Question

If the selling price of a mat is five times the discount offered and if the percentage of discount is equal to the percentage profit, find the ratio of the discount offered to the cost price.

Answer 1

Hint: In this question it is given that the selling price(S.P) of a mat is five times the discount offered and if the percentage of discount is equal to the percentage profit, we have to find the ratio of the discount offered to the cost price. So to find the solution we need to know that if we consider selling price as S.P, marked price as M.P and cost price as C.P then,
Percentage of discount = $$\dfrac{\text{M.P} -\text{S.P} }{\text{M.P} } \times 100\%$$.........(1)
Percentage of profit = $$\dfrac{\text{S.P} -\text{C.P} }{\text{C.P} } \times 100\%$$........(2)
By using the above condition we will get the solution.

Complete step-by-step solution:
Here it is given,
The selling price of a mat is five times the discount offered, for this we need to know that discount price is (M.P-S.P)
Therefore we can write the above statement as,
$$S.P=5\times \text{discount offered}$$
$$\Rightarrow S.P=5\times \left( M.P-S.P\right) $$
$$\Rightarrow S.P=5M.P-5S.P$$
$$\Rightarrow S.P+5S.P=5M.P$$
$$\Rightarrow 6S.P=5M.P$$
$$\Rightarrow 5M.P=6S.P$$
$$\Rightarrow M.P=\dfrac{6}{5} S.P$$.........(1)
Also it is given that percentage of discount is equal to the percentage profit, so we can write,
Percentage of discount = Percentage of profit
$$\Rightarrow \dfrac{M.P-S.P}{M.P} \times 100\% =\dfrac{S.P-C.P}{C.P} \times 100\%$$
$$\Rightarrow \dfrac{\dfrac{6}{5} S.P-S.P}{\dfrac{6}{5} S.P} \times 100\% =\dfrac{S.P-C.P}{C.P} \times 100\%$$ [by using (1)]
$$\Rightarrow \dfrac{\left( \dfrac{6}{5} -1\right) S.P}{\dfrac{6}{5} S.P} \times 100\% =\dfrac{S.P-C.P}{C.P} \times 100\%$$ [by taking S.P common from the numerator]
$$\Rightarrow \dfrac{\left( \dfrac{6}{5} -1\right) }{\dfrac{6}{5} } =\dfrac{S.P-C.P}{C.P}$$ [canceling S.P]
$$\Rightarrow \dfrac{\left( \dfrac{6-5}{5} \right) }{\dfrac{6}{5} } =\dfrac{S.P-C.P}{C.P}$$
$$\Rightarrow \dfrac{\dfrac{1}{5} }{\dfrac{6}{5} } =\dfrac{S.P-C.P}{C.P}$$
$$\Rightarrow \dfrac{1}{5} \times \dfrac{5}{6} =\dfrac{S.P-C.P}{C.P}$$ [$$\because \dfrac{(\dfrac{a}{b} )}{(\dfrac{c}{d} )} =\dfrac{a}{b} \times \dfrac{d}{c}$$]
$$\Rightarrow \dfrac{1}{6} =\dfrac{S.P-C.P}{C.P}$$
$$\Rightarrow C.P=6S.P-6C.P$$ [by cross multiply]
$$\Rightarrow C.P+6C.P=6S.P$$
$$\Rightarrow 7C.P=6S.P$$
$$\Rightarrow C.P=\dfrac{6}{7} S.P$$............(2) [dividing both side by 7]
Now as we know that selling price of a mat is five times the discount offered,
i.e, $$5\times \text{discount offered}=S.P$$
$$\Rightarrow \text{discount offered} =\dfrac{1}{5} S.P$$.......(3)

So from (2) and (3) we can write,
$$\dfrac{\text{discount offered} }{C.P} =\dfrac{\left( \dfrac{1}{5} \right) S.P}{\left( \dfrac{6}{7} \right) S.P}$$
$$\Rightarrow \dfrac{\text{discount offered} }{C.P} =\dfrac{\left( \dfrac{1}{5} \right) }{\left( \dfrac{6}{7} \right) }$$
$$\Rightarrow \dfrac{\text{discount offered} }{C.P} =\dfrac{1}{5} \times \dfrac{7}{6}$$
$$\Rightarrow \dfrac{\text{discount offered} }{C.P} =\dfrac{7}{5\times 6}$$
$$\Rightarrow \dfrac{\text{discount offered} }{C.P} =\dfrac{7}{30}$$
Therefore, $$\text{discount offered} \ \colon \text{C.P} =7\colon 30$$
i.e, the ratio of the discount offered to the cost price is 7:30.

Note: While calculating percentage you need to know that, whatever may be the final price is, you have to calculate the percentage of the change with respect to the initial price or value, so that is why in the formula (1) we have calculated the percentage of change w.r.t the initial price of the product which is marked price and in the formula (2) also we have calculated w.r.t the Cost price.