Question

Jamila sold a table and a chair for Rs. 1050, thereby making a profit of 10% on a table and 25% on the chair. If she had taken a profit of 25% on the table and 10% on the chair, she would have got Rs. 1065. Find the cost price of each.

Answer 1

Hint:
Here, we will find the cost price of the table and chair. We will use the profit formula and selling price formula to frame the equations for selling price from the given conditions and then by solving the equations we will find the cost price of a table and cost price of a chair.

Formula Used:
We will use the following formula:
1) \[{\rm{Profit}\% } = \dfrac{{{\rm{Profit}}}}{{C.P.}} \times 100\] where, the profit percentage is \[{\rm{Profit}}\% \] and cost price \[C.P.\].
2) Selling price is given by the formula \[S.P. = C.P. + {\rm{Profit}}\]

Complete Step by step Solution:
Let \[x\] be the cost of a table and \[y\] be the cost of a chair.
We are given that Jamila sells a table at \[10\% \] profit and a chair at \[25\% \] profit.
By using the profit formula, we get
The profit on a Table \[ = x \times 10\% \]
\[ \Rightarrow \] The profit on a Table \[ = x \times \dfrac{{10}}{{100}}\]
By using the selling price formula, we get
Now, the Selling Price on a Table \[ = C.P.\] of the table \[+ {\rm{Profit}}\]
 \[ \Rightarrow \] Selling Price on a Table \[ = x + \dfrac{{10x}}{{100}} = \dfrac{{110x}}{{100}}\]
\[ \Rightarrow \] Selling Price on a Table \[ = 1.1x\]
By using the profit formula, we get
The profit on a chair \[ = y \times 25\% \]
\[ \Rightarrow \] The profit on a chair\[ = y \times \dfrac{{25}}{{100}}\]
By using the selling price formula, we get
Now, the Selling Price on a Chair \[ = C.P.\] of the chair \[+ {\rm{Profit}}\]
 \[ \Rightarrow \] Selling Price on a chair \[ = y + \dfrac{{25y}}{{100}} = \dfrac{{125y}}{{100}}\]
\[ \Rightarrow \] Selling Price on a chair\[ = 1.25y\]
We are given that after selling a table at \[10\% \]profit and a chair at \[25\% \] profit, Jamila profits \[Rs.1050\] . So, we get
\[ \Rightarrow 1.1x + 1.25y = 1050\] ……………………………………………………………………………………\[\left( 1 \right)\]
We are given that Jamila sells a table at \[25\% \] profit and a chair at \[10\% \] profit.
By using the profit formula, we get
\[ \Rightarrow \] So, the profit on a Table \[ = x \times 25\% \]
\[ \Rightarrow \] So, the profit on a Table \[ = x \times \dfrac{{25}}{{100}}\]
By using the selling price formula, we get
Now, the new Selling Price on a Table \[ = C.P.\] of the table \[+ {\rm{Profit}}\]
 \[ \Rightarrow \] Selling Price on a Table \[ = x + \dfrac{{25x}}{{100}} = \dfrac{{125x}}{{100}}\]
\[ \Rightarrow \] Selling Price on a Table \[ = 1.25x\]
By using the profit formula, we get
The profit on a chair \[ = y \times 10\% \]
\[ \Rightarrow \] The profit on a chair \[ = y \times \dfrac{{10}}{{100}}\]
By using the selling price formula, we get
Now, the new Selling Price on a Chair \[ = C.P.\] of the chair \[+ {\rm{Profit}}\]
 \[ \Rightarrow \] Selling Price on a chair \[ = y + \dfrac{{10y}}{{100}} = \dfrac{{110y}}{{100}}\]
\[ \Rightarrow \] Selling Price on a chair \[ = 1.1y\]
We are given that after selling a table at \[25\% \] profit and a chair at \[10\% \] profit, Jamila profits \[Rs.1065\]. So, we get
\[1.25x + 1.1y = 1065\] …………………………………………………………………………………………….\[\left( 2 \right)\]
Multiplying the equation \[\left( 1 \right)\] by \[1.25\] and equation \[\left( 2 \right)\] by \[1.1\] , we get
\[ \Rightarrow \left( 1 \right) \times 1.25 \Rightarrow 1.375x + 1.5625y = 1312.5\]
\[ \Rightarrow \left( 2 \right) \times 1.1 \Rightarrow 1.375x + 1.21y = 1171.5\]
Subtracting these equations, the variable \[x\] gets cancelled since they are opposite in signs, we get
\[ \Rightarrow 0.3525y = 141\]
By dividing the equation, we get
\[ \Rightarrow y = \dfrac{{141}}{{0.3525}}\]
\[ \Rightarrow y = 400\]
Now, substituting \[y = 400\] in equation \[\left( 1 \right)\], we get
\[ \Rightarrow 1.1x + 1.25\left( {400} \right) = 1050\]
By multiplying the numbers, we get
\[ \Rightarrow 1.1x + 500 = 1050\]
By rewriting the equation, we get
\[ \Rightarrow 1.1x = 1050 - 500\]
Subtracting the terms, we get
\[ \Rightarrow 1.1x = 550\]
Dividing both sides by \[1.1\], we get
\[ \Rightarrow x = \dfrac{{550}}{{1.1}}\]
\[ \Rightarrow x = 500\]

Therefore, the cost price of a table is \[Rs.500\] and the cost price of a chair is \[Rs.400\].

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. Profit is denoted by positive sign and Loss is denoted by negative sign.