Question

# If the cost price of 18 chairs is equal to the selling price of 16 chairs. Find the gain or loss (in percent).

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Hint: We will assume that x is the selling price of 16 chairs so we get the cost price of 18 chairs also equal to x, and we will find SP and CP of 1 chair. Then we will find whether a gain or loss is taking place and find the gain or loss percentage accordingly. Gain can be calculated as S.P. - C.P. and loss as C.P. - S.P. We will also require the formula of gain percentage $gain\%=\dfrac{gain}{C.P}\times 100$ to get the gain% or the formula of loss percentage, $loss\%=\dfrac{loss}{CP}\times 100$ to find the loss%.

Let us take the selling price (S.P.) of 16 chairs to be Rs. x.
Now, from this we can say that the S.P. of 1 chair is
$=Rs.\ \dfrac{x}{16}$
Now, as given in the question, we know that the cost price(C.P.) of 18 chairs is also Rs. x.
So, from this, we can get the C.P. of 1 chair is
$=Rs.\ \dfrac{x}{18}$
So, we can see that the SP is greater than the CP, hence a profit is taking place. Now, the profit made by selling 16 chairs can be calculated as follows
Profit=selling price of 16 chairs -cost price of 16 chairs
Profit=x-16 times the cost price of 1 chair
(As assumed at the start of the question)
\begin{align} & =x-16\times \dfrac{x}{18} \\ & =\dfrac{2x}{18} \\ & =\dfrac{x}{9} \\ \end{align}
Hence, this is the profit that is experienced.
Now, for the profit percent, we can use the formula that is given in the hint as follows
\begin{align} & \Rightarrow profit\%=\dfrac{profit}{C.P.}\times 100 \\ & \Rightarrow profit\%=\dfrac{\dfrac{x}{9}}{16\times \dfrac{x}{18}}\times 100 \\ & \Rightarrow profit\%=\dfrac{1}{8}\times 100\% \\ & \Rightarrow profit\%=12.25\% \\ \end{align}

Hence the profit percentage = 12.25%

Note: The possible mistake that the students can make in this question is that, while comparing the SP and the CP, they may consider $\dfrac{x}{18}$ is greater than $\dfrac{x}{16}$ , hence will assume that it is a loss as they get CP is greater than SP, and find the loss percentage but it is wrong, as in case of a fraction, the fraction with the lower value in the denominator will be greater than the fraction with the higher value in the denominator.