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On selling a T.V. at $5\% $ gain and a fridge at $10\% $ gain, a shopkeeper gains Rs.2000. But if he sells the T.V at $10\% $ gain and the fridge at $5\% $ loss, he gains Rs.1500 on the transaction. Find the actual price of the T.V. and the fridge.

Answer
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Hint:

Such questions are easily solved by writing down the profit or loss in terms of linear equations and then solving those equations. In most cases, assume the cost price of the articles sold to be a variable(s) and then write down the equations. Now, let us approach the solution by writing down the linear equations.

Complete step-by-step answer:
Let the actual price (cost price) of the T.V. be Rs. X and the fridge be Rs. Y

CASE - $1$
Profit on T.V. sold at $5\% $ gain = $\frac{{5X}}{{100}}$
Profit on Fridge sold at $10\% $ gain = $\frac{{10Y}}{{100}}$ 
As per the question, the shopkeeper gains Rs. 2000 in the first case.
Hence, we can represent case - $1$ in equation form as
$\
   \Rightarrow \frac{{5X}}{{100}} + \frac{{10Y}}{{100}} = 2000 \\
   \Rightarrow \frac{X}{{20}} + \frac{Y}{{10}} = \frac{{2000}}{1} \\ 
\ $
By taking the L.C.M of denominators and on further simplification, we get
$\
   \Rightarrow \frac{{X + 2Y}}{{20}} = 2000 \\
   \Rightarrow X + 2Y = 40000 \to (1) \\ 
\ $

CASE - $2$
Profit on T.V. sold at $10\% $ gain = $\frac{{10X}}{{100}}$
Profit on Fridge sold at $5\% $ loss = $ - \frac{{5Y}}{{100}}$  (Here the loss is represented with a negative sign [‘-‘])

As per the question, the shopkeeper would gain Rs. 1500 in the second case.
Hence, we can represent case-2 in the equation form as
$\
   \Rightarrow \frac{{10X}}{{100}} - \frac{{5Y}}{{100}} = 1500 \\
   \Rightarrow \frac{X}{{10}} - \frac{Y}{{20}} = \frac{{1500}}{1} \\ 
\ $
On taking the L.C.M of denominators and on further simplification we get
$\
   \Rightarrow \frac{{2X - Y}}{{20}} = 1500 \\
   \Rightarrow 2X - Y = 30000 \to (2) \\ 
\ $
To simplify $(1)\& (2)$ equation, we have to multiply equation $(1) \times 2$ then we get
$\
   \Rightarrow [(X + 2Y) = 40000] \times 2 \\
   \Rightarrow 2X + 4Y = 80000 \to (3) \\ 
\ $
Now if we subtract $(2)$ from Equation $(3)$, we get
$\
   \Rightarrow 5Y = 50000 \\
   \Rightarrow Y = \frac{{50000}}{5} \\
   \Rightarrow Y = 10000 \\ 
\ $
Now putting $Y$ value in Equation $(1)$ , we get
$\
   \Rightarrow X + 2Y = 40000 \\
   \Rightarrow X + (2 \times 10000) = 40000 \\
   \Rightarrow X = 40000 - 20000 \\
   \Rightarrow X = 20000 \\ 
\ $  
Therefore we got both $X$ and $Y$ value which are the actual price values of T.V and Fridge.
Hence, the actual price of the T.V. is $X$=Rs.$20000$
And the actual price of the fridge is $Y$=Rs.$10000$ 

NOTE:
In this problem we have concentrated on the gain and loss values that means sign matters here for gain we use positive sign and for loss we use negative sign. Further on simplifying the given conditions we get the answer.