Question

A cycle is sold for $7880$at a loss of $20\% $. For how much should it be sold to gain $10\% $?

Answer 1

Hint: First, let the cost price of the cycle and evaluate it from the given selling price and the loss percent. Evaluate the profit on the cost price and add it to the cost price to get the selling price of the cycle.

Complete step by step answer:
We are given that cycle is sold for $7880$at a loss of $20\% $.
First, we let the cost price of the cycle be $x$.
Profit and loss are calculated on the cost price.
Evaluate the loss $20\% $ on the cost price.
Loss will be $20\% $ of $x$.
That is, $\dfrac{{20}}{{100}} \times x = \dfrac{x}{5}$
Cost price is equal to the sum of the loss and the selling price.
The selling price is $7880$ and the loss is $\dfrac{x}{5}$. Write the mathematical equation for the cost price.
\[x = 7880 + \dfrac{x}{5}\]
Solve for $x$.
\[
  x - \dfrac{x}{5} = 7880 \\
  \dfrac{{4x}}{5} = 7880 \\
  x = 9850 \\
 \]
Therefore, the cost price of the cycle is $9850$.
Evaluate the profit $10\% $ on the cost price.
Profit will be $10\% $ of $9850$.
That is, $\dfrac{{10}}{{100}} \times 9850 = 985$
Selling price is equal to the sum of the profit and the cost price.
Therefore, selling price will be $9850 + 985 = 10835$
Hence, the cycle should be sold at the price of $10835$to gain $10\% $.

Note: We can evaluate the cost price of the cycle by another method which is shown below:
Let the cost price of the cycle is $100$.
Since the loss is $20\% $, therefore the selling price will be $80$.
The value of $80$is $7880$ as the cycle is sold at $7880$
To evaluate the cost price, evaluate the value of $100$.
$
  80 \to 7880 \\
  1 \to \dfrac{{7880}}{{80}} \\
  100 \to \dfrac{{7880}}{{80}} \times 100 = 9850 \\
 $
Therefore, the cost price of the cycle is $9850$.