Understanding the Bicycle Sales Problem
The problem involves calculating the cost price (CP) of the first bicycle given the total cost price of two bicycles, the individual profit/loss percentages on each, and the fact that there was no overall profit or loss.
Defining Variables and Given Information
- Let the cost price of the first bicycle be CP1.
- Let the cost price of the second bicycle be CP2.
- The total cost price for both bicycles is ₹1,200. So, CP1 + CP2 = 1200.
- The first bicycle was sold at a loss of 13%.
- The second bicycle was sold at a gain of 11%.
- There was neither a loss nor a gain overall, meaning the total selling price (SP) equals the total cost price (CP). Total SP = ₹1,200.
Calculating Selling Prices
The formula for selling price (SP) when there is a loss is:
SP = CP × (1 - Loss%/100)
The formula for selling price (SP) when there is a gain is:
SP = CP × (1 + Gain%/100)
- Selling Price of the first bicycle (SP1):
SP1 = CP1 × (1 - 13/100) = CP1 × (1 - 0.13) = 0.87 × CP1 - Selling Price of the second bicycle (SP2):
SP2 = CP2 × (1 + 11/100) = CP2 × (1 + 0.11) = 1.11 × CP2
Setting up the Equation
Since there was no overall profit or loss, the sum of the selling prices equals the total cost price:
SP1 + SP2 = Total CP
Substituting the expressions for SP1 and SP2:
(0.87 × CP1) + (1.11 × CP2) = 1200
Solving for the Cost Price of the First Bicycle (CP1)
We know that CP2 = 1200 - CP1. Substitute this into the equation:
0.87 × CP1 + 1.11 × (1200 - CP1) = 1200
Now, solve for CP1:
- Distribute 1.11:
0.87 × CP1 + (1.11 × 1200) - (1.11 × CP1) = 1200
0.87 × CP1 + 1332 - 1.11 × CP1 = 1200 - Combine the terms with CP1:
(0.87 - 1.11) × CP1 + 1332 = 1200
-0.24 × CP1 + 1332 = 1200 - Isolate the term with CP1:
-0.24 × CP1 = 1200 - 1332
-0.24 × CP1 = -132 - Solve for CP1:
CP1 = -132 / -0.24
CP1 = 132 / 0.24
CP1 = 13200 / 24
CP1 = 550
Therefore, the cost price of the first bicycle is ₹550.
Final Answer Check
If CP1 = 550, then CP2 = 1200 - 550 = 650.
SP1 = 550 * (1 - 0.13) = 550 * 0.87 = 478.50
SP2 = 650 * (1 + 0.11) = 650 * 1.11 = 721.50
Total SP = SP1 + SP2 = 478.50 + 721.50 = 1200.
Total CP = 1200.
Since Total SP = Total CP, the calculation is correct.
Important Questions from Profit & Loss
A shopkeeper offers the following discount schemes for buyers.
I. Two successive discounts of 15% and 20%
II. Successive discount of 25% and 10%
III. A discount of 33%
The selling price will be minimum under the scheme:A shopkeeper sold 5/8 of his articles at a gain of 20% and the remaining at the cost price. What is his gain percentage in the whole transaction?
The marked price of 55 items was equal to the cost price of 99 items. The selling price of 56 items was equal to the marked price of 35 items. Calculate the profit or loss percentage from the sale of each item.
In an election between two candidates, P received 42% of the valid votes and Q won by 45,360 votes. 20% people did not cast their vote. If 10% of the votes cast were found invalid, what is the total number of votes registered in the poll booth?
- A dealer buys two articles X and Y for ₹1,500 each. He marks each of them at the same price. He sells X by giving two successive discounts of 25% and 44% and still earns 279 as profit. If he sells Y at a single discount of 58%, then what is the profit percentage on Y?
