Profit And Loss
Math MCQs

Question :    A shopkeeper loses 51% by selling an article for $1180.90. For how much should he sell it to get a profit of 50%?

Correct Answer  3615

Solution & Explanation

Strategy to solve the question

Step (1) Calculate the cost price at given loss percent and the sale price.

Step (2) After gettting the cost price find the sale price at given profit percent.

Solution

Method (1) Formula Method

Finding the Sale Price to earn a profit using formula when the Sale Price at a loss is given

Given, the sale price (SP) = $1180.9

And, the loss = 51%

Then, the Sale Price (SP) to earn a profit of 50% = ?

Formula to find the cost price when the sale price and loss percent are known

Cost Price (CP) = ¹⁰⁰/_{100 – gain%} × SP

Thus, the Cost Price (CP)

= ¹⁰⁰/_{100 – 51} × 1180.9

= ¹⁰⁰/₄₉ × 1180.9

= ^{100 × 1180.9}/₄₉

= ¹¹⁸⁰⁹⁰/₄₉ = 2410

Thus, the Cost Price (CP) = $2410

Finding the Sale Price using the formula when the cost price and Profit percent are known

Here now, CP = $2410

And, profit = 50%

Thus Sale Price (SP) = ?

Formula to find the Sale Price (SP) when the cost price (CP) and profit percent are known

Sale Price (SP) = ^{100 + gain%}/₁₀₀ × Cost Price (CP)

Thus, the Sale Price (SP)

= ^{100 + 50}/₁₀₀ × 2410

= ¹⁵⁰/₁₀₀ × 2410

= ^{150 × 2410}/₁₀₀

= ³⁶¹⁵⁰⁰/₁₀₀ = 3615

Thus, Sale Price (SP) at a profit of 50% = $3615 Answer

Method (2) Algebraic Method

Finding the Sale Price (SP) to earn the given profit using the Algebraic Method when the Sale Pric (SP) at a loss percent is given

Given, the sale price (SP) = $1180.9

And, the loss = 51%

Then, the Sale Price (SP) to earn a profit of 50% = ?

Let the Cost Price (CP) = m

Here given loss = 51%

Calculation of loss

loss is calculated on cost price (CP)

Formula to find the loss = loss% of CP

Thus, loss on the cost price of m

= 51% of m

= 51% × m

= ⁵¹/₁₀₀ × m

⇒ loss = ^{51 m}/₁₀₀

Calculation of Cost Price (CP)

The Sale Price (SP) = Cost Price (CP) – loss

⇒ 1180.9 = m – ^{51 m}/₁₀₀

⇒ 1180.9 = ^{100 m – 51 m}/₁₀₀

⇒ 1180.9 = ^{49 m}/₁₀₀

After cross multiplication, we get

1180.9 × 100 = 49 m

⇒ 49 m = 1180.9 × 100

⇒ m = ^{1180.9 × 100}/₄₉

⇒ m = ¹¹⁸⁰⁹⁰/₄₉ = 2410

⇒ m = 2410

Thus, the Cost Price (CP) = $2410

Finding the Sale Price (SP) when the Cost Price (CP) and profit percent are known

Now, here, Cost Price = 2410

And, the profit = 50%

Thus, Sale Price (SP)

= Cost Price (CP) + 50% of Cost Price (CP)

= 2410 + ⁵⁰/₁₀₀ × 2410

= 2410 + ^{50 × 2410}/₁₀₀

= 2410 + ¹²⁰⁵⁰⁰/₁₀₀

= 2410 + 1205 = 3615

Thus, Sale Price (SP) at a profit of 50% = $3615 Answer

Method (3) Unitary Method

Finding the Sale Price (SP) to earn a given profit when the Sale price (SP) at a loss is given

Given, the sale price (SP) = $1180.9

And, the loss = 51%

Then, the Sale Price (SP) to earn a profit of 50% = ?

Let, CP = 100

50% profit = 50% of CP

= 50% of 100 = 50

Thus, Sale Price to earn a profit of 50% = CP + 50

= 100 + 50 = 150

⇒ Sale Price to earn a profit of 50% = 150

51% loss = 51% of CP

= 51% of 100 = 51

Thus, Sale Price at a loss of 51% = CP – Loss

= 100 – 51 = 49

⇒ Sale Price at a loss of 51% = 49

∵ If the Sale Price is 49, then to earn a profit of 50% the Sale Price should be = 150

∴ If the Sale Price is 1, then to earn a profit of 50% the Sale Price should be

= ¹⁵⁰/₄₉

∴ If the Sale Price is 1180.9, then to earn a profit of 50% the Sale Price should be

= ¹⁵⁰/₄₉ × 1180.9

= ^{150 × 1180.9}/₄₉

= ¹⁷⁷¹³⁵/₄₉ = 3615

Thus, Sale Price to get a profit of 50% = $3615 Answer

Method (4) Shortcut Method

Shortcut Method to find the Sale Price at a given profit when Sale Price at a loss is given

Given, the sale price (SP) = $1180.9

And, the loss = 51%

Then, the Sale Price (SP) to earn a profit of 50% = ?

Here, the Sale Price at a loss of 51 = $1180.9

This means 49% of the Cost Price (CP) = 1180.9

Therefore, 1% of the Cost Price (CP) = ^1180.9/₄₉

Therefore, 100% of the Cost Price (CP)

= ^1180.9/₄₉ × 100 = 2410

Thus, the Cost Price (CP) = $2410

Now, we need to find the Sale Price at a profit of 50%

Thus, Sale Price at a profit of 50%

= 150% of the Cost Price (CP)

= 150% of 2410

= ¹⁵⁰/₁₀₀ × 2410 = 3615

Thus, the Cost Price at a profit of 50% = $3615 Answer

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