AFCAT Numerical Ability · Percentage

Profit, Loss and Discount with Percentages

Profit and loss are percentages of the cost price, discount is a percentage of the marked price: chain cost to marked to selling price, and combine sales made at different rates.

All Percentage notes

Why this matters

A third of this page is HARD, the most of any page in the chapter. The questions are short, but each one hides a choice of base: profit is on the cost price and discount is on the marked price, and mixing them up gives an answer that is always among the options.

Concept 1 of 5: Profit and loss on the cost price

A 15% profit means the selling price is 1.15 times the cost price. Every profit or loss question is that one multiplication, run forwards or backwards.

Definition

Profit % and loss % are on the cost price (CP). SP $=$ CP $\times \left(1 + \tfrac{p}{100}\right)$ for a profit, $\times \left(1 - \tfrac{l}{100}\right)$ for a loss.
CP from SP: divide by the same factor.
Two selling prices for two outcomes: their difference is the difference of the factors times CP. A 12% loss against a 9% gain is 21% of CP.
'The profit at ₹940 equals the loss at ₹700' puts CP halfway: $\tfrac{940 + 700}{2}$ .
CP : SP $= 4 : 5$ means a profit of 1 on 4, which is 25%.

Selling price from cost price

SP = CP\left(1 + \dfrac{p}{100}\right), \qquad CP = \dfrac{SP}{1 + p/100}

Worked example

A trader sells an article at a 12% loss. Had he sold it for ₹126 more, he would have gained 9%. Find the cost price.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Percentage · Profit, Loss and Discount

A dealer sells an article at

5\%

loss. Had he sold it for Rs.

104

more, he would have gained

8\%

. The cost price is:

Profit is on cost, not on selling price

Bought for ₹640 and sold for ₹720, the profit is 80 on 640, which is 12.5%. Dividing by the selling price gives 11.1%.

A loss and a gain add

Going from a 12% loss to a 9% gain covers 21% of the cost price. Subtracting the two, 3%, is the usual slip.

Drill 15 more on profit and loss on the cost price

Concept 2 of 5: Articles bought and sold in different lots

If the cost of 22 articles equals the selling price of 18, then 18 articles bring in what 22 cost. The profit is the 4 extra articles' worth, on the 18 that were sold.

Definition

CP of $a$ articles $=$ SP of $b$ articles: profit % $= \dfrac{a - b}{b} \times 100$ (a loss if $b > a$ ).
'Bought $x$ for ₹A, sold $y$ for ₹B': find the cost and the price of one article, then compare.
Per dozen and per item: bring both to the same count first.
A false weight: selling 900 g as a kilogram at the cost price gives 100 g of profit on 900 g sold.

Cost of a equals selling price of b

\text{profit } \% = \dfrac{a - b}{b} \times 100

Worked example

The cost price of 22 articles equals the selling price of 18. Find the profit percentage.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Percentage · Profit, Loss and Discount

The cost price of

20

articles is equal to selling price of

16

articles. Find the profit percent:

Divide by the articles sold

When the cost of 22 equals the selling price of 18, the profit is 4 on 18 (22.2%), not 4 on 22 (18.2%).

Drill 8 more on articles bought and sold in different lots

Concept 3 of 5: Marked price, discount and profit

A shopkeeper marks a price above cost and then takes a discount off it. Discount is a percentage of the marked price; profit is a percentage of the cost price. Chain the two factors: cost to marked to selling.

Definition

Discount is on the marked price (MP). SP $=$ MP $\times \left(1 - \tfrac{d}{100}\right)$ .
Marked $m\%$ above cost, then $d\%$ off: SP $=$ CP $\times \left(1 + \tfrac{m}{100}\right)\left(1 - \tfrac{d}{100}\right)$ .
Given CP, the profit wanted and the discount: SP from the profit, then MP $=$ SP $\div \left(1 - \tfrac{d}{100}\right)$ .
'Buy 4, get 1 free' gives 5 articles for the price of 4, a discount of 1 in 5, which is 20%.

From cost to marked to selling price

SP = CP\left(1 + \dfrac{m}{100}\right)\left(1 - \dfrac{d}{100}\right)

Worked example

A shopkeeper marks his goods 60% above cost and allows a 25% discount. Find his profit percentage.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Percentage · Profit, Loss and Discount

After giving

12\%

discount on an article, a trader gains

10\%

. If he gives

15\%

discount, his gain percent will be:

Discount on marked price, profit on cost

Marked 60% up and sold 25% off is not

60 - 25 = 35\%

profit. The 25% comes off 160, leaving 120, so the profit is 20%.

Drill 20 more on marked price, discount and profit

Concept 4 of 5: Successive discounts and chains of profit

Two discounts of 25% and 4% are two successive falls: the price is multiplied by

0.75 \times 0.96 = 0.72

, one discount of 28%. A chain of profits from maker to wholesaler to shop multiplies the same way.

Definition

Successive discounts $a\%$ and $b\%$ equal one discount of $a + b - \dfrac{ab}{100}$ per cent (the successive-change formula with two falls).
Three or more: multiply all the factors and take the result from 1.
A trade discount and then a cash discount 'on the balance' are successive discounts.
Maker to wholesaler to retailer to customer, each adding a profit: the customer's price is the first cost times every factor. Going back, divide.

Two successive discounts

\text{single discount} = a + b - \dfrac{ab}{100}

Worked example

Find the single discount equal to successive discounts of 25% and 4%.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 4 · Percentage · Profit, Loss and Discount

Three successive discounts of

10\%

20\%

and

30\%

are equivalent to a single discount of:

Successive discounts are less than their sum

25% and 4% make 28%, not 29%. The second discount is taken on the already reduced price.

Drill 8 more on successive discounts and chains of profit

Concept 5 of 5: Overall profit when parts sell at different rates

Overall profit is total selling price against total cost. When two items sell for the same price, one at a 20% gain and one at a 20% loss, the loss is on the dearer item, so the deal loses overall.

Definition

Overall profit % $= \dfrac{\text{total SP} - \text{total CP}}{\text{total CP}} \times 100$ . Never average the percentages.
Two articles sold at the same selling price, one at $r\%$ gain and one at $r\%$ loss: always a loss of $\dfrac{r^2}{100}\%$ .
Parts of one stock sold at different rates: take the cost of each part and apply its own rate (a weighted average of the rates, by cost).
Spoiled goods: the whole cost must come back from what is left to sell.

Same selling price, gain and loss of r%

\text{loss } \% = \dfrac{r^2}{100}

Worked example

Two items are sold for ₹2,400 each, one at a 20% gain and the other at a 20% loss. Find the overall gain or loss.

Practice this conceptself-check · 2 quick reps

The same idea in a practice question:

Practice questionModerate

Example 5 · Percentage · Profit, Loss and Discount

A man sells two watches for Rs.

1980

each. On one he gains

10

% and on the other he loses

10

%. Find his overall gain or loss percentage.

Equal gain and loss is not break-even

Sold at the same price, a 20% gain and a 20% loss lose 4% overall. The loss-making item cost more, so its loss outweighs the gain.

Drill 7 more on overall profit when parts sell at different rates

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 (5)

Profit and loss on the cost price
Selling price from cost price
$SP = CP\left(1 + \dfrac{p}{100}\right), \qquad CP = \dfrac{SP}{1 + p/100}$
Articles bought and sold in different lots
Cost of a equals selling price of b
$\text{profit } \% = \dfrac{a - b}{b} \times 100$
Marked price, discount and profit
From cost to marked to selling price
$SP = CP\left(1 + \dfrac{m}{100}\right)\left(1 - \dfrac{d}{100}\right)$
Successive discounts and chains of profit
Two successive discounts
$\text{single discount} = a + b - \dfrac{ab}{100}$
Overall profit when parts sell at different rates
Same selling price, gain and loss of r%
$\text{loss } \% = \dfrac{r^2}{100}$

Watch out for (6)

Profit is on cost, not on selling price
→ Profit and loss on the cost price
A loss and a gain add
→ Profit and loss on the cost price
Divide by the articles sold
→ Articles bought and sold in different lots
Discount on marked price, profit on cost
→ Marked price, discount and profit
Successive discounts are less than their sum
→ Successive discounts and chains of profit
Equal gain and loss is not break-even
→ Overall profit when parts sell at different rates

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