Profit and Loss, Discount and Tax — Exercise 7(A)

Find the gain or loss per cent, when:

C.P. = ₹ 750, S.P. = ₹ 875

Answer

Given:

C.P. = ₹ 750 and S.P. = ₹ 875

Since S.P. > C.P., so there is a gain.

Gain = S.P. - C.P. = ₹ 875 - ₹ 750 = ₹ 125.

Gain % = $\Big(\dfrac{\text{Gain}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{125}{750} \times 100\Big)$%

= $\dfrac{50}{3}$%

= $16\dfrac{2}{3}$%

Hence, the gain per cent is $\mathbf{16\dfrac{2}{3}}$%.

Find the gain or loss per cent, when:

C.P. = ₹ 126, S.P. = ₹ 94.50

Answer

Given:

C.P. = ₹ 126 and S.P. = ₹ 94.50

Since C.P. > S.P., so there is a loss.

Loss = C.P. - S.P. = ₹ 126 - ₹ 94.50 = ₹ 31.50

Loss % = $\Big(\dfrac{\text{Loss}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{31.50}{126} \times 100\Big)$%

= $\dfrac{3150}{126}$%

= 25%

Hence, the loss per cent is 25%.

Find the gain or loss per cent, when:

C.P. = ₹ 80.40, S.P. = ₹ 68.34

Answer

Given:

C.P. = ₹ 80.40 and S.P. = ₹ 68.34

Since C.P. > S.P., so there is a loss.

Loss = C.P. - S.P. = ₹ 80.40 - ₹ 68.34 = ₹ 12.06

Loss % = $\Big(\dfrac{\text{Loss}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{12.06}{80.40} \times 100\Big)$%

= $\dfrac{1206}{80.40}$%

= 15%

Hence, the loss per cent is 15%.

Find the gain or loss per cent, when:

C.P. = ₹ 58.75, S.P. = ₹ 51.70

Answer

Given:

C.P. = ₹ 58.75 and S.P. = ₹ 51.70

Since C.P. > S.P., so there is a loss.

Loss = C.P. - S.P. = ₹ 58.75 - ₹ 51.70 = ₹ 7.05

Loss % = $\Big(\dfrac{\text{Loss}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{7.05}{58.75} \times 100\Big)$%

= $\dfrac{705}{58.75}$%

= 12%

Hence, the loss per cent is 12%.

Ranjit purchased an almirah for ₹ 5,248 and paid ₹ 127 for its transportation. He sold it for ₹ 6,020. Find his gain or loss per cent.

Answer

Given:

C.P. of the almirah = ₹ 5,248

Overhead expenses = ₹ 127

Actual C.P. = C.P. + Overhead expenses = ₹ (5,248 + 127) = ₹ 5,375.

S.P. = ₹ 6,020

Since S.P. > C.P., so there is a gain.

Gain = S.P. - C.P. = ₹ (6,020 - 5,375) = ₹ 645.

Gain % = $\Big(\dfrac{\text{Gain}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{645}{5375} \times 100\Big)$%

= $\dfrac{64500}{5375}$%

= 12%

Hence, Ranjit's gain per cent is 12%.

Ahmed purchased an old scooter for ₹ 14,625 and spent ₹ 3,225 on its repairs. Then, he sold it for ₹ 16,422. Find his gain or loss per cent.

Answer

Given:

C.P. of the scooter = ₹ 14,625

Overhead expenses (repairs) = ₹ 3,225

Actual C.P. = C.P. + Overhead expenses = ₹ (14,625 + 3,225) = ₹ 17,850.

S.P. = ₹ 16,422

Since C.P. > S.P., so there is a loss.

Loss = C.P. - S.P. = ₹ (17,850 - 16,422) = ₹ 1,428.

Loss % = $\Big(\dfrac{\text{Loss}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{1428}{17850} \times 100\Big)$%

= $\dfrac{142800}{17850}$%

= 8%

Hence, Ahmed's loss per cent is 8%.

A man buys two cricket bats, one for ₹ 1,360 and the other for ₹ 1,040. He sells the first bat at a gain of 15% and the second one at a loss of 15%. Find his gain or loss per cent in the whole transaction.

Answer

Given:

C.P. of the first bat = ₹ 1,360, Gain % = 15%

C.P. of the second bat = ₹ 1,040, Loss % = 15%

For the first bat:

$\text{S.P.} = \dfrac{(100 + \text{gain percentage})}{100} \times \text{C.P.} \\[1em] = ₹ \Big(\dfrac{100 + 15}{100} \times 1360\Big) \\[1em] = ₹ \Big(\dfrac{115}{100} \times 1360\Big) \\[1em] = ₹ 1564$

For the second bat:

$\text{S.P.} = \dfrac{(100 - \text{loss percentage})}{100} \times \text{C.P.} \\[1em] = ₹ \Big(\dfrac{100 - 15}{100} \times 1040\Big) \\[1em] = ₹ \Big(\dfrac{85}{100} \times 1040\Big) \\[1em] = ₹ 884$

Total C.P. of the two bats = ₹ (1,360 + 1,040) = ₹ 2,400.

Total S.P. of the two bats = ₹ (1,564 + 884) = ₹ 2,448.

Since S.P. > C.P., so there is a gain.

Gain = ₹ (2,448 - 2,400) = ₹ 48.

Gain % = $\Big(\dfrac{\text{Gain}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{48}{2400} \times 100 \Big)$%

= $\dfrac{4800}{2400}$%

= 2%

Hence, the man gains 2% in the whole transaction.

Nandlal bought 20 dozen notebooks at ₹ 156 per dozen. He sold 8 dozen of them at 10% gain and the remaining 12 dozen at 20% gain. What is his gain per cent in the whole transaction?

Answer

Given:

Number of notebooks bought = 20 dozen, rate = ₹ 156 per dozen

Total C.P. = ₹ (156 × 20) = ₹ 3,120.

8 dozen are sold at a gain of 10%.

$\text{S.P. of 8 dozen} = ₹ \Big(\dfrac{156 \times 110 \times 8}{100}\Big) \\[1em] = ₹ \dfrac{137280}{100} \\[1em] = ₹ 1372.80$

The remaining 12 dozen are sold at a gain of 20%.

$\text{S.P. of 12 dozen} = ₹ \Big(\dfrac{156 \times 120 \times 12}{100}\Big) \\[1em] = ₹ \dfrac{224640}{100} \\[1em] = ₹ 2246.40$

Total S.P. = ₹ (1,372.80 + 2,246.40) = ₹ 3,619.20.

Since S.P. > C.P., so there is a gain.

Gain = ₹ (3,619.20 - 3,120) = ₹ 499.20.

Gain percentage = $\Big(\dfrac{\text{Gain}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{499.20}{3120} \times 100\Big)$%

= $\dfrac{49920}{3120}$%

= 16%

Hence, Nandlal's gain per cent in the whole transaction is 16%.

Heera bought 25 kg of rice at ₹ 48 per kg and 35 kg of rice at ₹ 60 per kg. He sold the mixture at ₹ 66 per kg. Find his gain per cent.

Answer

Given:

25 kg of rice at ₹ 48 per kg and 35 kg of rice at ₹ 60 per kg

C.P. of 25 kg of rice = ₹ (25 × 48) = ₹ 1,200.

C.P. of 35 kg of rice = ₹ (35 × 60) = ₹ 2,100.

Total C.P. of the mixture = ₹ (1,200 + 2,100) = ₹ 3,300.

Total weight of the mixture = (25 + 35) kg = 60 kg.

S.P. of the mixture = ₹ (60 × 66) = ₹ 3,960.

Since S.P. > C.P., so there is a gain.

Gain = ₹ (3,960 - 3,300) = ₹ 660.

Gain percentage = $\Big(\dfrac{\text{Gain}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{660}{3300} \times 100\Big)$%

= $\dfrac{66000}{3300}$%

= 20%

Hence, Heera's gain per cent is 20%.

If the selling price of an article is $\dfrac{4}{5}$th of its cost price, find the loss per cent.

Answer

Given:

S.P. = $\dfrac{4}{5}$ of C.P.

Since S.P. < C.P., so there is a loss.

Loss = C.P. - S.P.

= $\text{C.P.} - \dfrac{4}{5}\text{C.P.}$

= $\dfrac{\text{C.P.}}{5}$

Loss % = $\Big(\dfrac{\text{Loss}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{\dfrac{\text{C.P.}}{5}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{\text{C.P.}}{5} \times \dfrac{100}{\text{C.P.}}\Big)$%

= 20%

Hence, the loss per cent is 20%.

If the selling price of an article is $1\dfrac{1}{3}$ of its cost price, find the gain per cent.

Answer

Given:

S.P. = $1\dfrac{1}{3}$ of C.P. = $\dfrac{4}{3}$ of C.P.

Since S.P. > C.P., so there is a gain.

Gain = S.P. - C.P.

= $\dfrac{4}{3}\text{C.P.} - \text{C.P.}$

= $\dfrac{\text{C.P.}}{3}$

Gain % = $\Big(\dfrac{\text{Gain}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{\text{C.P.}}{3} \times \dfrac{100}{\text{C.P.}}\Big)$%

= $\dfrac{100}{3}$%

= $33\dfrac{1}{3}$%

Hence, the gain per cent is $\mathbf{33\dfrac{1}{3}}$%.

A man sold a table for ₹ 2,250 and gained one-ninth of its cost price. Find:

(i) the cost price of the table

(ii) the gain per cent earned by the man.

Answer

Given:

S.P. of the table = ₹ 2,250 and Gain = $\dfrac{1}{9}$ of C.P.

(i) Let the cost price of the table be ₹ $x$.

Then, Gain = ₹ $\dfrac{x}{9}$.

$\text{S.P.} = \text{C.P.} + \text{Gain} \\[1em] \Rightarrow 2250 = x + \dfrac{x}{9} \\[1em] \Rightarrow 2250 = \dfrac{10x}{9} \\[1em] \Rightarrow x = \dfrac{2250 \times 9}{10} \\[1em] \Rightarrow x = 2025$

Hence, the cost price of the table is ₹ 2,025.

(ii) Gain = ₹ (2,250 - 2,025) = ₹ 225.

Gain % = $\Big(\dfrac{\text{Gain}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{225}{2025} \times 100\Big)$%

= $\dfrac{100}{9}$%

= $11\dfrac{1}{9}$%

Hence, the gain per cent earned by the man is $\mathbf{11\dfrac{1}{9}}$%.

By selling a pen for ₹ 195, a man loses one-sixteenth of what it costs him. Find

(i) the cost price of the pen

(ii) loss per cent.

Answer

Given:

S.P. of the pen = ₹ 195 and Loss = $\dfrac{1}{16}$ of C.P.

(i) Let the cost price of the pen be ₹ $x$.

Then, Loss = ₹ $\dfrac{x}{16}$.

$\text{S.P.} = \text{C.P.} - \text{Loss} \\[1em] \Rightarrow 195 = x - \dfrac{x}{16} \\[1em] \Rightarrow 195 = \dfrac{15x}{16} \\[1em] \Rightarrow x = \dfrac{195 \times 16}{15} \\[1em] \Rightarrow x = 208$

Hence, the cost price of the pen is ₹ 208.

(ii) Loss = ₹ (208 - 195) = ₹ 13.

Loss % = $\Big(\dfrac{\text{Loss}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{13}{208} \times 100\Big)$%

= $\dfrac{100}{16}$%

= $\dfrac{25}{4}$%

= $6\dfrac{1}{4}$%

Hence, the loss per cent is $\mathbf{6\dfrac{1}{4}}$%.

A cycle was sold at a gain of 10%. Had it been sold for ₹ 99 more, the gain would have been 12%. Find the cost price of the cycle.

Answer

Let the cost price of the cycle be ₹ $x$.

When the gain is 10%, then S.P. = ₹ ($x$ + 10% of $x$) = ₹ $\dfrac{11x}{10}$.

When the gain is 12%, then S.P. = ₹ ($x$ + 12% of $x$) = ₹ $\dfrac{28x}{25}$.

Now, the difference between the two S.P.s = ₹ 99.

$\Rightarrow \dfrac{28x}{25} - \dfrac{11x}{10} = 99 \\[1em] \Rightarrow \dfrac{56x - 55x}{50} = 99 \\[1em] \Rightarrow \dfrac{x}{50} = 99 \\[1em] \Rightarrow x = 50 \times 99 \\[1em] \Rightarrow x = 4950$

Hence, the cost price of the cycle is ₹ 4,950.

A bucket was sold at a loss of 8%. Had it been sold for ₹ 56 more, there would have been a gain of 8%. What is the cost price of the bucket?

Answer

Let the cost price of the bucket be ₹ $x$.

When the loss is 8%, then S.P. = ₹ ($x$ - 8% of $x$) = ₹ $\dfrac{23x}{25}$.

When the gain is 8%, then S.P. = ₹ ($x$ + 8% of $x$) = ₹ $\dfrac{27x}{25}$.

Now, the difference between the two S.P.s = ₹ 56.

$\Rightarrow \dfrac{27x}{25} - \dfrac{23x}{25} = 56 \\[1em] \Rightarrow \dfrac{4x}{25} = 56 \\[1em] \Rightarrow x = \dfrac{56 \times 25}{4} \\[1em] \Rightarrow x = 350$

Hence, the cost price of the bucket is ₹ 350.

The selling price of 18 books is equal to the cost price of 21 books. Find the gain or loss per cent.

Answer

Let the cost price of 1 book be ₹ 1.

Then, C.P. of 21 books = ₹ 21.

S.P. of 18 books = C.P. of 21 books = ₹ 21.

⇒ S.P. of 1 book = ₹ $\dfrac{21}{18} = ₹ \dfrac{7}{6}$.

Since S.P. of 1 book > C.P. of 1 book, so there is a gain.

Gain on 1 book = ₹ $\Big(\dfrac{7}{6} - 1\Big) = ₹ \dfrac{1}{6}$.

Gain % = $\Big(\dfrac{\text{Gain}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{1}{6} \times \dfrac{100}{1}\Big)$%

= $\dfrac{50}{3}$%

= $16\dfrac{2}{3}$%

Hence, the gain per cent is $\mathbf{16\dfrac{2}{3}}$%.

The cost price of 12 fans is equal to the selling price of 16 fans. Find the gain or loss per cent.

Answer

Let the cost price of 1 fan be ₹ 1.

Then, C.P. of 12 fans = ₹ 12.

S.P. of 16 fans = C.P. of 12 fans = ₹ 12.

⇒ S.P. of 1 fan = ₹ $\dfrac{12}{16} = ₹ \dfrac{3}{4}$.

Since S.P. of 1 fan < C.P. of 1 fan, so there is a loss.

Loss on 1 fan = ₹ $\Big(1 - \dfrac{3}{4}\Big) = ₹ \dfrac{1}{4}$.

Loss % = $\Big(\dfrac{\text{Loss}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{1}{4} \times \dfrac{100}{1}\Big)$%

= 25%

Hence, the loss per cent is 25%.

On selling 250 cassettes, a man had a gain equal to the selling price of 25 cassettes. Find his gain per cent.

Answer

We have:

Gain on selling 250 cassettes = (S.P. of 250 cassettes) - (C.P. of 250 cassettes)

⇒ S.P. of 25 cassettes = (S.P. of 250 cassettes) - (C.P. of 250 cassettes)

⇒ C.P. of 250 cassettes = (S.P. of 250 cassettes) - (S.P. of 25 cassettes)

⇒ C.P. of 250 cassettes = S.P. of 225 cassettes

Let the selling price of 1 cassette be ₹ 1.

Then, C.P. of 250 cassettes = ₹ 225 and S.P. of 250 cassettes = ₹ 250.

Since S.P. > C.P., so there is a gain.

Gain = ₹ (250 - 225) = ₹ 25.

Gain % = $\Big(\dfrac{\text{Gain}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{25}{225} \times 100\Big)$%

= $\dfrac{100}{9}$%

= $11\dfrac{1}{9}$%

Hence, the gain per cent is $\mathbf{11\dfrac{1}{9}}$%.

On selling 36 oranges, a vendor loses the selling price of 4 oranges. Find his loss per cent.

Answer

We have:

Loss on selling 36 oranges = (C.P. of 36 oranges) - (S.P. of 36 oranges)

⇒ S.P. of 4 oranges = (C.P. of 36 oranges) - (S.P. of 36 oranges)

⇒ C.P. of 36 oranges = (S.P. of 36 oranges) + (S.P. of 4 oranges)

⇒ C.P. of 36 oranges = S.P. of 40 oranges

Let the selling price of 1 orange be ₹ 1.

Then, C.P. of 36 oranges = ₹ 40 and S.P. of 36 oranges = ₹ 36.

Since C.P. > S.P., so there is a loss.

Loss = ₹ (40 - 36) = ₹ 4.

Loss % = $\Big(\dfrac{\text{Loss}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{4}{40} \times 100\Big)$%

= 10%

Hence, the loss per cent is 10%.

Toffees are bought at 2 for a rupee and sold at 5 for ₹ 3. Find the gain or loss per cent.

Answer

Suppose the number of toffees bought = L.C.M. of 2 and 5 = 10.

C.P. of 2 toffees = ₹ 1.

C.P. of 1 toffee = ₹ $\dfrac{1}{2}$.

C.P. of 10 toffees = ₹ $\Big(\dfrac{1}{2} \times 10\Big)$ = ₹ 5.

S.P. of 5 toffees = ₹ 3.

S.P. of 1 toffee = ₹ $\dfrac{3}{5}$.

S.P. of 10 toffees = ₹ $\Big(\dfrac{3}{5} \times 10\Big)$ = ₹ 6.

Since S.P. > C.P., so there is a gain.

Gain = ₹ (6 - 5) = ₹ 1.

Gain % = $\Big(\dfrac{\text{Gain}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{1}{5} \times 100\Big)$%

= 20%

Hence, the gain per cent is 20%.

Coffee costing ₹ 450 per kg was mixed with Chicory costing ₹ 225 per kg in the ratio 5 : 2 for a certain blend. If the mixture was sold at ₹ 405 per kg; find the gain or loss per cent.

Answer

Let 5 kg of coffee be mixed with 2 kg of chicory.

C.P. of 5 kg of coffee = ₹ (450 × 5) = ₹ 2,250.

C.P. of 2 kg of chicory = ₹ (225 × 2) = ₹ 450.

Total C.P. of the blend = ₹ (2,250 + 450) = ₹ 2,700.

Total weight of the blend = (5 + 2) kg = 7 kg.

Total S.P. of the blend = ₹ (405 × 7) = ₹ 2,835.

Since S.P. > C.P., so there is a gain.

Gain = ₹ (2,835 - 2,700) = ₹ 135.

Gain % = $\Big(\dfrac{\text{Gain}}{\text{C.P.}} \times 100\Big)$%

= $\Big(\dfrac{135}{2700} \times 100\Big)$%

= $\dfrac{13500}{2700}$%

= 5%

Hence, the gain per cent is 5%.