Profit And Loss
Math MCQs

Question :    William loses 43% by selling an article for $1145.70. For how much should he sell it to get a profit of 42%?

Correct Answer  2854.2

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) = $1145.7

And, the loss = 43%

Then, the Sale Price (SP) to earn a profit of 42% = ?

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 – 43} × 1145.7

= ¹⁰⁰/₅₇ × 1145.7

= ^{100 × 1145.7}/₅₇

= ¹¹⁴⁵⁷⁰/₅₇ = 2010

Thus, the Cost Price (CP) = $2010

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

Here now, CP = $2010

And, profit = 42%

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 + 42}/₁₀₀ × 2010

= ¹⁴²/₁₀₀ × 2010

= ^{142 × 2010}/₁₀₀

= ²⁸⁵⁴²⁰/₁₀₀ = 2854.2

Thus, Sale Price (SP) at a profit of 42% = $2854.2 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) = $1145.7

And, the loss = 43%

Then, the Sale Price (SP) to earn a profit of 42% = ?

Let the Cost Price (CP) = m

Here given loss = 43%

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

= 43% of m

= 43% × m

= ⁴³/₁₀₀ × m

⇒ loss = ^{43 m}/₁₀₀

Calculation of Cost Price (CP)

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

⇒ 1145.7 = m – ^{43 m}/₁₀₀

⇒ 1145.7 = ^{100 m – 43 m}/₁₀₀

⇒ 1145.7 = ^{57 m}/₁₀₀

After cross multiplication, we get

1145.7 × 100 = 57 m

⇒ 57 m = 1145.7 × 100

⇒ m = ^{1145.7 × 100}/₅₇

⇒ m = ¹¹⁴⁵⁷⁰/₅₇ = 2010

⇒ m = 2010

Thus, the Cost Price (CP) = $2010

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

Now, here, Cost Price = 2010

And, the profit = 42%

Thus, Sale Price (SP)

= Cost Price (CP) + 42% of Cost Price (CP)

= 2010 + ⁴²/₁₀₀ × 2010

= 2010 + ^{42 × 2010}/₁₀₀

= 2010 + ⁸⁴⁴²⁰/₁₀₀

= 2010 + 844.2 = 2854.2

Thus, Sale Price (SP) at a profit of 42% = $2854.2 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) = $1145.7

And, the loss = 43%

Then, the Sale Price (SP) to earn a profit of 42% = ?

Let, CP = 100

42% profit = 42% of CP

= 42% of 100 = 42

Thus, Sale Price to earn a profit of 42% = CP + 42

= 100 + 42 = 142

⇒ Sale Price to earn a profit of 42% = 142

43% loss = 43% of CP

= 43% of 100 = 43

Thus, Sale Price at a loss of 43% = CP – Loss

= 100 – 43 = 57

⇒ Sale Price at a loss of 43% = 57

∵ If the Sale Price is 57, then to earn a profit of 42% the Sale Price should be = 142

∴ If the Sale Price is 1, then to earn a profit of 42% the Sale Price should be

= ¹⁴²/₅₇

∴ If the Sale Price is 1145.7, then to earn a profit of 42% the Sale Price should be

= ¹⁴²/₅₇ × 1145.7

= ^{142 × 1145.7}/₅₇

= ^162689.4/₅₇ = 2854.2

Thus, Sale Price to get a profit of 42% = $2854.2 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) = $1145.7

And, the loss = 43%

Then, the Sale Price (SP) to earn a profit of 42% = ?

Here, the Sale Price at a loss of 43 = $1145.7

This means 57% of the Cost Price (CP) = 1145.7

Therefore, 1% of the Cost Price (CP) = ^1145.7/₅₇

Therefore, 100% of the Cost Price (CP)

= ^1145.7/₅₇ × 100 = 2010

Thus, the Cost Price (CP) = $2010

Now, we need to find the Sale Price at a profit of 42%

Thus, Sale Price at a profit of 42%

= 142% of the Cost Price (CP)

= 142% of 2010

= ¹⁴²/₁₀₀ × 2010 = 2854.2

Thus, the Cost Price at a profit of 42% = $2854.2 Answer

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