AFCAT Numerical Ability · Profit and Loss

Finding the Cost Price, Then Repricing

Turn the given sale into the cost price first, then apply the new percentage; when two sales differ by a known sum, that sum is the gap between their percentages of the cost.

All Profit and Loss notes

Why this matters

A large page, and three AFCAT past-year questions sit on it, from 2016 and 2018. Nearly every row is two steps through the cost price, and the marks go on jumping from one selling price straight to the other.

Concept 1 of 3: Through the cost price to a new selling price

A question that gives one sale and asks about another never jumps across directly. The first sale fixes the cost price, and the second percentage is taken on that cost.

Definition

Step 1: $CP = SP_1 \div \left(1 \pm \tfrac{x}{100}\right)$ .
Step 2: $SP_2 = CP \times \left(1 \pm \tfrac{y}{100}\right)$ .
In one line: $SP_2 = SP_1 \times \dfrac{100 \pm y}{100 \pm x}$ .
Given a second selling price instead of a second percentage, compare it with the cost price to get the new profit or loss.

Repricing in one line

SP_2 = SP_1 \times \dfrac{100 \pm y}{100 \pm x}

Worked example

By selling a fan for ₹1,890, a trader loses

10\%

. At what price should he sell it to gain

20\%

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2018 · AFCAT (II) 2018 — Numerical Ability · 02-209Easy

Example 1 · Profit and Loss · Finding Cost Price from Conditions

By selling a chair for ₹368, a man lost 8%. For how much should he have sold it to gain 15%?

The percentages do not add

From a

10\%

loss to a

20\%

gain is not 'add

30\%

to the selling price':

1890 \times 1.3 = 2457

, while the right answer is 2520. Both percentages are on the cost.

Divide to reach the cost

₹1,890 at a

10\%

loss means a cost of

1890 \div 0.9 = 2100

, not

1890 + 189 = 2079

Drill 44 more on through the cost price to a new selling price

Concept 2 of 3: When two prices differ by a known sum

If one sale is at

x\%

and another at

y\%

, the two selling prices differ by the gap between those percentages of the same cost price, counting a loss as negative. That rupee difference fixes the cost at once.

Definition

A loss of $x\%$ and a gain of $y\%$ : the gap is $(x + y)\%$ of CP.
Two gains (or two losses) of $x\%$ and $y\%$ : the gap is $(y - x)\%$ of CP.
$CP = \dfrac{\text{gap in rupees}}{\text{gap in per cent}} \times 100$ .
Per-kg and per-unit wording works the same way.

Cost from a price gap

CP = \dfrac{\text{difference in rupees}}{\text{difference in per cent}} \times 100

Worked example

A cooler is sold at a loss of

6\%

. Had it been sold for ₹540 more, there would have been a gain of

12\%

. Find the cost price.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Profit and Loss · Finding Cost Price from Conditions

A man sells an article at 10% loss. If the article was sold for ₹270 more, he would have gained 5%. What is the cost price?

A loss and a gain add

From a

6\%

loss to a

12\%

gain is

18\%

of the cost, not

6\%

. Count the loss as below zero.

The gap is a share of the cost

The rupee gap is a percentage of the cost price, so divide it by that percentage. Dividing it by either selling price gives nothing useful.

Drill 10 more on when two prices differ by a known sum

Concept 3 of 3: Setting up an equation in the cost price

When a change touches the cost price as well, or the profit is described in words, call the cost

c

and write the condition as an equation. One line of algebra replaces the guessing.

Definition

Equal profit at one price and loss at another: $SP_1 - c = c - SP_2$ , so $c$ is the average of the two prices.
Both cost and selling price change by ₹ $d$ : write the new sale against the new cost, $SP + d = \left(1 + \tfrac{r}{100}\right)(c + d)$ .
'If the selling price were doubled, the profit would be $k$ times': $2S - C = k(S - C)$ .
A new cost with the same selling price: take the old cost as 100.

Equal profit and loss

SP_1 - c = c - SP_2 \;\Rightarrow\; c = \dfrac{SP_1 + SP_2}{2}

Worked example

An article is sold at a profit of

20\%

. If both its cost price and its selling price were ₹100 more, the profit would be

15\%

. Find the cost price.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2018 · AFCAT (II) 2018 — Numerical Ability · 02-211Moderate

Example 3 · Profit and Loss · Finding Cost Price from Conditions

A horse is sold at a profit of 25%. If both the cost price and the selling price were ₹200 less, the profit would be 5% more. The cost price is

The new percentage is on the new cost

If both prices rise by ₹100, the profit in rupees is the same but its percentage is taken on

c + 100

. Dividing by the old cost gives the wrong equation.

Midway only for equal rupee amounts

The cost is the average of two prices only when the profit at one equals the loss at the other in rupees.

Drill 8 more on setting up an equation in the cost 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)

Through the cost price to a new selling price
Repricing in one line
$SP_2 = SP_1 \times \dfrac{100 \pm y}{100 \pm x}$
When two prices differ by a known sum
Cost from a price gap
$CP = \dfrac{\text{difference in rupees}}{\text{difference in per cent}} \times 100$
Setting up an equation in the cost price
Equal profit and loss
$SP_1 - c = c - SP_2 \;\Rightarrow\; c = \dfrac{SP_1 + SP_2}{2}$

Watch out for (6)

The percentages do not add
→ Through the cost price to a new selling price
Divide to reach the cost
→ Through the cost price to a new selling price
A loss and a gain add
→ When two prices differ by a known sum
The gap is a share of the cost
→ When two prices differ by a known sum
The new percentage is on the new cost
→ Setting up an equation in the cost price
Midway only for equal rupee amounts
→ Setting up an equation in the cost price

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