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 dealer buys two articles X and Y for ₹2,000 each. He marks each of them at the same price. He sells X by giving two successive discounts of 44% and 37% and still earns ₹989 as profit. If he sells Y at a single discount of 64%, then what is the profit percentage on Y?
- 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?
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.
Raman fixes the sale price of his goods at 16% above the cost price. He sells his goods at 12% less than the fixed price. Find the profit percentage correct to two places of decimal.
- A dishonest dealer professes to sell grains at cost price, but he uses a weight of 925 g for 1 kg weight. Find his gain percentage. (Approximate to two decimals.)
