Profit, Loss & Discount

high importance~2 Q in Tier 124 formulas⚡ 14 shortcuts5 subtopics

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Same Selling Price: One Profit, One Loss

⏱ 4 min read🧩 5 question types🎯 13 practice Q

The idea in one minute

Two articles sell at the same price. One makes $x\%$ profit, the other $x\%$ loss. The result is always a net loss of $\dfrac{x^2}{100}$ per cent.

The loss article's cost price is the bigger one, so its loss outweighs the profit. For rupee answers, rebuild each cost price by dividing the common SP by its chip.

The setup

Two articles are sold for the same price. One carries a profit of $x\%$ , the other a loss of $x\%$ .

Most students guess "no profit, no loss". The answer is always a net loss.

Rule: Net loss $= \dfrac{x^2}{100}$ per cent of the total cost price.

Quick values worth memorising

$x = 10$ : loss $1\%$
$x = 20$ : loss $4\%$
$x = 25$ : loss $6.25\%$
$x = 16\dfrac{2}{3}\% = \dfrac{1}{6}$ : loss $\left(\dfrac{1}{6}\right)^2 = \dfrac{1}{36} = 2\dfrac{7}{9}\%$

Watch: "No profit, no loss" is the trap option. It is always wrong here.

Why the loss always wins

Take a common SP of Rs 100 and $x = 20$ .

Profit article: $CP = 100 \div \dfrac{6}{5} = 83\dfrac{1}{3}$ , gain $= 16\dfrac{2}{3}$ .
Loss article: $CP = 100 \div \dfrac{4}{5} = 125$ , loss $= 25$ .

The loss (Rs 25) beats the gain (Rs $16\dfrac{2}{3}$ ). The gap, Rs $8\dfrac{1}{3}$ , is exactly $4\%$ of the total CP Rs $208\dfrac{1}{3}$ . Dividing by a chip below 1 always moves further than dividing by one above 1.

Full reconstruction in rupees

When rupees are asked, rebuild both cost prices.

Two watches sell at Rs 960 each, at $20\%$ profit and $20\%$ loss:

$CP_1 = 960 \times \dfrac{5}{6} = 800$ , $CP_2 = 960 \times \dfrac{5}{4} = 1200$ .
Total CP $= 2000$ , total SP $= 1920$ .
Loss $= 80 = 4\%$ of 2000. The formula agrees.

Other quick cases: Rs 1,680 each gives CPs 1,400 and 2,100, loss Rs 140. Rs 1,980 each at $\pm 10\%$ gives CPs 1,800 and 2,200, loss Rs 40.

Tip: The loss per cent depends only on $x$ . Change the SP and the per cent does not move.

Unequal percentages break the shortcut

One article at $10\%$ profit, the other at $17.5\%$ loss, both at Rs 1,155.

$CP$ s: $1155 \times \dfrac{10}{11} = 1050$ and $1155 \times \dfrac{40}{33} = 1400$ . Total CP 2,450, total SP 2,310, loss Rs 140.

Careful: Use $\dfrac{x^2}{100}$ only when the two percents are equal in size.

Working backwards from the loss

Total loss Rs 50 with $\pm 20\%$ on the same SP:

\frac{5S}{6} + \frac{5S}{4} = \frac{25S}{12}, \quad \text{loss} = \frac{S}{12} = 50 \Rightarrow S = 600

Given only the loss per cent, find $x$ : $x^2 = 100 \times \text{loss\%}$ , so $x = 10\sqrt{\text{loss\%}}$ . A $4\%$ loss means $x = 20$ ; a $6.25\%$ loss means $x = 25$ .

Note: The loss item's CP is always the bigger one. If your rebuild says otherwise, you inverted a chip.

How the options are built

"No profit, no loss" - the instinct answer, always wrong.
$\dfrac{x^2}{200}\%$ - half the correct loss.
$\dfrac{2x^2}{100}\%$ - double the correct loss.
$\dfrac{x^2}{100}\%$ - the correct answer.

One rupee option usually applies the loss per cent to the SP instead of the total CP. Check the base before choosing.

Question types you will see

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

Type 1very common3 practice Q

Net result for equal gain and loss on one SP

How to spot it:

'Two articles sold at the same price, one at x% profit and the other at x% loss. Overall result?'

\text{net loss\%} = \frac{x^2}{100}

Method

Square x and divide by 100.
State it as a loss, never a gain.
Reject the 'no profit no loss' option.

Why it works:

The loss side has the bigger CP, so its per cent dominates.

Try this

Two articles are sold at the same price, one at 25% profit and the other at 25% loss. What is the net result?

Show solution

Loss $= \dfrac{25^2}{100} = \dfrac{625}{100}$ .
$= 6.25\%$ of the total CP.

Answer

6.25% loss

Type 2very common4 practice Q

Loss in rupees by rebuilding both CPs

How to spot it:

The common SP is given in rupees; the total loss or total CP is asked.

\text{loss} = \left(\frac{S}{1 + x/100} + \frac{S}{1 - x/100}\right) - 2S

Method

Divide the SP by the profit chip for one CP.
Divide by the loss chip for the other.
Add the CPs, subtract twice the SP.
Check: loss over total CP equals x squared over 100.

Why it works:

Rupee answers need the actual cost prices, which the SP recovers.

Try this

Two mobiles are sold at Rs 960 each, one at 20% profit and the other at 20% loss. Find the total loss.

Show solution

CPs: $960 \times \dfrac{5}{6} = 800$ and $960 \times \dfrac{5}{4} = 1200$ .
Total CP $= 2000$ , total SP $= 1920$ .
Loss $= 80 = 4\%$ of 2000.

Answer

Rs 80 (4% loss)

Type 3common2 practice Q

Same SP with unequal percents

How to spot it:

One at a% profit, the other at b% loss with a not equal to b, same SP.

\text{loss} = \left(\frac{S}{1 + a/100} + \frac{S}{1 - b/100}\right) - 2S

Method

Build each CP from the SP with its own chip.
Add the CPs and compare with twice the SP.
Divide by the total CP for the per cent.

Why it works:

Unequal percents break the x-squared shortcut, so rebuild.

Try this

Two tables are sold at Rs 1,155 each, one at a 10% profit and the other at a 17.5% loss. Find the net result.

Show solution

CPs: $1155 \times \dfrac{10}{11} = 1050$ and $1155 \times \dfrac{40}{33} = 1400$ .
Total CP $= 2450$ , total SP $= 2310$ .
Loss $= 140$ .

Answer

Rs 140 loss

Type 4occasional2 practice Q

Find the common SP from the loss

How to spot it:

The total loss in rupees is given; the common selling price is asked.

\text{loss} = \frac{x^2 S}{10000 - x^2}

Method

Write both CPs in terms of S.
Total CP minus 2S is the loss.
Set it equal to the rupee loss and solve for S.

Why it works:

The rupee loss is a fixed fraction of the total CP, so it fixes S.

Try this

Two articles are sold at the same price, one at 20% profit and the other at 20% loss. The total loss is Rs 50. Find the selling price of each article.

Show solution

CP total $= \dfrac{5S}{6} + \dfrac{5S}{4} = \dfrac{25S}{12}$ .
Loss $= \dfrac{S}{12} = 50$ .
$S = 600$ .

Answer

Rs 600

Type 5occasional

Find x from the loss per cent

How to spot it:

The overall loss per cent is given and the equal profit and loss per cent x is asked.

x = 10\sqrt{\text{loss\%}}

Method

Write x squared over 100 equal to the loss per cent.
Take the square root.
Multiply by 10.

Why it works:

The loss per cent is x squared over 100, so the square root reverses it.

Try this

Two articles sold at the same price, one at x% profit and the other at x% loss, give an overall loss of 4%. Find x.

Show solution

$\dfrac{x^2}{100} = 4$ .
$x^2 = 400$ .
$x = 20$ .

Answer

20%

Formula sheet

Net loss for equal plus-minus x%

\text{net loss\%} = \frac{x^2}{100}

Per cent of total CP.

Reconstructed costs

CP_1 = \frac{S}{1 + x/100}, \quad CP_2 = \frac{S}{1 - x/100}

S = common selling price.

Loss in rupees

\text{loss} = CP_1 + CP_2 - 2S

x from the loss per cent

x = 10\sqrt{\text{loss\%}}

Shortcuts that save time

⚡ The x squared over 100 reflex

Same SP, opposite equal percents: write the loss per cent straight away.

Example

Two mobiles are sold for Rs 1,200 each, one at 25% profit and the other at 25% loss. Find the overall loss per cent.

Show solution

Loss $= \dfrac{25 \times 25}{100}$ .
$= 6.25\%$ . (CPs 960 and 1,600; loss Rs 160 of 2,560.)

Answer

6.25% loss

⚡ Rebuild both cost prices

Divide the common SP by each chip, add the CPs, compare with twice the SP.

Example

Two articles are sold for Rs 1,680 each, one at 20% profit and the other at 20% loss. Find the net loss in rupees.

Show solution

CPs: $1680 \times \dfrac{5}{6} = 1400$ and $1680 \times \dfrac{5}{4} = 2100$ .
Total CP 3,500 vs total SP 3,360.
Loss $= 140$ , which is $4\%$ of 3,500.

Answer

Rs 140 (4%)

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Answering 'no profit no loss' because $+x$ and $-x$ cancel.

The percents act on different CPs; the result is always a loss of $\dfrac{x^2}{100}\%$ .

Mistake 02

Using $\dfrac{x^2}{100}$ when the percents are unequal.

Rebuild both CPs from the SP and add.

Mistake 03

Reporting the loss per cent as the rupee loss.

Rupee loss $=$ total CP $-$ total SP; the per cent is on total CP.

Mistake 04

Dividing both SPs by the same chip.

Profit item by $1 + \dfrac{x}{100}$ ; loss item by $1 - \dfrac{x}{100}$ .

Mistake 05

Applying the loss per cent to the total SP.

Loss per cent is measured on the total cost price.

Quick revision

Read this the night before the exam.

Equal $\pm x\%$ on the same SP: loss $\dfrac{x^2}{100}\%$ .
10 $\to$ 1, 20 $\to$ 4, 25 $\to$ 6.25 per cent loss.
CPs: divide the common SP by $1 \pm \dfrac{x}{100}$ .
Loss item's CP is always the bigger one.
Unequal percents: rebuild both CPs, no shortcut.
From loss per cent back to x: $x = 10\sqrt{\text{loss\%}}$ .

Practice: 13 questions

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

Topic test · 13 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.

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