Problems on profit and loss

You know the for­mu­las for profit, loss and dis­count. Now we use them on the kinds of prob­lems that come up in tests. In each one, first decide: what is the cost price, and what is the sell­ing price? Once you have both, the rest is just the for­mu­las. When a prob­lem gives no amounts at all, pick an easy num­ber, such as ₹1 for one item, or call the unknown amount $x$.

Exam­ple 1: Los­ing the price of 2 choco­lates#

By sell­ing 18 choco­lates, a ven­dor loses the sell­ing price of 2 choco­lates. Find his loss per cent.

No prices are given, so choose one. Let the sell­ing price of 1 choco­late be ₹1.

S.P. of 18 choco­lates $= 18$ rupees. The loss is the S.P. of 2 choco­lates, which is ₹2.

$$\begin{aligned} \text{loss} &= \text{C.P.} - \text{S.P.} \\ 2 &= \text{C.P.} - 18 \\ \text{C.P.} &= 20 \end{aligned}$$

$$\displaystyle \text{loss}\% = \frac{2}{20} \times 100 = 10\%$$

Exam­ple 2: A watch#

A shop­keeper bought a watch for ₹200 and sold it for ₹250. What is his profit per cent?

C.P. $= 200$ and S.P. $= 250$. S.P. is more, so there is a profit of $250 - 200 = 50$ rupees.

$$\displaystyle \text{profit}\% = \left(\frac{50}{200} \times 100\right)\% = 25\%$$

Exam­ple 3: Com­mis­sion#

An agent gets a com­mis­sion of 2.5% on the cloth he sells. One day his com­mis­sion is ₹12.50. What is the value of the cloth he sold that day?

A com­mis­sion is a pay­ment to an agent, worked out as a per­cent­age of the sales.

Let the value of the cloth be $x$ rupees. Then 2.5% of $x$ is 12.50.

$$\displaystyle \begin{aligned} \frac{2.5}{100} \times x &= 12.50 \\ x &= \frac{12.50 \times 100}{2.5} = 500 \end{aligned}$$

He sold cloth worth ₹500.

Exam­ple 4: Profit equals loss#

The profit when an arti­cle is sold for ₹425 is the same as the loss when it is sold for ₹355. Find the cost price.

Let the cost price be $x$ rupees.

Profit when sold for ₹425 $= 425 - x$. Loss when sold for ₹355 $= x - 355$.

$$\begin{aligned} 425 - x &= x - 355 \\ 425 + 355 &= x + x \\ 780 &= 2x \\ x &= 390 \end{aligned}$$

The cost price is ₹390. The cost is exactly halfway between the two sell­ing prices.

Exam­ple 5: Cost of 19 equals sell­ing price of 16#

The cost price of 19 chairs is equal to the sell­ing price of 16 chairs. Find the gain per cent.

Let the C.P. of each chair be $x$ and the S.P. of each chair be $y$. Then $19x = 16y$, so $\displaystyle \frac{y}{x} = \frac{19}{16}$.

$$\displaystyle \begin{aligned} \text{gain}\% &= \frac{y - x}{x} \times 100 \\ &= \left(\frac{y}{x} - 1\right) \times 100 \\ &= \left(\frac{19}{16} - 1\right) \times 100 \\ &= \frac{3}{16} \times 100 = \frac{75}{4} = 18\frac{3}{4} \end{aligned}$$

The gain is $\displaystyle 18\frac{3}{4}\%$.

Exam­ple 6: A false mea­sure#

A trader says he sells at cost price, but he gives only 900 ml for every litre. Find his profit per cent.

This is the false weight result from the last les­son. The error is $1000 - 900 = 100$ ml.

$$\displaystyle \text{profit}\% = \frac{100}{1000 - 100} \times 100 = \frac{100}{900} \times 100 = \frac{100}{9} = 11\frac{1}{9}$$

His profit is $\displaystyle 11\frac{1}{9}\%$.

Exam­ple 7: Oranges at two prices#

Asha buys some oranges at 12 for ₹9 and the same num­ber at 18 for ₹9. She sells all of them at 18 for ₹15. Does she gain or lose, and by what per cent?

Let her buy $x$ oranges at each price, so $2x$ oranges in all.

First lot: 12 oranges cost ₹9, so $x$ oranges cost $\displaystyle \frac{9x}{12} = \frac{3x}{4}$ rupees.

Sec­ond lot: 18 oranges cost ₹9, so $x$ oranges cost $\displaystyle \frac{9x}{18} = \frac{x}{2}$ rupees.

$$\displaystyle \text{total C.P.} = \frac{3x}{4} + \frac{x}{2} = \frac{3x}{4} + \frac{2x}{4} = \frac{5x}{4}$$

She sells 18 for ₹15, so $2x$ oranges bring $\displaystyle 2x \times \frac{15}{18} = \frac{5x}{3}$ rupees.

S.P. is more than C.P., so she gains.

$$\displaystyle \begin{aligned} \text{gain}\% &= \frac{\frac{5x}{3} - \frac{5x}{4}}{\frac{5x}{4}} \times 100 \\ &= \frac{\frac{5x}{12}}{\frac{5x}{4}} \times 100 \\ &= \frac{1}{3} \times 100 = 33\frac{1}{3} \end{aligned}$$

She gains $\displaystyle 33\frac{1}{3}\%$.

Exam­ple 8: Spoilt eggs#

A shop­keeper buys 100 eggs at ₹1.20 each. 4 eggs break on the way. He sells the rest at ₹15 a dozen. Find his profit or loss.

C.P. of 100 eggs $= 100 \times 1.20 = 120$ rupees.

12 eggs sell for ₹15, so 1 egg sells for $15 \div 12 = 1.25$ rupees.

Eggs left $= 100 - 4 = 96$. S.P. of 96 eggs $= 96 \times 1.25 = 120$ rupees.

S.P. equals C.P., so there is no profit and no loss.

Exam­ple 9: Sugar by the quin­tal#

A dealer sold 600 quin­tals of sugar at a profit of 7%. One quin­tal cost him ₹1600. Find his total profit and the sell­ing price.

C.P. of 600 quin­tals $= 600 \times 1600 = 960000$ rupees, that is ₹9,60,000.

$$\displaystyle \text{profit} = \frac{7}{100} \times 960000 = 67200$$

$$\displaystyle \text{S.P.} = \frac{100 + 7}{100} \times 960000 = 107 \times 9600 = 1027200$$

The profit is ₹67,200 and the sell­ing price is ₹10,27,200. Check: $960000 + 67200 = 1027200$.

Exam­ple 10: Wheat with extra costs#

A dealer buys 200 quin­tals of wheat at ₹1200 a quin­tal. He spends ₹10,000 on trans­port and stor­age. He sells the wheat at ₹13 per kg. Find his profit or loss, and the per cent.

Extra costs such as trans­port are added to the cost price.

C.P. $= 200 \times 1200 + 10000 = 240000 + 10000 = 250000$ rupees.

1 quin­tal = 100 kg, so 200 quin­tals = 20,000 kg.

S.P. $= 13 \times 100 \times 200 = 260000$ rupees.

S.P. is more, so there is a profit of $260000 - 250000 = 10000$ rupees.

$$\displaystyle \text{profit}\% = \left(\frac{10000}{250000} \times 100\right)\% = 4\%$$

Exam­ple 11: A tele­vi­sion#

By sell­ing a colour tele­vi­sion for ₹23,520, a dealer loses 4%. What did the tele­vi­sion cost him? At what price should he sell it to gain 8%?

$$\displaystyle \text{C.P.} = \frac{100}{100 - 4} \times 23520 = \frac{100}{96} \times 23520 = 24500$$

$$\displaystyle \text{S.P. for an 8\% gain} = \frac{108}{100} \times 24500 = 26460$$

The tele­vi­sion cost ₹24,500. To gain 8% he should sell it for ₹26,460.

Exam­ple 12: Cost of 15 equals sell­ing price of 20#

The cost price of 15 apples is the same as the sell­ing price of 20 apples. Find the gain or loss per cent.

Let the C.P. of 1 apple be $x$ and the S.P. be $y$. Then $15x = 20y$, so $\displaystyle \frac{y}{x} = \frac{15}{20} = \frac{3}{4}$.

S.P. is less than C.P., so there is a loss.

$$\displaystyle \text{loss}\% = \frac{x - y}{x} \times 100 = \left(1 - \frac{3}{4}\right) \times 100 = 25\%$$

A quick way to see it: on 20 apples the shop gets only what 15 cost, so it loses the cost of 5 apples out of 20, which is 25%.

Prac­tice#

1. By sell­ing 25 pens, a shop loses the sell­ing price of 5 pens. Find the loss per cent.
2. By sell­ing 20 bags, a trader gains the sell­ing price of 4 bags. Find the gain per cent.
3. An agent gets 5% com­mis­sion on his sales. One month he earns ₹1250 as com­mis­sion. What were his sales?
4. The profit when an arti­cle is sold for ₹840 is equal to the loss when it is sold for ₹660. Find the cost price.
5. The cost price of 12 books is equal to the sell­ing price of 10 books. Find the gain per cent.
6. Bananas are bought at 6 for ₹10 and sold at 4 for ₹10. Find the gain per cent.
7. A ven­dor buys 50 kg of toma­toes at ₹24 per kg. 5 kg go bad. He sells the rest at ₹30 per kg. Find his profit per cent.
8. A fridge is sold for ₹17,100 at a loss of 5%. Find its cost price, and the price at which it should be sold to gain 10%.
9. A dealer buys 50 quin­tals of rice at ₹3000 a quin­tal and spends ₹10,000 on trans­port. He sells the rice at ₹34 per kg. Find his profit per cent.
10. The cost price of 8 pens is equal to the sell­ing price of 10 pens. Find the gain or loss per cent.

Answers#