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On selling a tea set at $5\% $ loss and a lemon set at $15\% $ gain, a crockery seller gains Rs. 7. If he sells the tea set at $5\% $ gain and lemon set at $10\% $ gain, he gains Rs. 13. Find the actual price of the tea set and the lemon set:
A. Tea set Rs. 180, Lemon set Rs. 120
B. Tea set Rs. 130, Lemon set Rs. 70
C. Tea set Rs. 90, Lemon set Rs. 100
D. Tea set Rs. 100, Lemon set Rs. 80

Answer Verified Verified
Hint- In this question seller gain depends upon the tea set or lemon set, if he sells tea set or lemon set with gain then we will use positive sign and if he sells tea set or lemon set with loss we will use negative sign to make equations. So by the help of the question we will make two equations with two unknown variables x and y, then proceed further by solving it.

“Complete step-by-step answer:”
Let the cost of tea set \[ = {\text{ }}x\] and the cost of lemon set \[ = {\text{ }}y\]
If he sells a tea set at $5\% $ loss and the lemon set at $15\% $ gain
Loss on a tea set $ = x \times \dfrac{5}{{100}}$
Gain on a lemon set $ = y \times \dfrac{{15}}{{100}}$
The total gain of the seller is
= gain on lemon set – loss on tea set
$
   \Rightarrow \dfrac{{15y}}{{100}} - \dfrac{{5x}}{{100}} = 7 \\
   \Rightarrow 15y - 5x = 700 \\
   \Rightarrow - x + 3y = 140.....................(1) \\
 $
If he sells a tea set at $5\% $ gain and the lemon set at $10\% $ gain
Gain on a tea set $ = x \times \dfrac{5}{{100}}$
Gain on a lemon set $ = y \times \dfrac{{10}}{{100}}$
The total gain of the seller is
= gain on lemon set + gain on tea set
$
   \Rightarrow \dfrac{{10y}}{{100}} + \dfrac{{5x}}{{100}} = 13 \\
   \Rightarrow 10y + 5x = 1300 \\
   \Rightarrow x + 2y = 260.....................(2) \\
 $
Now adding equation (1) and (2) we get
$
   \Rightarrow - x + 3y + x + 2y = 140 + 260 \\
   \Rightarrow 5y = 400 \\
   \Rightarrow y = 80 \\
 $
Substituting the value of y=80 in equation (1), we get
$
   \Rightarrow - x + 3 \times 80 = 140 \\
   \Rightarrow - x + 240 = 140 \\
   \Rightarrow - x = - 100 \\
   \Rightarrow x = 100 \\
 $
Hence, the actual price of tea set is Rs. 100 and the actual price of lemon set is Rs. 80

Note- To solve this question, form conditions from the given statement in form of variables and remember the number of unknown is equal to the number of equations. So find all the equations from the statement needed for solving the questions. The equations may be linear, or quadratic. Solve these equations to find the answer.

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