Question:
A shopkeeper loses 46% by selling an article for $1166.40. For how much should he sell it to get a profit of 45%?
Correct Answer
3132
Solution And 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) = $1166.4
And, the loss = 46%
Then, the Sale Price (SP) to earn a profit of 45% = ?
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 – 46 × 1166.4
= 100/54 × 1166.4
= 100 × 1166.4/54
= 116640/54 = 2160
Thus, the Cost Price (CP) = $2160
Finding the Sale Price using the formula when the cost price and Profit percent are known
Here now, CP = $2160
And, profit = 45%
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 + 45/100 × 2160
= 145/100 × 2160
= 145 × 2160/100
= 313200/100 = 3132
Thus, Sale Price (SP) at a profit of 45% = $3132 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) = $1166.4
And, the loss = 46%
Then, the Sale Price (SP) to earn a profit of 45% = ?
Let the Cost Price (CP) = m
Here given loss = 46%
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
= 46% of m
= 46% × m
= 46/100 × m
⇒ loss = 46 m/100
Calculation of Cost Price (CP)
The Sale Price (SP) = Cost Price (CP) – loss
⇒ 1166.4 = m – 46 m/100
⇒ 1166.4 = 100 m – 46 m/100
⇒ 1166.4 = 54 m/100
After cross multiplication, we get
1166.4 × 100 = 54 m
⇒ 54 m = 1166.4 × 100
⇒ m = 1166.4 × 100/54
⇒ m = 116640/54 = 2160
⇒ m = 2160
Thus, the Cost Price (CP) = $2160
Finding the Sale Price (SP) when the Cost Price (CP) and profit percent are known
Now, here, Cost Price = 2160
And, the profit = 45%
Thus, Sale Price (SP)
= Cost Price (CP) + 45% of Cost Price (CP)
= 2160 + 45/100 × 2160
= 2160 + 45 × 2160/100
= 2160 + 97200/100
= 2160 + 972 = 3132
Thus, Sale Price (SP) at a profit of 45% = $3132 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) = $1166.4
And, the loss = 46%
Then, the Sale Price (SP) to earn a profit of 45% = ?
Let, CP = 100
45% profit = 45% of CP
= 45% of 100 = 45
Thus, Sale Price to earn a profit of 45% = CP + 45
= 100 + 45 = 145
⇒ Sale Price to earn a profit of 45% = 145
46% loss = 46% of CP
= 46% of 100 = 46
Thus, Sale Price at a loss of 46% = CP – Loss
= 100 – 46 = 54
⇒ Sale Price at a loss of 46% = 54
∵ If the Sale Price is 54, then to earn a profit of 45% the Sale Price should be = 145
∴ If the Sale Price is 1, then to earn a profit of 45% the Sale Price should be
= 145/54
∴ If the Sale Price is 1166.4, then to earn a profit of 45% the Sale Price should be
= 145/54 × 1166.4
= 145 × 1166.4/54
= 169128/54 = 3132
Thus, Sale Price to get a profit of 45% = $3132 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) = $1166.4
And, the loss = 46%
Then, the Sale Price (SP) to earn a profit of 45% = ?
Here, the Sale Price at a loss of 46 = $1166.4
This means 54% of the Cost Price (CP) = 1166.4
Therefore, 1% of the Cost Price (CP) = 1166.4/54
Therefore, 100% of the Cost Price (CP)
= 1166.4/54 × 100 = 2160
Thus, the Cost Price (CP) = $2160
Now, we need to find the Sale Price at a profit of 45%
Thus, Sale Price at a profit of 45%
= 145% of the Cost Price (CP)
= 145% of 2160
= 145/100 × 2160 = 3132
Thus, the Cost Price at a profit of 45% = $3132 Answer
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