Question

On selling a tea set at \[5\%\] loss and a lemon set at \[15\%\] gain, a shopkeeper gains Rs.70. If he sells the tea set at \[5\%\] gain and lemon set at \[10\%\] gain, he gains Rs. 130. Find the cost price of the lemon set.

Answer 1

Hint:
Here, we will assume the cost of tea set and lemon set to be some variable. We will use the profit and loss formula to frame the equations for profit from the given conditions. Then by solving the equations we will find the cost price of the lemon set and the cost price of the tea set.

Formula Used:
We will use the following formula:
1. If we are given the profit percentage \[{\rm{profit}}\% \] of the cost price\[C.P.\] , then the profit is given by \[{\rm{profit}} = {\rm{profit}}\% \times C.P.\]
2. If we are given the loss percentage \[{\rm{loss}}\% \] of the cost price\[C.P.\] , then the loss is given by \[{\rm{loss}} = {\rm{loss}}\% \times C.P.\]

Complete Step by step Solution:
Let \[x\] be the cost of tea set and \[y\] be the cost of the lemon set.
We are given that the shopkeeper sells a tea set at \[5\% \] loss and a lemon set at \[15\% \] gain.
By using the profit and loss formula, we get
The loss on a Tea Set \[ = x \times 5\% \]
\[ \Rightarrow \] The loss on a Tea Set \[ = x \times \dfrac{5}{{100}}\]
The gain on a lemon Set \[ = y \times 15\% \]
\[ \Rightarrow \] The gain on a lemon Set \[ = y \times \dfrac{{15}}{{100}}\]
We are given that after selling a tea set at \[5\% \] loss and a lemon set at \[15\% \] gain, the shopkeeper gains \[{\rm{Rs}}.70\] . So, we get
\[\dfrac{{15y}}{{100}} - \dfrac{{5x}}{{100}} = 70\]
By cross multiplying, we get
\[ \Rightarrow 15y - 5x = 7000\]
\[ \Rightarrow - 5x + 15y = 7000\] ……………………………………………………………………………………\[\left( 1 \right)\]
We are given that the shopkeeper sells a tea set at \[5\% \] gain and a lemon set at \[10\% \] gain.
By using the profit formula, we get
The gain on a Tea Set \[ = x \times 5\% \]
\[ \Rightarrow \] The gain on a Tea Set \[ = x \times \dfrac{5}{{100}}\]
The gain on a lemon Set \[ = y \times 10\% \]
\[ \Rightarrow \] So, the gain on a lemon Set\[ = y \times \dfrac{{10}}{{100}}\]
We are given that after selling a tea set at \[5\% \] gain and a lemon set at \[10\% \] gain, the shopkeeper gains \[{\rm{Rs}}.130\] . So, we get
\[\dfrac{{5x}}{{100}} + \dfrac{{10y}}{{100}} = 130\]
By cross multiplying, we get
\[ \Rightarrow 5x + 10y = 13000\] …………………………………………………………………………………………….\[\left( 2 \right)\]
By adding equation \[\left( 1 \right)\] and equation \[\left( 2 \right)\], we get
\[25y = 20000\]
By dividing the equation, we get
\[ \Rightarrow y = \dfrac{{20000}}{{25}}\]
\[ \Rightarrow y = 800\]
Now, substituting \[y = 800\] in equation \[\left( 1 \right)\], we get
\[ - 5x + 15\left( {800} \right) = 7000\]
By multiplying the numbers, we get
\[ \Rightarrow - 5x + 12000 = 7000\]
By rewriting the equation, we get
\[ \Rightarrow 5x = 12000 - 7000\]
Subtracting the terms, we get
\[ \Rightarrow 5x = 5000\]
Dividing both sides by 5, we get
\[ \Rightarrow x = \dfrac{{5000}}{5}\]
\[ \Rightarrow x = 1000\]
Therefore, the cost price of a tea set is Rs 1000 and the cost price of a lemon set is Rs 800.

Therefore, the cost price of a lemon set is Rs 800.

Note:
We know that the cost price is the price of an item at which an item is bought. The selling price is the price of an item at which an item is sold. If the selling price is greater than the cost price, then there is a profit. If the selling price is less than the cost price, then there is a loss. Profit or loss percentage is calculated only for the same number of items. Both the percentages are calculated over the cost price of an item.