Profit, Loss & Discount

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Profit, Loss and CP–SP Basics

⏱ 5 min read🧩 6 question types🎯 16 practice Q

The idea in one minute

Cost price (CP) is what the seller paid; selling price (SP) is what the buyer pays. Profit means SP is above CP; loss means SP is below CP.

Profit and loss per cent are always measured on CP. Turn each per cent into a multiplier: a profit of $20\%$ means $SP = CP \times \dfrac{6}{5}$ .

The words you need

Cost price (CP): what the seller paid. Add transport or repair costs (overheads) to it.
Selling price (SP): what the buyer pays the seller.
Profit = SP - CP when SP is bigger. Loss = CP - SP when SP is smaller.

A fan bought for Rs 1,250 and sold for Rs 1,400 gives a profit of Rs 150.

Rule: Profit and loss per cent are always calculated on CP, unless the question clearly says otherwise.

Profit and loss per cent

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

For the fan: $\dfrac{150}{1250} \times 100 = 12\%$ . Dividing by SP instead gives $10\dfrac{5}{7}\%$ , the classic wrong option.

Overheads join the CP first: bought at 2,400 plus 150 transport makes CP 2,550.

Multipliers, called chips

Profit of $x\%$ means $SP = CP \times \dfrac{100+x}{100}$ . Loss of $x\%$ means $SP = CP \times \dfrac{100-x}{100}$ .

Change	Chip	Change	Chip
$20\%$ profit	$6/5$	$10\%$ loss	$9/10$
$25\%$ profit	$5/4$	$12.5\%$ loss	$7/8$
$16\dfrac{2}{3}\%$ profit	$7/6$	$5\%$ loss	$19/20$

Going back from SP to CP, divide by the chip. A table sold for 1,140 at a $5\%$ loss: $CP = 1140 \times \dfrac{20}{19} = 1200$ .

Tip: Write every per cent as a chip fraction before touching the rupees.

Two selling prices, one cost price

"Sold at $8\%$ loss; had it been sold for Rs 336 more, the gain would be $6\%$ ."

The two SPs are $92\%$ and $106\%$ of the same CP.
Their gap is $8 + 6 = 14\%$ of CP, and it equals Rs 336.
$14\%$ of CP $= 336$ , so CP $= 2400$ .

Example: Selling at Rs 720 gives as much profit as the loss on selling at Rs 560. CP sits exactly midway: $\dfrac{720+560}{2} = 640$ .

Watch: Loss and gain percents add when one SP is below CP and the other above it; two gains subtract.

Counting articles instead of rupees

"The CP of 15 articles equals the SP of 12." The seller receives money for 15 by giving goods that cost him 12. Profit $= \dfrac{15-12}{12} \times 100 = 25\%$ .

"Buys 4 for Rs 15 and sells 5 for Rs 24": take 20 articles, the common count. CP $= 75$ , SP $= 96$ , profit $= 28\%$ .

Watch: Divide by the articles he gives away, not the number named first.

Changed cost and changed price together

"A man sells at $10\%$ profit. Had he bought at $20\%$ less and sold for Rs 60 less, he would gain $25\%$ . Find the CP."

Let CP $= C$ . First story: $SP = 1.1C$ .
Second story: $CP = 0.8C$ and $SP = 0.8C \times 1.25 = C$ .
So $1.1C - 60 = C$ , giving $0.1C = 60$ and $C = 600$ .

Careful: Always rebuild the SP from your CP as a final check. Exam numbers come out whole; if yours do not, re-read the question.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Profit or loss per cent from CP and SP

How to spot it:

CP and SP are given, sometimes with a transport or repair cost, and the per cent is asked.

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

Method

Add any overheads to the CP.
Find profit or loss = SP - CP.
Divide by CP, never by SP.
Multiply by 100.

Why it works:

Profit is measured against the money the seller actually invested.

Try this

A fan bought for Rs 1,250 is sold for Rs 1,400. Find the profit per cent.

Show solution

Profit $= 1400 - 1250 = 150$ .
$\dfrac{150}{1250} \times 100 = 12$ .

Answer

12%

Type 2very common2 practice Q

Find CP or SP from a profit or loss per cent

How to spot it:

One price and the per cent are given; the other price is asked.

SP = CP \times \frac{100 \pm x}{100}, \quad CP = SP \times \frac{100}{100 \pm x}

Method

Write the chip for the per cent.
CP to SP: multiply. SP to CP: divide.
Reduce the chip to a small fraction for speed.

Why it works:

SP is CP scaled by the chip, so reversing the journey divides.

Try this

A table sold for Rs 1,140 gives a loss of 5%. Find its cost price.

Show solution

Loss $5\%$ : chip $= \dfrac{19}{20}$ .
$CP = 1140 \times \dfrac{20}{19} = 1200$ .

Answer

Rs 1,200

Type 3very common2 practice Q

Two selling prices around one CP

How to spot it:

'Sold at a% loss; had it been sold for Rs k more, the gain would be b%.'

CP = \frac{k \times 100}{a + b}

Method

Write both SPs as percents of the CP.
Add the percents (loss and gain on opposite sides).
That per cent of CP equals the rupee gap k.
Divide to find CP.

Why it works:

Both SPs are percents of the same CP, so only the per cent gap moves the SP.

Try this

An article sold at 8% loss would have given a 6% gain if sold for Rs 336 more. Find the cost price.

Show solution

Percents: $92\%$ and $106\%$ ; gap $= 14\%$ of CP.
$14\%$ of CP $= 336$ .
$CP = 336 \times \dfrac{100}{14} = 2400$ .

Answer

Rs 2,400

Type 4common3 practice Q

Article-count questions

How to spot it:

'CP of 15 articles equals SP of 12', or rates like 'buys 4 for Rs 15 and sells 5 for Rs 24'.

\text{Profit\%} = \frac{\text{articles bought} - \text{articles sold}}{\text{articles sold}} \times 100

Method

For 'CP of a = SP of b', take the price of one article as 1.
Profit per cent = (a - b) over b, times 100.
For rates like 4-for-15, take the LCM of the counts so both sides cover the same number of articles.

Why it works:

Fixing a common count turns the story into an ordinary CP-SP comparison.

Try this

The cost price of 15 articles is equal to the selling price of 12 articles. Find the profit per cent.

Show solution

He receives 15 article-prices for goods costing 12.
$\dfrac{15 - 12}{12} \times 100 = 25$ .

Answer

25%

Type 5common2 practice Q

New selling price for a different gain

How to spot it:

'Sold for Rs 1,530 at 10% loss; at what price should it be sold to gain 10%?'

SP_2 = SP_1 \times \frac{100 + b}{100 - a}

Method

Go back to the CP by dividing by the first chip.
Multiply by the new chip for the wanted gain.
For a changed CP as well, write both stories with CP as the unknown and equate.

Why it works:

The CP is the anchor that links every selling story.

Try this

An article sold for Rs 1,530 gives a loss of 10%. At what price should it be sold to gain 10%?

Show solution

$CP = 1530 \times \dfrac{10}{9} = 1700$ .
$SP = 1700 \times \dfrac{11}{10} = 1870$ .

Answer

Rs 1,870

Type 6common

Equal profit and loss at two prices

How to spot it:

'Selling at Rs 720 gives as much profit as the loss on selling at Rs 560.' The CP is asked.

CP = \frac{S_1 + S_2}{2}

Method

Note the two selling prices, one above and one below the CP.
Equal profit and loss puts the CP exactly midway.
Average the two prices.

Why it works:

The two prices sit at equal distances from the CP, so the CP is their average.

Try this

Selling an article at Rs 720 gives as much profit as the loss on selling it at Rs 560. Find the cost price.

Show solution

CP is midway between 560 and 720.
$CP = \dfrac{560 + 720}{2} = 640$ .
Check: $720 - 640 = 80 = 640 - 560$ .

Answer

Rs 640

Formula sheet

Profit / loss per cent

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

Always on CP.

Selling price

SP = CP \times \frac{100 \pm x}{100}

Plus for profit, minus for loss.

Cost price from SP

CP = SP \times \frac{100}{100 \pm x}

Divide by the chip.

Equal profit and loss at two prices

CP = \frac{S_1 + S_2}{2}

When profit at S1 equals loss at S2.

Shortcuts that save time

⚡ Fraction swap for the per cent

Turn the profit into a fraction of CP and the per cent falls out. $170/850$ is clearly $1/5$ .

Example

A trader sells a cycle for Rs 1,020 that cost him Rs 850. Find the profit per cent.

Show solution

Profit $= 1020 - 850 = 170$ .
$\dfrac{170}{850} = \dfrac{1}{5} = 20\%$ .

Answer

20%

⚡ Divide by the chip to get CP

A loss of $4\%$ means the chip $96/100$ . Divide the SP by the chip, nothing else.

Example

A machine is sold for Rs 624 at a loss of 4%. Find its cost price.

Show solution

Loss $4\%$ : chip $= \dfrac{96}{100} = \dfrac{24}{25}$ .
$CP = 624 \times \dfrac{25}{24} = 650$ .

Answer

Rs 650

⚡ Same per cent scales linearly

When the profit per cent is fixed, CP and SP scale together. Article counts often hide a clean ratio.

Example

By selling 15 pens a shopkeeper recovers the cost of 12 pens. Find his gain per cent.

Show solution

He gets money for 15 pens by giving 12.
Gain $= \dfrac{3}{12} = 25\%$ .

Answer

25%

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Working out the profit per cent on SP.

Profit and loss per cent are always on CP.

Mistake 02

Recovering CP as SP minus $x\%$ of SP.

Undo the chip: $CP = SP \times \dfrac{100}{100 \pm x}$ .

Mistake 03

Forgetting transport or repair in the CP.

Add all overheads to the buying price first.

Mistake 04

In article-count sums, dividing by the wrong count.

Divide the extra articles by the articles given away.

Mistake 05

Reading 'gain of $25\%$ ' and 'marked $25\%$ up' as the same thing.

Gain compares SP with CP; markup compares MP with CP. Different quantities.

Quick revision

Read this the night before the exam.

Profit $=$ SP $-$ CP; loss $=$ CP $-$ SP.
Profit and loss per cent are always on CP (plus overheads).
SP $=$ CP $\times$ chip; CP $=$ SP $\div$ chip.
Two SPs: rupee gap $=$ per cent gap of CP.
Equal profit and loss at two prices: CP is the average.
CP of $a$ articles $=$ SP of $b$ : profit $= \dfrac{a-b}{b} \times 100$ .

Practice: 16 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

Suggested time 5 min · wrong answers go to your mistake notebook automatically.

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