Question

Mr. Singha sold two tape recorders for Rs.990 each; gaining 10% on one and losing 10% on the other. Find his total loss or gain as percent on the whole transaction.

Answer 1

Hint: Here we assume the original price of each tape recorder as different variables and find the total transaction amount. Using the concept of percentage we find the value of gain and loss with respect to the original prices of the two tape recorders. We find the new transaction amount by adding the values obtained after adding the gain and subtracting the loss by one value. Calculate the difference in amounts from both transactions and find the percentage change.
* x percentage of m can be calculated as\[\dfrac{x}{{100}} \times m\].

Complete step-by-step answer:
Let us assume the price of the first tape recorder as A and the price of the second tape recorder as B.
Then we can write \[A + B = 990 + 990\]
\[ \Rightarrow A + B = 1980\] … (1)
Now we know Mr. Singha gained 10% on tape recorder A
Therefore, the new price on which he sold the tape recorder A is greater by 10%.
Let the new price be A’, then we can write
\[ \Rightarrow A' = A + 10\% A\]
Calculate the percentage on RHS by writing it in fraction form
\[ \Rightarrow A' = 990 + \dfrac{{10}}{{100}} \times 990\]
Cancel the same terms from numerator and denominator in RHS
\[ \Rightarrow A' = 990 + 99\]
\[ \Rightarrow A' = 1089\]
Similarly, we know Mr. Singha loss 10% on tape recorder B
Therefore, the new price on which he sold the tape recorder B is lesser by 10%.
Let the new price be B’, then we can write
\[ \Rightarrow B' = B - 10\% B\]
Calculate the percentage on RHS by writing it in fraction form
\[ \Rightarrow B' = 990 - \dfrac{{10}}{{100}} \times 990\]
Cancel the same terms from numerator and denominator in RHS
\[ \Rightarrow B' = 990 + 99\]
\[ \Rightarrow B' = 891\]
Therefore, total of selling tape recorders A and B \[ = A' + B'\]
Substitute the values of A’ and B’
\[ \Rightarrow A' + B' = 1089 + 891\]
\[ \Rightarrow A' + B' = 1980\] … (2)
Now we calculate the difference between two transactions by subtracting equation (1) from equation (2).
\[ \Rightarrow (A' + B') - (A + B) = 1980 - 1980 = 0\]
Since, there is no difference of amount between the two transactions. Therefore, there is no loss or no gain. Therefore, there is 0% gain and 0% loss in the whole transaction.

Note: Alternate method:
We are given prices of two tape recorders A and B as Rs.990 each
We know initial transaction will be CP of tape recorder A +CP of tape recorder B
Selling Price \[ = 990 + 990 = 1980\]
We can also use the formula of selling price for gain i.e. \[SP = [\dfrac{{100 + G\% }}{{100}}]CP\]
We are given Cost price CP as Rs.990
Gain percentage, G as 10
Therefore, substituting the values in the formula we get
\[ \Rightarrow SP = [\dfrac{{100 + 10}}{{100}}] \times 990\]
Add the terms in numerator
\[ \Rightarrow SP = \dfrac{{110}}{{100}} \times 990\]
Cancel the same terms from numerator and denominator.
\[ \Rightarrow SP = 11 \times 99\]
\[ \Rightarrow SP = 1089\] … (1)
Similarly, we can use formula for selling price for loss, i.e. \[SP = [\dfrac{{100 - L\% }}{{100}}]CP\]
We are given Cost price CP as Rs.990
Gain percentage, L as 10
Therefore, substituting the values in the formula we get
\[ \Rightarrow SP = [\dfrac{{100 - 10}}{{100}}] \times 990\]
Add the terms in numerator
\[ \Rightarrow SP = \dfrac{{90}}{{100}} \times 990\]
Cancel the same terms from numerator and denominator.
\[ \Rightarrow SP = 9 \times 99\]
\[ \Rightarrow SP = 891\] … (2)
Total selling price after 10% gain and 10% loss is given by addition of selling prices from equation (1) and (2)
New selling price \[ = 1089 + 891 = 1980\]
Since, there is no difference of amount between the two transactions. Therefore, there is no loss or no gain.
Therefore, there is 0% gain and 0% loss.