Percentage, Profit and Loss — SSC CGL Formulas and Traps

Percentage Basics

Percentage of a quantity and finding the whole

Part and whole

x\% \text{ of } N = \frac{x}{100} \times N \qquad \text{whole} = \frac{\text{part} \times 100}{x}

Equal percentages of two numbers give their ratio

Equal percentages and differences

a\% \text{ of } A = b\% \text{ of } B \implies A : B = b : a \qquad x\% \text{ of } N - y\% \text{ of } N = (x - y)\% \text{ of } N

More than, less than and the changing base

Turning a comparison around

\text{less by } \frac{r}{100 + r} \times 100\% \qquad \text{more by } \frac{r}{100 - r} \times 100\%

$r$ the percentage in the original comparison

Common percentages as fractions

Fraction	Percentage
$\frac{1}{2}$	$50\%$
$\frac{1}{3}$	$33\frac{1}{3}\%$
$\frac{2}{3}$	$66\frac{2}{3}\%$
$\frac{1}{4}$	$25\%$
$\frac{1}{5}$	$20\%$
$\frac{1}{6}$	$16\frac{2}{3}\%$
$\frac{1}{7}$	$14\frac{2}{7}\%$
$\frac{1}{8}$	$12\frac{1}{2}\%$
$\frac{3}{8}$	$37\frac{1}{2}\%$
$\frac{5}{8}$	$62\frac{1}{2}\%$
$\frac{1}{9}$	$11\frac{1}{9}\%$
$\frac{1}{11}$	$9\frac{1}{11}\%$
$\frac{1}{12}$	$8\frac{1}{3}\%$ Easy to confuse with one eighth, which is 12½%.
$\frac{1}{16}$	$6\frac{1}{4}\%$
$\frac{1}{20}$	$5\%$
$\frac{1}{25}$	$4\%$

To use a row, divide by the bottom of the fraction and multiply by the top.

Common traps

Answering the part that was given

When a question asks for the part left over (empty, saved, spent on other things), take the given shares away from 100% first. A 5 L can holding 4 L is 80% full, so the empty share asked for is 20%.

Mixing units

A percentage compares like with like. 0.002 kilolitres is 2,000 millilitres, so change every amount to one unit before dividing.

Rounding a recurring percentage too early

33\%

of 900 is 297, but

33\frac{1}{3}\%

of 900 is exactly 300. Use the fraction

\frac{1}{3}

, not 33.

Writing the ratio the wrong way round

When 40% of A equals 24% of B, A is the smaller number, so A : B is 24 : 40, which is 3 : 5. Writing 40 : 24 gives the reverse ratio, and SSC prints it as an option.

Using the same percentage both ways

If A is 20% more than B, B is not 20% less than A. With B = 100 and A = 120, B is 20 less on a base of 120, which is

16\frac{2}{3}\%

.

Comparing against the wrong number

In 'by what per cent is the wrong result more than the correct one', the base is the correct result. A wrong 150 against a correct 120 is 25% more, not 20%.

Successive Percentage Change

Learn this subtopic in the notes

Successive changes as a product of factors

Net effect of two changes

\text{net}\% = a + b + \frac{ab}{100} \qquad \text{equal rise and fall: } -\frac{r^2}{100}\%

$a, b$ the two changes in per cent, a fall written as negative
$r$ the size of an equal rise and fall

Spending a share of what is left

Amount left after spending out of the remainder

\text{left} = S\left(1 - \frac{x}{100}\right)\left(1 - \frac{y}{100}\right)\left(1 - \frac{z}{100}\right)

Working back to the original value

Back to the start, and growth over years

\text{original} = \frac{\text{final}}{\left(1 + \frac{a}{100}\right)\left(1 + \frac{b}{100}\right)} \qquad P_n = P_0\left(1 + \frac{r}{100}\right)^n

Price and consumption when spending is fixed

Cut in consumption to keep spending fixed

\text{cut} = \frac{r}{100 + r} \times 100\% \qquad \text{spending} = \text{price} \times \text{quantity}

Common traps

Adding the percentages

A rise of 30% followed by a fall of 30% is not 'no change'.

1.3 \times 0.7 = 0.91

, a 9% fall.

Dropping the sign in the product term

For a rise of 25% and a fall of 12%, b is

-12

, so the product term is

\frac{25 \times (-12)}{100} = -3

. The net change is

25 - 12 - 3 = 10\%

, not 16%.

Adding shares of different amounts

20% of the income, then 50% of the rest, then 40% of the rest leaves

0.8 \times 0.5 \times 0.6 = 0.24

of the income. Adding 20, 50 and 40 gives 110%, which means nothing.

Undoing a change with the same percentage

A value that rose by 30% does not return by falling 30%. From 130, it must fall by 30 on a base of 130, which is about 23.08%.

Cutting consumption by the price rise

A price rise of 25% does not need a 25% cut. A 20% cut is enough, because

1.25 \times 0.8 = 1

.

Mixing up the new price and the old price

When a price cut lets you buy extra kilograms, the money saved buys them at the new, lower price. Divide that by the price factor to get the old price.

Percentage Word Problems

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Elections: voters, votes cast, valid votes and the margin

Margin and valid votes

\text{margin} = (2w - 100)\% \text{ of valid votes} \qquad \text{valid} = N \times \frac{p}{100} \times \frac{q}{100}

$w$ winner's share of the valid votes, in per cent
$N$ voters on the list
$p$ percentage of voters who voted
$q$ percentage of votes cast that were valid

Pass marks and marks scored

Pass mark

\frac{p}{100} \times M = \text{score} + \text{shortfall}

Income, expenditure and savings

Savings and their change

\text{income} = \text{expenditure} + \text{savings} \qquad s = \frac{100\,i - E\,e}{S}

$i, e, s$ percentage changes in income, expenditure and savings
$E, S$ expenditure and savings when income is 100

Combined percentage of several groups

Weighted percentage and alligation

\text{overall} = \frac{A\,a + B\,b}{A + B} \qquad A : B = (b - \text{overall}) : (\text{overall} - a)

Common traps

Taking the winner's share as the margin

If the winner gets 56% of the valid votes, the margin is

56 - 44 = 12\%

of them, not 56%.

Taking invalid votes out of the whole list

'10% of the votes cast were invalid' is 10% of the people who voted, not of everyone on the list. Apply the turnout factor first, then the invalid share.

Subtracting the shortfall

'Failed by 30 marks' means the pass mark is 30 above the score. Add the shortfall to the score; do not take it away.

Averaging percentages from papers of different sizes

80% on a 50-mark paper and 60% on a 150-mark paper is 40 + 90 = 130 out of 200, which is 65%, not the 70% a simple average gives.

Averaging the two changes

Income up 20% and spending up 12% does not make savings rise by anything in between. Savings are what is left, so work out old and new savings and compare those.

Taking a plain average of the rates

30% of 40 boys and 50% of 60 girls is 42% of the class, not 40%. Weight each percentage by the size of its group.

Profit and Loss

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Cost price, selling price and profit percentage

Profit percentage and the two prices

\text{profit}\% = \frac{SP - CP}{CP} \times 100 \qquad SP = CP\left(1 + \frac{p}{100}\right) \qquad CP = \frac{SP}{1 + \frac{p}{100}}

Two selling prices of the same article

Swing between two selling prices

SP_2 - SP_1 = \frac{p_2 - p_1}{100} \times CP \qquad CP = \frac{SP_1 + SP_2}{2} \text{ (equal profit and loss)}

An article sold through several hands

Price after two sales

\text{final} = C\left(1 + \frac{p_1}{100}\right)\left(1 + \frac{p_2}{100}\right)

Overall profit on several items or lots

Overall profit

\text{overall}\% = \frac{\sum SP - \sum CP}{\sum CP} \times 100

Common traps

Taking the profit on the selling price

Profit percentage uses the cost price as its base. A profit of one-sixth of the selling price is one-fifth of the cost, so it is 20%, not

16\frac{2}{3}\%

.

Leaving out the overheads

Transport or repair money is part of what the trader paid. Add it to the cost price first, or the profit comes out too large.

Subtracting a loss from a profit

From a 16% loss to a 20% gain the price moves by 36% of the cost, not 4%. A loss sits on the other side of zero.

Adding the dealers' profits

Profits of 20% and then 40% make a factor of

1.2 \times 1.4 = 1.68

, a total of 68%, not 60%.

Averaging percentages on different costs

Two articles sold at the same price, one at a 20% gain and one at a 20% loss, always give a 4% loss overall, not zero, because the loss-making one cost more.

Leaving the spoilt stock out of the cost

Broken eggs or withered flowers were paid for. Their cost stays in the total cost; only the sales shrink.

Marked Price and Discount

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Discount on the marked price

Selling price after a discount

SP = MP\left(1 - \frac{d}{100}\right) \qquad d\% = \frac{MP - SP}{MP} \times 100

Profit after a mark-up and a discount

Profit after mark-up and discount

\text{profit}\% = m - d - \frac{m\,d}{100}

$m$ mark-up above cost, in per cent
$d$ discount on the marked price, in per cent

Finding the cost or marked price from the profit

The two routes to the selling price

CP\left(1 + \frac{p}{100}\right) = MP\left(1 - \frac{d}{100}\right) \qquad \frac{CP}{MP} = \frac{100 - d}{100 + p}

Several stages and fixed rupee amounts

Price after several stages

\text{final} = C\left(1 + \frac{m_1}{100}\right)\left(1 - \frac{d_1}{100}\right)\left(1 + \frac{m_2}{100}\right)\left(1 - \frac{d_2}{100}\right)

Common traps

Adding the discount back on the selling price

After a 20% discount the selling price is 0.8 of the marked price, so the marked price is SP divided by 0.8. Multiplying the SP by 1.2 gives too small an answer.

Treating a rupee discount like a percentage

₹500 off and then 7% off a ₹10,000 bill leaves

9500 \times 0.93 = 8835

; 7% off first and then ₹500 off leaves 8,800. Apply them in the order the question gives.

Subtracting the discount from the mark-up

A 40% mark-up with a 15% discount is a profit of

40 - 15 - 6 = 19\%

, not 25%. The discount is taken on the larger marked price.

Marking up by profit plus discount

To gain 20% after a 25% discount, the marked price must be

\frac{120}{75} = 1.6

times the cost, that is 60% above cost, not 45%.

Taking the discount as a share of the cost

A discount percentage is of the marked price. With CP 100 and MP 140, a discount of 28 rupees is 20%, not 28%.

Putting a rupee discount into the product

A cash discount of ₹150 is not a percentage, so it cannot be multiplied in. Finish the percentage steps, then subtract the rupees.

Successive Discounts and Offers

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Single discount equal to successive discounts

Equal single discount

D = a + b - \frac{ab}{100}

$a, b$ the two successive discounts, in per cent

Finding a missing discount or the bill

Bill from the gap between two schemes

\text{bill} = \frac{\text{rupee gap}}{(D - D_{eq}) / 100}

Comparing schemes and free-item offers

Discount from free items

\text{buy } x \text{ get } y \text{ free: discount} = \frac{y}{x + y} \times 100\%

Common traps

Adding successive discounts

Discounts of 30% and 20% one after the other are a single discount of 44%, not 50%.

Giving the share paid as the discount

If the buyer pays 0.528 of the marked price, the single discount is 47.2%, not 52.8%. Subtract the kept share from 1.

Taking the second discount on the marked price

The second discount is a share of the price after the first one. ₹1,200 down to ₹900 with a first 10% leaves a second discount of 180 out of 1,080, which is

16\frac{2}{3}\%

, not 180 out of 1,200.

Dividing the free items by the items paid for

'Buy 5, get 2 free' is a discount of

\frac{2}{7}

, about 28.6%, because the buyer takes away 7 items. Dividing 2 by 5 gives 40%, which is wrong.

Answering the opposite of the question

A question that asks under which scheme the selling price is highest wants the scheme with the smallest discount, not the largest.

Faulty Weights and Article Counts

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Faulty weight at the cost price

Gain from a short weight

\text{gain}\% = \frac{\text{true} - \text{false}}{\text{false}} \times 100

Faulty weight with a profit or loss on price

Real profit with a short weight

1 + \frac{\text{gain}}{100} = \left(1 + \frac{p}{100}\right) \times \frac{\text{true}}{\text{false}}

Cost price of some articles equal to the selling price of others

Profit when two counts are equal in value

\text{profit}\% = \frac{x - y}{y} \times 100 \text{ when CP of } x = \text{SP of } y

Common traps

Dividing by the true weight

50 g short on every kilogram is a gain of

\frac{50}{950} \times 100 \approx 5.26\%

, not 5%. The base is the 950 g handed over.

Adding the two gains

A 10% mark-up with a weight 10% short is not a 20% gain. The factors multiply:

1.1 \times \frac{10}{9} = \frac{11}{9}

, a gain of about 22.2%.

Reading 'weight 20% less' as a 20% gain

If the weight is 20% less than true, the seller gives 0.8 of what he charges for, so the weight factor is

\frac{1}{0.8} = 1.25

, a 25% gain from the weight alone.

Dividing by the larger count

When the cost price of 28 equals the selling price of 24, the gain is 4 on the 24 sold, so

\frac{4}{24} = 16\frac{2}{3}\%

, not

\frac{4}{28}

.

Mixing up 'gains the SP of n' and 'gains the CP of n'

Gaining the cost price of 3 articles on 12 sold is a gain of

\frac{3}{12} = 25\%

. Gaining the selling price of 3 on 12 sold means SP of 9 equals CP of 12, a gain of

33\frac{1}{3}\%

.

Percentage, Profit and Loss formulas

Percentage Basics

Percentage of a quantity and finding the whole

Equal percentages of two numbers give their ratio

More than, less than and the changing base

Common percentages as fractions

Common traps

Successive Percentage Change

Successive changes as a product of factors

Spending a share of what is left

Working back to the original value

Price and consumption when spending is fixed

Common traps

Percentage Word Problems

Elections: voters, votes cast, valid votes and the margin

Pass marks and marks scored

Income, expenditure and savings

Combined percentage of several groups

Common traps

Profit and Loss

Cost price, selling price and profit percentage

Two selling prices of the same article

An article sold through several hands

Overall profit on several items or lots

Common traps

Marked Price and Discount

Discount on the marked price

Profit after a mark-up and a discount

Finding the cost or marked price from the profit

Several stages and fixed rupee amounts

Common traps

Successive Discounts and Offers

Single discount equal to successive discounts

Finding a missing discount or the bill

Comparing schemes and free-item offers

Common traps

Faulty Weights and Article Counts

Faulty weight at the cost price

Faulty weight with a profit or loss on price

Cost price of some articles equal to the selling price of others

Common traps

More SSC CGL Quantitative Aptitude formula sheets