Question:
For how much should he sell the bicycle to get a profit of 19%, if a shopkeeper loses 20% by selling it for $688?
Correct Answer
1023.4
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) = $688
And, the loss = 20%
Then, the Sale Price (SP) to earn a profit of 19% = ?
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 – 20 × 688
= 100/80 × 688
= 100 × 688/80
= 68800/80 = 860
Thus, the Cost Price (CP) = $860
Finding the Sale Price using the formula when the cost price and Profit percent are known
Here now, CP = $860
And, profit = 19%
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 + 19/100 × 860
= 119/100 × 860
= 119 × 860/100
= 102340/100 = 1023.4
Thus, Sale Price (SP) at a profit of 19% = $1023.4 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) = $688
And, the loss = 20%
Then, the Sale Price (SP) to earn a profit of 19% = ?
Let the Cost Price (CP) = m
Here given loss = 20%
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
= 20% of m
= 20% × m
= 20/100 × m
⇒ loss = 20 m/100
Calculation of Cost Price (CP)
The Sale Price (SP) = Cost Price (CP) – loss
⇒ 688 = m – 20 m/100
⇒ 688 = 100 m – 20 m/100
⇒ 688 = 80 m/100
After cross multiplication, we get
688 × 100 = 80 m
⇒ 80 m = 688 × 100
⇒ m = 688 × 100/80
⇒ m = 68800/80 = 860
⇒ m = 860
Thus, the Cost Price (CP) = $860
Finding the Sale Price (SP) when the Cost Price (CP) and profit percent are known
Now, here, Cost Price = 860
And, the profit = 19%
Thus, Sale Price (SP)
= Cost Price (CP) + 19% of Cost Price (CP)
= 860 + 19/100 × 860
= 860 + 19 × 860/100
= 860 + 16340/100
= 860 + 163.4 = 1023.4
Thus, Sale Price (SP) at a profit of 19% = $1023.4 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) = $688
And, the loss = 20%
Then, the Sale Price (SP) to earn a profit of 19% = ?
Let, CP = 100
19% profit = 19% of CP
= 19% of 100 = 19
Thus, Sale Price to earn a profit of 19% = CP + 19
= 100 + 19 = 119
⇒ Sale Price to earn a profit of 19% = 119
20% loss = 20% of CP
= 20% of 100 = 20
Thus, Sale Price at a loss of 20% = CP – Loss
= 100 – 20 = 80
⇒ Sale Price at a loss of 20% = 80
∵ If the Sale Price is 80, then to earn a profit of 19% the Sale Price should be = 119
∴ If the Sale Price is 1, then to earn a profit of 19% the Sale Price should be
= 119/80
∴ If the Sale Price is 688, then to earn a profit of 19% the Sale Price should be
= 119/80 × 688
= 119 × 688/80
= 81872/80 = 1023.4
Thus, Sale Price to get a profit of 19% = $1023.4 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) = $688
And, the loss = 20%
Then, the Sale Price (SP) to earn a profit of 19% = ?
Here, the Sale Price at a loss of 20 = $688
This means 80% of the Cost Price (CP) = 688
Therefore, 1% of the Cost Price (CP) = 688/80
Therefore, 100% of the Cost Price (CP)
= 688/80 × 100 = 860
Thus, the Cost Price (CP) = $860
Now, we need to find the Sale Price at a profit of 19%
Thus, Sale Price at a profit of 19%
= 119% of the Cost Price (CP)
= 119% of 860
= 119/100 × 860 = 1023.4
Thus, the Cost Price at a profit of 19% = $1023.4 Answer
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