Question

A man buys two cycles for a total cost of Rs.900. By selling one cycle at a loss of 20% and the other at a profit of 25%, he makes a profit of Rs.90 on the whole transaction. Find the cost price of each cycle.
(a) Rs.200; Rs.500
(b) Rs.300; Rs600
(c) Rs.400; Rs.500
(d) Rs.100; Rs.600

Answer 1

Hint: In this question, we need to assume the cost price of each cycle as some variables. Now, the sum of their cost prices should be equated to Rs.900. Then the difference between the sum of the selling prices after profit and loss and the sum of cost prices should be equated to Rs.90 in which the formula for selling price after profit is given by \[SP=\dfrac{100+x}{100}\times CP\] and for selling price of loss is given by \[SP=\dfrac{100-y}{100}\times CP\]

Complete step by step solution:
Let us assume the cost prices of both the cycles as \[C{{P}_{1}}, C{{P}_{2}}\] and their selling prices as \[S{{P}_{1}}, S{{P}_{2}}\]
Now, from the given conditions in the question, we have the sum of cost prices of both the cycles as Rs.900 which can be written as
\[C{{P}_{1}}+C{{P}_{2}}=900.............\left( 1 \right)\]
As we already know that if profit is x%, then
\[SP=\dfrac{100+x}{100}\times CP\]
Here, we also know that if the loss is y%, then
\[SP=\dfrac{100-y}{100}\times CP\]
Here, form the given conditions in the question we have
\[x=25,y=20\]
Now, let us find the selling price of the cycle after the loss of 20%
\[S{{P}_{1}}=\dfrac{100-20}{100}\times C{{P}_{1}}\]
Now, this can be further written in the simplified form as
\[S{{P}_{1}}=\dfrac{4}{5}C{{P}_{1}}\]
Now, let us find the selling price of the other cycle after a profit of 25%
\[S{{P}_{2}}=\dfrac{100+25}{100}\times C{{P}_{2}}\]
Now, this can be further written in the simplified form as
\[S{{P}_{2}}=\dfrac{5}{4}C{{P}_{2}}\]
Now, given in the question that the difference between the sum of the selling prices of both the cycles and the sum of the cost prices is Rs.90 which can be further written as
\[\Rightarrow S{{P}_{2}}+S{{P}_{1}}-\left( C{{P}_{1}}+C{{P}_{2}} \right)=90\]
Now, but substituting the respective values in terms of cost price we get,
\[\Rightarrow \dfrac{5}{4}C{{P}_{2}}+\dfrac{4}{5}C{{P}_{1}}-900=90\]
Now, on rearranging the terms we get,
\[\Rightarrow \dfrac{5}{4}C{{P}_{2}}+\dfrac{4}{5}C{{P}_{1}}=990\]
Now, this can be further written in the simplified form as
\[\Rightarrow 25C{{P}_{2}}+16C{{P}_{1}}=19800......\left( 2 \right)\]
Now, on multiplying the equation (1) with 16 we get,
\[\Rightarrow 16C{{P}_{2}}+16C{{P}_{1}}=14400\]
Now, form the equation (2) we have
\[\begin{align}
  & \Rightarrow 25C{{P}_{2}}+16C{{P}_{1}}=19800 \\
 & \Rightarrow 16C{{P}_{2}}+16C{{P}_{1}}=14400 \\
\end{align}\]
Now, on subtracting these both equations we get,
\[\Rightarrow 9C{{P}_{2}}=5400\]
Now, on dividing both sides with 9 we get,
\[\therefore C{{P}_{2}}=600\]
Now, from equation (1) we have
\[\Rightarrow C{{P}_{1}}+C{{P}_{2}}=900\]
Now, on substituting the respective value we get,
\[\Rightarrow C{{P}_{1}}+600=900\]
Now, on further simplification we get,
\[\therefore C{{P}_{1}}=300\]
Hence, the correct option is (b).

Note:
It is important to note that the profit obtained after selling the cycles should be equated to the difference of the sum of selling price and sum of cost price but no difference between the two selling prices. Because if considered that way then the result will be completely incorrect.
It is also to be noted that the profit and loss should be calculated on the cost price which gives the selling price instead if we consider cost price in place of selling price then the result will be completely incorrect.