Question

There are two watches of cost Rs.800. One is sold at a profit of 16% and other at a loss of 8%. If there is no loss or no gain in the whole transaction, the cost price of the watch on which the shopkeeper gains is?

Answer 1

Hint:
The important condition given is no loss or no gain in the whole transaction. Cost price of both watches together is 800. This is the selling price only. But the only thing is he gains on one watch and losses on the other. Let’s solve it!

Complete step by step solution:
Let x be the C.P. of one watch that gains profit and 800-x be the C.P. of another watch that has loss.
Now for profit transaction:
C.P.=x
%profit= 16%
So the formula will be,
\[\% p = \dfrac{{S.P. - C.P.}}{{C.P.}}\]
\[ \Rightarrow \dfrac{{16}}{{100}} = \dfrac{{S.P. - x}}{x}\]
Cross multiplying x and taking x terms on one side
\[ \Rightarrow \dfrac{{16}}{{100}}x + x = S.P.\]
Taking x common
\[ \Rightarrow x\left( {\dfrac{{16}}{{100}} + 1} \right) = S.P.\]
Taking LCM,
\[ \Rightarrow x\left( {\dfrac{{116}}{{100}}} \right) = S.P........ \to 1\]
Now similarly for the watch that suffered a loss
%L=8%
C.P.=800-x
So the formula will be
\[\% l = \dfrac{{C.P. - S.P.}}{{C.P.}}\]
\[ \Rightarrow \dfrac{8}{{100}} = \dfrac{{(800 - x) - S.P.}}{{800 - x}}\]
On cross multiplying
\[ \Rightarrow \dfrac{{8(800 - x)}}{{100}} = (800 - x) - S.P.\]
\[ \Rightarrow S.P. = (800 - x) - \dfrac{{8\left( {800 - x} \right)}}{{100}}\]
Taking \[800 - x\] common from terms on left side
\[ \Rightarrow S.P. = (800 - x)\left[ {1 - \dfrac{8}{{100}}} \right]\]
Taking LCM , in bracket
\[ \Rightarrow S.P. = (800 - x)\left[ {\dfrac{{100 - 8}}{{100}}} \right]\]
\[ \Rightarrow S.P. = (800 - x)\left[ {\dfrac{{92}}{{100}}} \right]........ \to 2\]
Now adding 1 and 2 we get
\[ \Rightarrow S.P. + S.P = x\left( {\dfrac{{116}}{{100}}} \right) + (800 - x)\left[ {\dfrac{{92}}{{100}}} \right]\]
But since cost price and selling price are same addition of selling prices will be 800.
Multiplying both sides by 100,
\[ \Rightarrow 800 \times 100 = x116 + (800 - x)92\]
Multiply the bracket by 92,
\[ \Rightarrow 800 \times 100 = 116x + 800 \times 92 - 92x\]
Taking terms of x on one side and subtracting them
\[ \Rightarrow 800 \times 100 = 24x + 800 \times 92\]
\[ \Rightarrow 800 \times 100 - 800 \times 92 = 24x\]
Taking 800 common,
\[ \Rightarrow 800(100 - 92) = 24x\]
\[ \Rightarrow 800 \times 8 = 24x\]
Dividing both sides by 8,
\[ \Rightarrow 100 \times 8 = 3x\]
\[\begin{gathered}
   \Rightarrow 800 = 3x \\
   \Rightarrow x = \dfrac{{800}}{3} \\
   \Rightarrow x = 266.6 \\
\end{gathered} \]
This is the C.P. of the watch that is sold for profit .

Thus the C.P. of the watch on which the shopkeeper gains is \[Rs.266.6\].

Note:
Here students can directly write and form the equation as 116 for 16% profit and 92 for 8% loss. This will save the time.
\[x\dfrac{{116}}{{100}} + (800 - x)\left[ {\dfrac{{92}}{{100}}} \right] = 800\]
Remaining conditions are the same. Simple!