Profit And Loss
Math MCQs

Question :    By selling a bicycle for $443.70 a shopkeeper loses 13%. For how much should he sell it to get a profit of 12%?

Correct Answer  571.2

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) = $443.7

And, the loss = 13%

Then, the Sale Price (SP) to earn a profit of 12% = ?

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 – 13} × 443.7

= ¹⁰⁰/₈₇ × 443.7

= ^{100 × 443.7}/₈₇

= ⁴⁴³⁷⁰/₈₇ = 510

Thus, the Cost Price (CP) = $510

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

Here now, CP = $510

And, profit = 12%

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 + 12}/₁₀₀ × 510

= ¹¹²/₁₀₀ × 510

= ^{112 × 510}/₁₀₀

= ⁵⁷¹²⁰/₁₀₀ = 571.2

Thus, Sale Price (SP) at a profit of 12% = $571.2 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) = $443.7

And, the loss = 13%

Then, the Sale Price (SP) to earn a profit of 12% = ?

Let the Cost Price (CP) = m

Here given loss = 13%

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

= 13% of m

= 13% × m

= ¹³/₁₀₀ × m

⇒ loss = ^{13 m}/₁₀₀

Calculation of Cost Price (CP)

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

⇒ 443.7 = m – ^{13 m}/₁₀₀

⇒ 443.7 = ^{100 m – 13 m}/₁₀₀

⇒ 443.7 = ^{87 m}/₁₀₀

After cross multiplication, we get

443.7 × 100 = 87 m

⇒ 87 m = 443.7 × 100

⇒ m = ^{443.7 × 100}/₈₇

⇒ m = ⁴⁴³⁷⁰/₈₇ = 510

⇒ m = 510

Thus, the Cost Price (CP) = $510

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

Now, here, Cost Price = 510

And, the profit = 12%

Thus, Sale Price (SP)

= Cost Price (CP) + 12% of Cost Price (CP)

= 510 + ¹²/₁₀₀ × 510

= 510 + ^{12 × 510}/₁₀₀

= 510 + ⁶¹²⁰/₁₀₀

= 510 + 61.2 = 571.2

Thus, Sale Price (SP) at a profit of 12% = $571.2 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) = $443.7

And, the loss = 13%

Then, the Sale Price (SP) to earn a profit of 12% = ?

Let, CP = 100

12% profit = 12% of CP

= 12% of 100 = 12

Thus, Sale Price to earn a profit of 12% = CP + 12

= 100 + 12 = 112

⇒ Sale Price to earn a profit of 12% = 112

13% loss = 13% of CP

= 13% of 100 = 13

Thus, Sale Price at a loss of 13% = CP – Loss

= 100 – 13 = 87

⇒ Sale Price at a loss of 13% = 87

∵ If the Sale Price is 87, then to earn a profit of 12% the Sale Price should be = 112

∴ If the Sale Price is 1, then to earn a profit of 12% the Sale Price should be

= ¹¹²/₈₇

∴ If the Sale Price is 443.7, then to earn a profit of 12% the Sale Price should be

= ¹¹²/₈₇ × 443.7

= ^{112 × 443.7}/₈₇

= ^49694.4/₈₇ = 571.2

Thus, Sale Price to get a profit of 12% = $571.2 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) = $443.7

And, the loss = 13%

Then, the Sale Price (SP) to earn a profit of 12% = ?

Here, the Sale Price at a loss of 13 = $443.7

This means 87% of the Cost Price (CP) = 443.7

Therefore, 1% of the Cost Price (CP) = ^443.7/₈₇

Therefore, 100% of the Cost Price (CP)

= ^443.7/₈₇ × 100 = 510

Thus, the Cost Price (CP) = $510

Now, we need to find the Sale Price at a profit of 12%

Thus, Sale Price at a profit of 12%

= 112% of the Cost Price (CP)

= 112% of 510

= ¹¹²/₁₀₀ × 510 = 571.2

Thus, the Cost Price at a profit of 12% = $571.2 Answer

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