Profit and Loss With solved examples including shortcut methods

April 10, 2019Lalit Tomar income profit

Profit and loss with solved examples

Profit and Loss

Profit and loss are always calculated on the cost price, which is taken as 100%.

Basic terms and formulae

C.P (Cost price)The price at which an article is purchased.
S.P (Selling price)The price at which an article is sold.
Profit or gainProfit = S.P − C.P
LossLoss = C.P − S.P
Profit %(Profit / C.P) × 100
Loss %(Loss / C.P) × 100
RememberProfit and loss are always counted on C.P, and C.P is always 100%.

Observations

20% profit on an article100% represents the C.P, 120% the S.P and 20% the profit.
10% loss on an article100% represents the C.P, 90% the S.P and 10% the loss.
In both casesThe relation is always between C.P and S.P.

Important points

When C.P and profit or loss % are given S.P = C.P × (100 + Profit%) ÷ 100
S.P = C.P × (100 − Loss%) ÷ 100
When S.P and profit or loss % are given C.P = S.P × 100 ÷ (100 + Profit%)
C.P = S.P × 100 ÷ (100 − Loss%)
Marked price (M.P)The M.P of an article is also called the marked price or labelled price.

Discount

The reduction made on the marked price is known as a discount.

DiscountDiscount = M.P − S.P
Discount %(Discount ÷ M.P) × 100
S.P from M.P and discount %S.P = M.P × (100 − Discount%) ÷ 100
Relation between C.P and M.PM.P = [C.P × (100 ± Profit or Loss%)] ÷ (100 − Discount%)
Use + for profit and − for loss.

Successive discount

Two discounts of x% and y%Equivalent discount = (x + y − xy/100) %
Three discounts of x%, y% and z%First find p = (x + y − xy/100) %, then the equivalent discount = (p + z − pz/100) %

Solved examples based on Profit and Loss

Ex.1A fruit seller makes a profit of 20% by selling mangoes at a certain price. If he charges ₹2 more for each mango, his profit becomes 40%. Find the C.P of 1 mango.

Sol. Let the C.P of a mango be 100. A 20% profit means S.P = 120, and a 40% profit means S.P = 140.

The ₹2 increase moves the S.P from 120 to 140, so 20 = ₹2. Then 100 = 2 × (100/20) = ₹10. Ans: ₹10

Ex.2A man purchased 2 fans for ₹1200 each. He sold one at a 10% profit and the other at a 5% loss. Find the overall profit.

Method 1. Total C.P = 1200 + 1200 = ₹2400.

Net profit = (1200 × 10/100) − (1200 × 5/100) = 120 − 60 = ₹60. Profit % = (60/2400) × 100 = 2.5%.

Method 2. When the articles have the same C.P: (profit% − loss%) ÷ number of articles = (10 − 5)/2 = 2.5%.

Ans: 2.5% profit

Ex.3The cost price of an article is ₹1900. 5/7 of it is sold at a profit of 6% and the remaining part at a loss of 8%. Find the overall profit or loss %.

Sol. When parts are given as a fraction, the denominator gives the total number of articles. So there are 7 articles: 5 are sold at 6% profit and 2 at 8% loss.

(5 × 6% − 2 × 8%) ÷ 7 = (30 − 16)/7 = 2%. Ans: 2% profit

Ex.4The cost price of an article is ₹2500. 2/3 of it is sold at a profit of 4%. At what profit % should the rest be sold to gain 5% overall?

Sol. Let the total number of articles be 3. Two are sold at 4% profit and one at x%.

(2 × 4 + x)/3 = 5, so 8 + x = 15 and x = 7. Ans: 7%

Ex.5By selling 32 oranges for ₹30, a man loses 25%. Find the selling price of 4 oranges so that he gains 20%.

Sol. 32 oranges = ₹30, so 4 oranges = (30/32) × 4 = ₹15/4. This is the S.P at a 25% loss, i.e. 75% of the C.P.

75% = 15/4, so 1% = (15/4) × 1/75, and 120% = (15/4) × 120/75 = ₹6. Ans: ₹6

Ex.6A man sold 2 articles at the same price. On the first he gains 25% and on the other he loses 20%. Find his overall profit or loss %.

Method 1. 25% = 1/4 and 20% = 1/5.

For the first article C.P = 4 and S.P = 5. For the second, C.P = 5 and S.P = 4. To make the S.P equal, multiply the first by 4 and the second by 5.

Total C.P = 16 + 25 = 41 and total S.P = 20 + 20 = 40, so loss = 1. Loss % = (1/41) × 100 = 100/41%.

Method 2. C.P₁ : C.P₂ = 16 : 25, so the total C.P = 41.

Profit or loss = 16 × (25/100) − 25 × (20/100) = 4 − 5 = −1. Loss % = (1/41) × 100 = 100/41%.

Method 3 (formula). [100(P − L) − 2PL] ÷ [(100 + P) + (100 − L)]

= [100 × 5 − 2 × 25 × 20] ÷ (125 + 80) = (500 − 1000)/205 = −100/41.

Ans: Loss of 100/41 % ≈ 2.44%

Ex.7A man sold 2 articles at the same price. On the first he gains 20% and on the other he loses 20%. Find the overall profit or loss.

Method 1. S.P₁ = S.P₂, so 120 C.P₁ = 80 C.P₂ and C.P₁/C.P₂ = 80/120 = 2/3. Total C.P = 2 + 3 = 5.

Profit or loss = 2 × (20/100) − 3 × (20/100) = −1/5, a loss. Loss % = (1/5)/5 × 100 = 4%.

Method 2 (formula). When the S.P is the same and the profit and loss are both r%, loss % = r²/100 = 20²/100 = 4%.

Ans: 4% loss

Ex.8A man sells a book at a profit of 20%. Had he bought it for 20% less and sold it for ₹18 less, he would have gained 25%. Find the cost price of the book.

Method 1. C.P₁ = 100%, S.P₁ = 120%. C.P₂ = 80%, S.P₂ = 80 × 125/100 = 100%.

The difference between S.P₁ and S.P₂ is 20%, so 20% = ₹18 and 100% = 18 × (100/20) = ₹90.

Method 2. 20% = 1/5 and 25% = 1/4. Taking C.P₁ = 5, S.P₁ = 6, C.P₂ = 4 and S.P₂ = 5, the difference between the S.P values is 1.

So 1 = ₹18 and 5 = ₹90. Ans: ₹90

Ex.9The profit earned by selling an article for ₹1920 equals the loss from selling it for ₹1280. At what price should it be sold to gain 25%?

Sol. Profit = loss, so 1920 − C.P = C.P − 1280. Then 2 C.P = 3200 and C.P = ₹1600.

S.P for 25% profit = 1600 × 125/100 = ₹2000. Ans: ₹2000

Ex.10If the S.P of 20 articles equals the C.P of 25 articles, find the profit or loss %.

Sol. 20 S.P = 25 C.P, so S.P/C.P = 25/20 = 5/4.

Profit = (1/4) × 100 = 25%. Ans: 25% profit

Ex.11After selling 72 articles, a man earns a profit equal to the S.P of 9 articles. Find the profit %.

Sol. Profit = S.P − C.P, so 9 S.P = 72 S.P − 72 C.P. Then 72 C.P = 63 S.P, i.e. 8 C.P = 7 S.P and S.P/C.P = 8/7.