Percentage and its Applications — Exercise 7.3

Find the discount and the selling price, when:

(i) the marked price = ₹575, discount = 12%

(ii) the printed price = ₹12,750, discount = $8\dfrac{1}{3}$%

Answer

(i) M.P. of the article = ₹575

Rate of discount = 12%

Discount = 12% of ₹575

$= \dfrac{12}{100} \times 575 \\[1em] = \dfrac{6900}{100} \\[1em] = ₹69$

S.P. of the article = M.P. - Discount

= ₹575 - ₹69

= ₹506

Hence, the discount is ₹69 and the selling price is ₹506.

(ii) Printed price of the article = ₹12,750

Rate of discount = $8\dfrac{1}{3}$% = $\dfrac{25}{3}$%

Discount = $\dfrac{25}{3}$% of ₹12,750

$= \dfrac{25}{3} \times \dfrac{1}{100} \times 12750 \\[1em] = \dfrac{318750}{300} \\[1em] = ₹1{,}062.50$

S.P. of the article = Printed price - Discount

= ₹12,750 - ₹1,062.50

= ₹11,687.50

Hence, the discount is ₹1,062.50 and the selling price is ₹11,687.50.

Find the discount and the discount percentage, when:

(i) marked price = ₹780, selling price = ₹721.50

(ii) advertised price = ₹28,500, selling price = ₹24,510

Answer

(i) M.P. of the article = ₹780

S.P. of the article = ₹721.50

Discount = M.P. - S.P.

= ₹780 - ₹721.50

= ₹58.50

Discount percentage = $\dfrac{\text{Discount}}{\text{M.P.}} \times 100$%

= $\dfrac{58.50}{780} \times 100$%

= $\dfrac{5850}{780}$%

= 7.5%

Hence, the discount is ₹58.50 and the discount percentage is 7.5%.

(ii) Advertised price of the article = ₹28,500

S.P. of the article = ₹24,510

Discount = Advertised price - S.P.

= ₹28,500 - ₹24,510

= ₹3,990

Discount percentage = $\dfrac{\text{Discount}}{\text{Advertised price}} \times 100$%

= $\dfrac{3990}{28500} \times 100$%

= $\dfrac{399000}{28500}$%

= 14%

Hence, the discount is ₹3,990 and the discount percentage is 14%.

A notebook is marked at ₹30. Find the price a student pays for a dozen notebooks if he gets 15% discount.

Answer

Given:

M.P. of one notebook = ₹30

Rate of discount = 15%

M.P. of a dozen notebooks = 12 x ₹30 = ₹360

S.P. = $\Big(1 - \dfrac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

$\text{S.P. of a dozen notebooks} = \Big(1 - \dfrac{15}{100}\Big)\text{ of M.P.} \\[1em] = \dfrac{85}{100} \times 360 \\[1em] = \dfrac{30600}{100} \\[1em] = ₹306$

Hence, the student pays ₹306 for a dozen notebooks.

A dealer gave 9% discount on sunglasses and charges ₹728 from the customer. Find the marked price of the sunglasses.

Answer

Given:

S.P. of the sunglasses = ₹728

Rate of discount = 9%

Let the marked price of the sunglasses be ₹$x$.

S.P. = $\Big(1 - \dfrac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

$\text{S.P.} = \Big(1 - \dfrac{9}{100}\Big)\text{ of M.P.} \\[1em] \Rightarrow 728 = \dfrac{91}{100} \times x \\[1em] \Rightarrow x = \dfrac{728 \times 100}{91} \\[1em] \Rightarrow x = \dfrac{72800}{91} \\[1em] \Rightarrow x = 800$

Hence, the marked price of the sunglasses is ₹800.

The list price of an article is ₹800 and a dealer is selling it at a discount of 20%. Find:

(i) the selling price of the article.

(ii) the cost price of the article if he makes 25% profit on selling it.

Answer

Given:

List price of the article = ₹800

Rate of discount = 20%

Profit = 25%

(i) S.P. = $\Big(1 - \dfrac{\text{Discount Percentage}}{100}\Big) \times$ List price

S.P. of the article = $\Big(1 - \dfrac{20}{100}\Big)$ of list price

$= \dfrac{80}{100} \times 800 \\[1em] = \dfrac{64000}{100} \\[1em] = ₹640$

Hence, the selling price of the article is ₹640.

(ii) S.P. of the article = ₹640, profit = 25%

S.P. = $\Big(1 + \dfrac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$\Rightarrow 640 = \Big(1 + \dfrac{25}{100}\Big)\text{ of C.P.} \\[1em] \Rightarrow 640 = \dfrac{125}{100}\text{ of C.P.} \\[1em] \Rightarrow \text{C.P.} = \dfrac{640 \times 100}{125} \\[1em] \Rightarrow \text{C.P.} = \dfrac{64000}{125} \\[1em] \Rightarrow \text{C.P.} = ₹512$

Hence, the cost price of the article is ₹512.

A shopkeeper marks his goods at such a price that would give him a profit of 10% after allowing a discount of 12%. If an article is marked at ₹2,250, find its:

(i) selling price

(ii) cost price

Answer

Given:

M.P. of the article = ₹2,250

Rate of discount = 12%

Profit = 10%

(i) S.P. = $\Big(1 - \dfrac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

S.P. of the article = $\Big(1 - \dfrac{12}{100}\Big)$ of M.P.

$= \dfrac{88}{100} \times 2250 \\[1em] = \dfrac{198000}{100} \\[1em] = ₹1{,}980$

Hence, the selling price of the article is ₹1,980.

(ii) S.P. of the article = ₹1,980, profit = 10%

S.P. = $\Big(1 + \dfrac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$\Rightarrow 1980 = \Big(1 + \dfrac{10}{100}\Big)\text{ of C.P.} \\[1em] \Rightarrow 1980 = \dfrac{110}{100}\text{ of C.P.} \\[1em] \Rightarrow \text{C.P.} = \dfrac{1980 \times 100}{110} \\[1em] \Rightarrow \text{C.P.} = \dfrac{198000}{110} \\[1em] \Rightarrow \text{C.P.} = ₹1{,}800$

Hence, the cost price of the article is ₹1,800.

A shopkeeper purchased a badminton racket for ₹650. He sells it at a discount of 20% and still makes a profit of 20%. Find:

(i) the selling price

(ii) marked price.

Answer

Given:

C.P. of the badminton racket = ₹650

Rate of discount = 20%

Profit = 20%

(i) S.P. = $\Big(1 + \dfrac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

S.P. of the badminton racket = $\Big(1 + \dfrac{20}{100}\Big)$ of C.P.

$= \dfrac{120}{100} \times 650 \\[1em] = \dfrac{78000}{100} \\[1em] = ₹780$

Hence, the selling price of the badminton racket is ₹780.

(ii) S.P. of the badminton racket = ₹780, discount = 20%

Let the marked price of the badminton racket be ₹$x$.

S.P. = $\Big(1 - \dfrac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

$\Rightarrow 780 = \Big(1 - \dfrac{20}{100}\Big) \times x \\[1em] \Rightarrow 780 = \dfrac{80}{100} \times x \\[1em] \Rightarrow x = \dfrac{780 \times 100}{80} \\[1em] \Rightarrow x = \dfrac{78000}{80} \\[1em] \Rightarrow x = 975$

Hence, the marked price of the badminton racket is ₹975.

A shopkeeper buys a brass vase for ₹1,200 and marks it 80% above the cost price. If he gives 15% discount on it, find:

(i) the marked price

(ii) the selling price

(iii) his profit percentage

Answer

Given:

C.P. of the brass vase = ₹1,200

The vase is marked 80% above the cost price.

Rate of discount = 15%

(i) M.P. of the brass vase = C.P. + 80% of C.P.

$= 1200 + \dfrac{80}{100} \times 1200 \\[1em] = 1200 + \dfrac{96000}{100} \\[1em] = 1200 + 960 \\[1em] = ₹2{,}160$

Hence, the marked price of the brass vase is ₹2,160.

(ii) S.P. = $\Big(1 - \dfrac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

S.P. of the brass vase = $\Big(1 - \dfrac{15}{100}\Big)$ of M.P.

$= \dfrac{85}{100} \times 2160 \\[1em] = \dfrac{183600}{100} \\[1em] = ₹1{,}836$

Hence, the selling price of the brass vase is ₹1,836.

(iii) Profit = S.P. - C.P.

= ₹1,836 - ₹1,200

= ₹636

Profit percentage = $\dfrac{\text{Profit}}{\text{C.P.}} \times 100$%

= $\dfrac{636}{1200} \times 100$%

= $\dfrac{63600}{1200}$%

= 53%

Hence, his profit percentage is 53%.

The cost price of an article is ₹1,600, which is 20% below the marked price. If the article is sold at a discount of 16%, find:

(i) the marked price

(ii) the selling price

(iii) profit percentage

Answer

Given:

C.P. of the article = ₹1,600, which is 20% below the marked price.

Rate of discount = 16%

(i) Since the cost price is 20% below the marked price,

C.P. = M.P. - 20% of M.P.

$\Rightarrow 1600 = \Big(1 - \dfrac{20}{100}\Big)\text{ of M.P.} \\[1em] \Rightarrow 1600 = \dfrac{80}{100}\text{ of M.P.} \\[1em] \Rightarrow \text{M.P.} = \dfrac{1600 \times 100}{80} \\[1em] \Rightarrow \text{M.P.} = \dfrac{160000}{80} \\[1em] \Rightarrow \text{M.P.} = ₹2{,}000$

Hence, the marked price of the article is ₹2,000.

(ii) S.P. = $\Big(1 - \dfrac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

S.P. of the article = $\Big(1 - \dfrac{16}{100}\Big)$ of M.P.

$= \dfrac{84}{100} \times 2000 \\[1em] = \dfrac{168000}{100} \\[1em] = ₹1{,}680$

Hence, the selling price of the article is ₹1,680.

(iii) Profit = S.P. - C.P.

= ₹1,680 - ₹1,600

= ₹80

Profit percentage = $\dfrac{\text{Profit}}{\text{C.P.}} \times 100$%

= $\dfrac{80}{1600} \times 100$%

= $\dfrac{8000}{1600}$%

= 5%

Hence, the profit percentage is 5%.

A shopkeeper allows 20% discount on his goods and still earns a profit of 20%. If an article is sold for ₹360, find:

(i) the marked price

(ii) the cost price

Answer

Given:

S.P. of the article = ₹360

Rate of discount = 20%

Profit = 20%

(i) S.P. = $\Big(1 - \dfrac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

Let the marked price of the article be ₹$x$.

$\text{S.P.} = \Big(1 - \dfrac{20}{100}\Big)\text{ of M.P.} \\[1em] \Rightarrow 360 = \dfrac{80}{100} \times x \\[1em] \Rightarrow x = \dfrac{360 \times 100}{80} \\[1em] \Rightarrow x = \dfrac{36000}{80} \\[1em] \Rightarrow x = 450$

Hence, the marked price of the article is ₹450.

(ii) S.P. of the article = ₹360, profit = 20%

S.P. = $\Big(1 + \dfrac{\text{Profit Percentage}}{100}\Big) \times$ C.P.

$\Rightarrow 360 = \Big(1 + \dfrac{20}{100}\Big)\text{ of C.P.} \\[1em] \Rightarrow 360 = \dfrac{120}{100}\text{ of C.P.} \\[1em] \Rightarrow \text{C.P.} = \dfrac{360 \times 100}{120} \\[1em] \Rightarrow \text{C.P.} = \dfrac{36000}{120} \\[1em] \Rightarrow \text{C.P.} = ₹300$

Hence, the cost price of the article is ₹300.

The printed price of a refrigerator is ₹28,600. A dealer allows two successive discounts of 10% and 5%. Find the price which a customer has to pay for the refrigerator.

Answer

Given:

Printed price of the refrigerator = ₹28,600

Two successive discounts = 10% and 5%

S.P. = $\Big(1 - \dfrac{d_1}{100}\Big)\Big(1 - \dfrac{d_2}{100}\Big) \times$ M.P., where $d_1$% and $d_2$% are the two successive discounts.

$\text{S.P. of the refrigerator} = \Big(1 - \dfrac{10}{100}\Big)\Big(1 - \dfrac{5}{100}\Big)\text{ of M.P.} \\[1em] = \dfrac{90}{100} \times \dfrac{95}{100} \times 28600 \\[1em] = \dfrac{9}{10} \times \dfrac{19}{20} \times 28600 \\[1em] = \dfrac{4890600}{200} \\[1em] = ₹24{,}453$

Hence, the customer has to pay ₹24,453 for the refrigerator.

Two dealers have marked an article at the same price. The first dealer allows two successive discounts of 15% and 5%. The other allows a discount of 20%. Which is the better offer?

Answer

Given:

Both the dealers have marked the article at the same price.

First dealer allows two successive discounts of 15% and 5%.

Second dealer allows a single discount of 20%.

Let the marked price of the article be ₹100.

For the first dealer :

S.P. = $\Big(1 - \dfrac{d_1}{100}\Big)\Big(1 - \dfrac{d_2}{100}\Big) \times$ M.P., where $d_1$% and $d_2$% are the two successive discounts.

$\text{S.P. of the article} = \Big(1 - \dfrac{15}{100}\Big)\Big(1 - \dfrac{5}{100}\Big) \times 100 \\[1em] = \dfrac{85}{100} \times \dfrac{95}{100} \times 100 \\[1em] = \dfrac{8075}{100} \\[1em] = ₹80.75$

For the second dealer :

S.P. = $\Big(1 - \dfrac{\text{Discount Percentage}}{100}\Big) \times$ M.P.

$\text{S.P. of the article} = \Big(1 - \dfrac{20}{100}\Big) \times 100 \\[1em] = \dfrac{80}{100} \times 100 \\[1em] = ₹80$

As ₹80 < ₹80.75, the article is cheaper with the second dealer.

Hence, the offer of the second dealer is the better offer.

Find a single discount equivalent to two successive discounts of 30% and 10%.

Answer

Given:

Two successive discounts = 30% and 10%

Let the marked price of the article be ₹$x$ and a single discount of $d$% be equivalent to the two given successive discounts of 30% and 10%.

Both the selling prices must be equal.

$\Rightarrow \Big(1 - \dfrac{d}{100}\Big)\text{ of }x = \Big(1 - \dfrac{30}{100}\Big)\Big(1 - \dfrac{10}{100}\Big)\text{ of }x \\[1em] \Rightarrow 1 - \dfrac{d}{100} = \dfrac{70}{100} \times \dfrac{90}{100} \\[1em] \Rightarrow 1 - \dfrac{d}{100} = \dfrac{7}{10} \times \dfrac{9}{10} \\[1em] \Rightarrow 1 - \dfrac{d}{100} = \dfrac{63}{100} \\[1em] \Rightarrow \dfrac{100 - d}{100} = \dfrac{63}{100} \\[1em] \Rightarrow 100 - d = 63 \\[1em] \Rightarrow d = 100 - 63 \\[1em] \Rightarrow d = 37$

Hence, a single discount of 37% is equivalent to the two given successive discounts.