Profit And Loss
Math MCQs

Question :    Mrs. Mary loses 19% by selling an article for $656.10. For how much should she sell it to get a profit of 18%?

Correct Answer  955.8

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) = $656.1

And, the loss = 19%

Then, the Sale Price (SP) to earn a profit of 18% = ?

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 – 19} × 656.1

= ¹⁰⁰/₈₁ × 656.1

= ^{100 × 656.1}/₈₁

= ⁶⁵⁶¹⁰/₈₁ = 810

Thus, the Cost Price (CP) = $810

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

Here now, CP = $810

And, profit = 18%

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 + 18}/₁₀₀ × 810

= ¹¹⁸/₁₀₀ × 810

= ^{118 × 810}/₁₀₀

= ⁹⁵⁵⁸⁰/₁₀₀ = 955.8

Thus, Sale Price (SP) at a profit of 18% = $955.8 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) = $656.1

And, the loss = 19%

Then, the Sale Price (SP) to earn a profit of 18% = ?

Let the Cost Price (CP) = m

Here given loss = 19%

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

= 19% of m

= 19% × m

= ¹⁹/₁₀₀ × m

⇒ loss = ^{19 m}/₁₀₀

Calculation of Cost Price (CP)

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

⇒ 656.1 = m – ^{19 m}/₁₀₀

⇒ 656.1 = ^{100 m – 19 m}/₁₀₀

⇒ 656.1 = ^{81 m}/₁₀₀

After cross multiplication, we get

656.1 × 100 = 81 m

⇒ 81 m = 656.1 × 100

⇒ m = ^{656.1 × 100}/₈₁

⇒ m = ⁶⁵⁶¹⁰/₈₁ = 810

⇒ m = 810

Thus, the Cost Price (CP) = $810

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

Now, here, Cost Price = 810

And, the profit = 18%

Thus, Sale Price (SP)

= Cost Price (CP) + 18% of Cost Price (CP)

= 810 + ¹⁸/₁₀₀ × 810

= 810 + ^{18 × 810}/₁₀₀

= 810 + ¹⁴⁵⁸⁰/₁₀₀

= 810 + 145.8 = 955.8

Thus, Sale Price (SP) at a profit of 18% = $955.8 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) = $656.1

And, the loss = 19%

Then, the Sale Price (SP) to earn a profit of 18% = ?

Let, CP = 100

18% profit = 18% of CP

= 18% of 100 = 18

Thus, Sale Price to earn a profit of 18% = CP + 18

= 100 + 18 = 118

⇒ Sale Price to earn a profit of 18% = 118

19% loss = 19% of CP

= 19% of 100 = 19

Thus, Sale Price at a loss of 19% = CP – Loss

= 100 – 19 = 81

⇒ Sale Price at a loss of 19% = 81

∵ If the Sale Price is 81, then to earn a profit of 18% the Sale Price should be = 118

∴ If the Sale Price is 1, then to earn a profit of 18% the Sale Price should be

= ¹¹⁸/₈₁

∴ If the Sale Price is 656.1, then to earn a profit of 18% the Sale Price should be

= ¹¹⁸/₈₁ × 656.1

= ^{118 × 656.1}/₈₁

= ^77419.8/₈₁ = 955.8

Thus, Sale Price to get a profit of 18% = $955.8 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) = $656.1

And, the loss = 19%

Then, the Sale Price (SP) to earn a profit of 18% = ?

Here, the Sale Price at a loss of 19 = $656.1

This means 81% of the Cost Price (CP) = 656.1

Therefore, 1% of the Cost Price (CP) = ^656.1/₈₁

Therefore, 100% of the Cost Price (CP)

= ^656.1/₈₁ × 100 = 810

Thus, the Cost Price (CP) = $810

Now, we need to find the Sale Price at a profit of 18%

Thus, Sale Price at a profit of 18%

= 118% of the Cost Price (CP)

= 118% of 810

= ¹¹⁸/₁₀₀ × 810 = 955.8

Thus, the Cost Price at a profit of 18% = $955.8 Answer

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