Profit And Loss
Math MCQs

Question :    Mr. Joseph loses 5% by selling an article for $104.50. For how much should he sell it to get a profit of 4%?

Correct Answer  114.4

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) = $104.5

And, the loss = 5%

Then, the Sale Price (SP) to earn a profit of 4% = ?

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 – 5} × 104.5

= ¹⁰⁰/₉₅ × 104.5

= ^{100 × 104.5}/₉₅

= ¹⁰⁴⁵⁰/₉₅ = 110

Thus, the Cost Price (CP) = $110

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

Here now, CP = $110

And, profit = 4%

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 + 4}/₁₀₀ × 110

= ¹⁰⁴/₁₀₀ × 110

= ^{104 × 110}/₁₀₀

= ¹¹⁴⁴⁰/₁₀₀ = 114.4

Thus, Sale Price (SP) at a profit of 4% = $114.4 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) = $104.5

And, the loss = 5%

Then, the Sale Price (SP) to earn a profit of 4% = ?

Let the Cost Price (CP) = m

Here given loss = 5%

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

= 5% of m

= 5% × m

= ⁵/₁₀₀ × m

⇒ loss = ^{5 m}/₁₀₀

Calculation of Cost Price (CP)

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

⇒ 104.5 = m – ^{5 m}/₁₀₀

⇒ 104.5 = ^{100 m – 5 m}/₁₀₀

⇒ 104.5 = ^{95 m}/₁₀₀

After cross multiplication, we get

104.5 × 100 = 95 m

⇒ 95 m = 104.5 × 100

⇒ m = ^{104.5 × 100}/₉₅

⇒ m = ¹⁰⁴⁵⁰/₉₅ = 110

⇒ m = 110

Thus, the Cost Price (CP) = $110

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

Now, here, Cost Price = 110

And, the profit = 4%

Thus, Sale Price (SP)

= Cost Price (CP) + 4% of Cost Price (CP)

= 110 + ⁴/₁₀₀ × 110

= 110 + ^{4 × 110}/₁₀₀

= 110 + ⁴⁴⁰/₁₀₀

= 110 + 4.4 = 114.4

Thus, Sale Price (SP) at a profit of 4% = $114.4 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) = $104.5

And, the loss = 5%

Then, the Sale Price (SP) to earn a profit of 4% = ?

Let, CP = 100

4% profit = 4% of CP

= 4% of 100 = 4

Thus, Sale Price to earn a profit of 4% = CP + 4

= 100 + 4 = 104

⇒ Sale Price to earn a profit of 4% = 104

5% loss = 5% of CP

= 5% of 100 = 5

Thus, Sale Price at a loss of 5% = CP – Loss

= 100 – 5 = 95

⇒ Sale Price at a loss of 5% = 95

∵ If the Sale Price is 95, then to earn a profit of 4% the Sale Price should be = 104

∴ If the Sale Price is 1, then to earn a profit of 4% the Sale Price should be

= ¹⁰⁴/₉₅

∴ If the Sale Price is 104.5, then to earn a profit of 4% the Sale Price should be

= ¹⁰⁴/₉₅ × 104.5

= ^{104 × 104.5}/₉₅

= ¹⁰⁸⁶⁸/₉₅ = 114.4

Thus, Sale Price to get a profit of 4% = $114.4 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) = $104.5

And, the loss = 5%

Then, the Sale Price (SP) to earn a profit of 4% = ?

Here, the Sale Price at a loss of 5 = $104.5

This means 95% of the Cost Price (CP) = 104.5

Therefore, 1% of the Cost Price (CP) = ^104.5/₉₅

Therefore, 100% of the Cost Price (CP)

= ^104.5/₉₅ × 100 = 110

Thus, the Cost Price (CP) = $110

Now, we need to find the Sale Price at a profit of 4%

Thus, Sale Price at a profit of 4%

= 104% of the Cost Price (CP)

= 104% of 110

= ¹⁰⁴/₁₀₀ × 110 = 114.4

Thus, the Cost Price at a profit of 4% = $114.4 Answer

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