AFCAT Numerical Ability · Profit and Loss

Cost Price, Selling Price and Profit Per Cent

Profit or loss is the gap between the selling price and the cost price, and its percentage is taken on the cost price unless the question says otherwise.

All Profit and Loss notes

Why this matters

Every later page comes down to two moves made here: a percentage taken on the cost price, and a multiplier such as 1.2 or 0.85. No row on this page is HARD; the marks go on dividing by the selling price instead of the cost, or on adding a percentage where the question needs a division.

Concept 1 of 3: Profit, loss and their percentages

The cost price (CP) is what the seller paid and the selling price (SP) is what he got. If SP is more, the difference is a profit; if it is less, a loss. A percentage needs a base, and in profit and loss the base is the cost price.

Definition

Profit $= SP - CP$ ; loss $= CP - SP$ .
Profit % $= \dfrac{\text{profit}}{CP} \times 100$ ; loss % $= \dfrac{\text{loss}}{CP} \times 100$ .
A markup of $x\%$ means a price $x\%$ above the cost price, so a markup percentage is also taken on the cost.
Per-unit and whole-lot figures give the same percentage: bought at ₹24 a kg and sold at ₹30 a kg is a profit of 6 on 24, that is $25\%$ , however many kg are sold.
Given the SP and the profit in rupees, subtract first: the cost is SP minus profit.

Profit per cent

\text{Profit}\% = \dfrac{SP - CP}{CP} \times 100

Worked example

A shopkeeper buys a lamp for ₹640 and sells it for ₹800. Find his profit and his profit per cent.

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The same idea in a practice question:

Practice questionEasy

Example 1 · Profit and Loss · Profit and Loss Percentage

Dhanush buys a watch for ₹2000 and sells it for ₹2200. Find his profit/loss per cent.

Divide by the cost price

Bought for ₹400 and sold for ₹500: the profit of 100 is

25\%

of the cost. Dividing by the selling price gives

20\%

, which is the wrong base.

Find the cost before the percentage

When a question gives the selling price and the gain in rupees, the cost is SP minus gain. Taking the gain as a share of the selling price is the usual slip.

Drill 8 more on profit, loss and their percentages

Concept 2 of 3: Selling price and cost price as multiples

30\%

profit makes the selling price 1.3 times the cost; a

15\%

loss makes it 0.85 times the cost. Going from cost to selling price multiplies, and going back divides by the same number.

Definition

Profit of $x\%$ : $SP = CP\left(1 + \tfrac{x}{100}\right)$ . Loss of $x\%$ : $SP = CP\left(1 - \tfrac{x}{100}\right)$ .
Given the SP, divide: $CP = SP \div \left(1 \pm \tfrac{x}{100}\right)$ .
A price change works the same way: after a $25\%$ fall a price is 0.75 of the old one, so the old price is the new one divided by 0.75.
Common multipliers: a $10\%$ gain is 1.1, a $12.5\%$ gain is $1.125 = \tfrac 98$ , a $25\%$ loss is $0.75 = \tfrac 34$ .

Selling price from cost price

SP = CP\left(1 \pm \dfrac{x}{100}\right)

Worked example

A watch is sold for ₹1,955 at a loss of

15\%

. Find its cost price.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Profit and Loss · Profit and Loss Percentage

An article is sold for ₹480 at a loss of 20%. What was the cost price?

The percentage is of the cost, not of the selling price

Sold for ₹1,320 at a

10\%

profit: the cost is

1320 \div 1.1 = 1200

, not

1320 - 132 = 1188

Undo a fall by dividing

After a

25\%

fall to ₹36, the old price is

36 \div 0.75 = 48

. Adding

25\%

to 36 gives 45, which is too little.

Drill 7 more on selling price and cost price as multiples

Concept 3 of 3: Profit from a ratio of cost price to selling price

A ratio

CP : SP = a : b

lets you take the cost as

a

and the selling price as

b

. The profit is

b - a

parts out of

a

Definition

$CP : SP = a : b$ gives profit % $= \dfrac{b - a}{a} \times 100$ ; it is a loss when $b < a$ .
'SP is $k$ times CP' is the ratio $1 : k$ . An SP double the CP is a $100\%$ profit.
'CP is $p\%$ of SP': take SP as 100 and CP as $p$ .
A loss stated as a share of the selling price must be converted. A loss of $0.1\,SP$ makes $CP = 1.1\,SP$ , so the loss is $\dfrac{0.1}{1.1} = 9\tfrac{1}{11}\%$ of the cost.

Profit from a ratio

CP : SP = a : b \;\Rightarrow\; \text{Profit}\% = \dfrac{b - a}{a} \times 100

Worked example

The cost price and the selling price of an article are in the ratio

8 : 11

. Find the profit per cent.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Profit and Loss · Profit and Loss Percentage

The ratio of cost price to selling price is 4:5. What is the profit percentage?

The base is the cost

CP : SP = 3 : 2

is a loss of 1 on 3, that is

33\tfrac 13\%

. Dividing by the 2 gives

50\%

A loss on the selling price is a smaller loss on the cost

A loss of

10\%

of the selling price is

9\tfrac{1}{11}\%

of the cost. The option that repeats the

10\%

answers a different question.

Drill 7 more on profit from a ratio of cost price to selling price

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 (3)

Profit, loss and their percentages
Profit per cent
$\text{Profit}\% = \dfrac{SP - CP}{CP} \times 100$
Selling price and cost price as multiples
Selling price from cost price
$SP = CP\left(1 \pm \dfrac{x}{100}\right)$
Profit from a ratio of cost price to selling price
Profit from a ratio
$CP : SP = a : b \;\Rightarrow\; \text{Profit}\% = \dfrac{b - a}{a} \times 100$

Watch out for (6)

Divide by the cost price
→ Profit, loss and their percentages
Find the cost before the percentage
→ Profit, loss and their percentages
The percentage is of the cost, not of the selling price
→ Selling price and cost price as multiples
Undo a fall by dividing
→ Selling price and cost price as multiples
The base is the cost
→ Profit from a ratio of cost price to selling price
A loss on the selling price is a smaller loss on the cost
→ Profit from a ratio of cost price to selling price

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