac{a}{100}\right)\left(1 - \tfrac{b}{100}\right)

When income and spending both change

New savings

\text{new savings} = 100\left(1 + \tfrac{a}{100}\right) - s\left(1 + \tfrac{b}{100}\right)

Combining groups with different percentages

Weighted percentage

\text{overall } \% = \dfrac{n_1 p_1 + n_2 p_2}{n_1 + n_2}

Overlapping groups: failed in one, both or neither

Two overlapping groups

n(A \cup B) = n(A) + n(B) - n(A \cap B)

Common traps

Answer the part asked

With 62.5% boys and 30 girls, the class is 80. If the question asks for boys, 80 is the distractor; the answer is 50.

Divide by the savings share, not the spending

If spending is 78% and savings ₹5,500, the income is

5500 \div 0.22

. Dividing by 0.78 treats the savings as the spending.

'Of the rest' is not of the whole

25% and then 40% of the rest leaves 45%, not 35%. Adding the two percentages treats the 40% as a share of the whole salary.

Savings swing more than income

With income up 20% and spending up 10%, savings rise 60%, not 10% or 20%. Work it out on 100; never guess from the two percentages.

Do not average the percentages

With boys and girls 3 : 1, 40% and 60% give 45% overall, not 50%. A plain average is right only when the groups are equal.

Take the overlap off once

40% and 32% failing with 12% in both is 60% failing at least one. Adding 40 and 32 counts the 12% twice.

Marks, Pass Marks and Elections

Learn this subtopic in the notes

Scores, questions and negative marking

Score with negative marking

\text{score} = m \times \text{right} - p \times \text{wrong}

The pass mark from one student's score

Pass mark

\text{pass mark} = \text{score} + \text{shortfall} = \dfrac{p}{100} \times \text{maximum}

Two scores either side of the pass mark

Maximum from two candidates

\dfrac{b - a}{100}\,M = f + s

Overall percentage across papers

Weighted overall percentage

\text{overall} = \dfrac{\sum w_i p_i}{\sum w_i}

Elections: shares, margins and invalid votes

Total from the margin

\text{valid votes} = \dfrac{\text{margin} \times 100}{\text{winner}\% - \text{loser}\%}

Common traps

Wrong answers cost twice

Moving one answer from right to wrong with +4 and −1 marking drops the score by 5, not 4. That is why the equation becomes

5x

.

Failing by means add, passing by means subtract

Scored 113 and failed by 22: the pass mark is 135. Subtracting gives 91, and that wrong pass mark leads to a wrong maximum among the options.

Add the shortfall and the surplus

One student 33 below the pass mark and another 15 above it are

33 + 15 = 48

marks apart. Subtracting,

33 - 15

, is only right when both students fail.

Unequal maxima are not averaged

40 out of 50 is 80% and 66 out of 150 is 44%, but the overall is

\tfrac{106}{200} = 53\%

, not 62%. The bigger paper counts for more.

The margin is a share of the votes cast

If 80% vote and the margin is 10% of the votes cast, it is 8% of the voters' list. Dividing the margin by 0.1 gives the votes cast, not the list.

Profit, Loss and Discount with Percentages

Learn this subtopic in the notes

Profit and loss on the cost price

Selling price from cost price

SP = CP\left(1 + \dfrac{p}{100}\right), \qquad CP = \dfrac{SP}{1 + p/100}

Articles bought and sold in different lots

Cost of a equals selling price of b

\text{profit } \% = \dfrac{a - b}{b} \times 100

Marked price, discount and profit

From cost to marked to selling price

SP = CP\left(1 + \dfrac{m}{100}\right)\left(1 - \dfrac{d}{100}\right)

Successive discounts and chains of profit

Two successive discounts

\text{single discount} = a + b - \dfrac{ab}{100}

Overall profit when parts sell at different rates

Same selling price, gain and loss of r%

\text{loss } \% = \dfrac{r^2}{100}

Common traps

Profit is on cost, not on selling price

Bought for ₹640 and sold for ₹720, the profit is 80 on 640, which is 12.5%. Dividing by the selling price gives 11.1%.

A loss and a gain add

Going from a 12% loss to a 9% gain covers 21% of the cost price. Subtracting the two, 3%, is the usual slip.

Divide by the articles sold

When the cost of 22 equals the selling price of 18, the profit is 4 on 18 (22.2%), not 4 on 22 (18.2%).

Discount on marked price, profit on cost

Marked 60% up and sold 25% off is not

60 - 25 = 35\%

profit. The 25% comes off 160, leaving 120, so the profit is 20%.

Successive discounts are less than their sum

25% and 4% make 28%, not 29%. The second discount is taken on the already reduced price.

Equal gain and loss is not break-even

Sold at the same price, a 20% gain and a 20% loss lose 4% overall. The loss-making item cost more, so its loss outweighs the gain.

Mixtures and Solutions in Percentages

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Strength of a solution when water is added or removed

New strength

\text{new } \% = \dfrac{\text{unchanged amount}}{\text{new total}} \times 100

How much of the pure component to add

Pure component to add

\dfrac{a + x}{T + x} = \dfrac{\text{target}}{100}

Changing a ratio by adding one component

Amount to add

\text{added} = \text{fixed} \times \dfrac{\text{new term of the other}}{\text{new term of the fixed}} - \text{other now}

Mixing two strengths, and alligation

Alligation

\dfrac{q_1}{q_2} = \dfrac{s_2 - m}{m - s_1}

Drawing off and replacing

Repeated replacement with water

\text{left} = V\left(1 - \dfrac{x}{V}\right)^n

Common traps

Keep the right thing fixed

Adding water fixes the salt, not the water. Working with the water's percentage gives the right answer only if you remember the water itself changed.

The total grows too

To lift 75 L from 12% to 34% alcohol, adding 22% of 75 L is not enough: what you add also enlarges the total. Fix the water and solve.

Find the parts before adding

In 72 L at

5 : 4

, one part is 8 L. Adding the change in ratio terms directly (2 parts) only works if you scale by that 8 L, which gives the 16 L.

The ratio is crossed

To get 32% from 20% and 50%, the 20% solution takes the larger share,

50 - 32 = 18

parts, because the target is nearer to it. Pairing each strength with its own distance reverses the answer.

The second draw takes less milk

After the first round the can is only 75% milk, so the second 10 L takes out 7.5 L of milk, not 10. That is why the factor is squared rather than doubled.

Percentages in Interest, Work and Speed

Learn this subtopic in the notes

Simple and compound interest

SI = \dfrac{PRT}{100}, \qquad A = P\left(1 + \dfrac{R}{100}\right)^T

Work done as a percentage of a job

Rates add

\dfrac{1}{T} = \dfrac{1}{a} + \dfrac{1}{b}

Speed changed by a percentage

Time after a change in speed

t_{\text{new}} = \dfrac{t_{\text{old}}}{1 + r/100}

Speed, distance and time: the basics used here

Speed, distance and time

d = s \times t, \qquad \text{average speed} = \dfrac{\text{total distance}}{\text{total time}}

Common traps

Amount is not interest

₹4,000 at 10% compound for 2 years amounts to ₹4,840, but the interest is ₹840. Read whether the question asks for the amount or the interest.

More efficient does not mean that much less time

40% more efficient takes

\tfrac{1}{1.4}

of the time, about 29% less, not 40% less. Rates and times are inverse.

Time does not fall by the same percentage

20% faster saves

16\tfrac 23\%

of the time, not 20%. The time is divided by 1.2, not multiplied by 0.8.

Average speed is not the average of speeds

30 km at 10 km/h and 30 km at 15 km/h average 12 km/h, not 12.5. More time is spent at the slower speed.

Percentage formulas

Percentage Basics: Of, What Percent, and the Whole

x% of a number, and what percent one number is of another

A percentage of a percentage

Finding the whole from a part

Two percentages of the same number

Fractions and their percentages

Common traps

Percentage Change, and Comparing Two Quantities

Percentage change measured on the old value

The value before a percentage change

More than and less than: the base changes

Two quantities measured against a third

When x% of one equals y% of another

Common traps

Successive Changes, Price and Consumption

The net effect of successive changes

Undoing a change: the percentage needed to get back

When a quantity is a product: area, volume, takings

Keeping expenditure fixed when the price changes

A cheaper price buys more for the same money

Common traps

Population Growth and Depreciation

Growth and depreciation year after year

Going back in time

A different rate each year

Finding the rate of growth

The make-up of a population

Common traps

Shares of a Whole: Budgets, Savings and Groups

Parts of a whole, and the rest

Income, spending and savings

A share of what is left

When income and spending both change

Combining groups with different percentages

Overlapping groups: failed in one, both or neither

Common traps

Marks, Pass Marks and Elections

Scores, questions and negative marking

The pass mark from one student's score

Two scores either side of the pass mark

Overall percentage across papers

Elections: shares, margins and invalid votes

Common traps

Profit, Loss and Discount with Percentages

Profit and loss on the cost price

Articles bought and sold in different lots

Marked price, discount and profit

Successive discounts and chains of profit

Overall profit when parts sell at different rates

Common traps

Mixtures and Solutions in Percentages

Strength of a solution when water is added or removed

How much of the pure component to add

Changing a ratio by adding one component

Mixing two strengths, and alligation

Drawing off and replacing

Common traps

Percentages in Interest, Work and Speed

Work done as a percentage of a job

Speed changed by a percentage

Speed, distance and time: the basics used here

Common traps

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