Question

Entry fee in an exhibition was Rs1 Later this was reduced by \[25\% \] which increased the sale by \[20\% \] then found the percentage of slump in business.

Answer 1

Hint:
First, we will assume total sales to be 100, hence the total number of visitors would be 100. Then we will calculate the new Entry fee which is \[25\% \] less than the original one and then we will calculate the new sales which have increased by \[20\% \] according to the new entry fee. After that, we will calculate the increase in visitors after applying lowered entry fees and increased total sales. Then, we would find the percentage by which visitors have increased.

Complete step by step solution:
Let us the total original sale be $'S' = Rs.\;100$
According to the question, Entry fee was $'E' = Rs.\;1$
Therefore, the Original number of visitors $'V' = \;100$
We know that sales are equal to the product of rate and the total number of visitors, that is
 $S = E \times V$ …(1)
We know that when \[x\% \] is reduced from y, then new y is given by
 ${y_{new}} = y - \left( {\dfrac{{x \times y}}{{100}}} \right)$
Now, according to the question, the fees is reduced by \[25\% \] ,


As the fee reduces \[25\% \] , the new Entry fee is given by,
  $ \Rightarrow {E_{new}} = {E_{old}} - \left( {\dfrac{{25 \times {E_{old}}}}{{100}}} \right)$
Now, substituting ${E_{old}} = Rs.\;1$ , we get
 $ \Rightarrow {E_{new}} = 1 - \left( {\dfrac{{25 \times 1}}{{100}}} \right)$
Now taking LCM at RHS, we get
 $ \Rightarrow {E_{new}} = \dfrac{{100 - 25}}{{100}}$
On simplification we get,
 $ \Rightarrow {E_{new}} = \dfrac{{75}}{{100}}$
Hence we have,
 $ \Rightarrow {E_{new}} = 0.75$
We know that when \[x\% \] is gained in y, then new y is given by
 ${y_{new}} = y + \left( {\dfrac{{x \times y}}{{100}}} \right)$
Now, according to the question, reduction in Entry fees resulted in a \[20\% \] gain in total sales, hence new sales
 $ \Rightarrow {S_{new}} = {S_{old}} + \left( {\dfrac{{20 \times {S_{old}}}}{{100}}} \right)$

Now, substituting ${S_{old}} = Rs.\;100$ , we get
 $ \Rightarrow {S_{new}} = 100 + \left( {\dfrac{{20 \times 100}}{{100}}} \right)$
Now taking LCM at RHS, we get
 $ \Rightarrow {S_{new}} = \dfrac{{10000 + 2000}}{{100}}$
On simplification we get,
 $ \Rightarrow {S_{new}} = \dfrac{{12000}}{{100}}$
Hence we get,
 $ \Rightarrow {S_{new}} = 120$
Now we got the new entry fee and new sales, using (1),
 $ \Rightarrow {S_{new}} = {E_{new}} \times {V_{new}}$
On substituting, ${E_{new}} = 0.75$ and ${S_{new}} = 120$ , we get
 $ \Rightarrow 120 = 0.75 \times {V_{new}}$
On dividing the equation by 0.75 we get,
 $ \Rightarrow {V_{new}} = \dfrac{{120}}{{0.75}}$
Hence on simplification we get,
 $ \Rightarrow {V_{new}} = 160$
Hence the new number of visitors is 160
Now, to calculate the number of visitors increased,
 $ \Rightarrow {V_{inc}} = {V_{new}} - {V_{old}}$

Now, substituting ${V_{old}} = \;100$ and ${V_{new}} = 160$ , we get
 $ \Rightarrow {V_{inc}} = 160 - 100$
On simplification we get,
 $ \Rightarrow {V_{inc}} = 60$
We know that to find gain percentage,
 $gain\% = \dfrac{{gain}}{{Old\,value}} \times 100$
Hence, the gain percentage in visitors is given by
 $ \Rightarrow {V_{gain\% }} = \dfrac{{{V_{inc}}}}{{{V_{old}}}} \times 100$
Now, substituting ${V_{inc}} = 60$ and ${V_{old}} = \;100$ , we get,
 $ \Rightarrow {V_{gain\% }} = \dfrac{{60}}{{100}} \times 100$
On simplification we get,
 $ \Rightarrow {V_{gain\% }} = 60$

Hence, \[Increase\;\% = 60\% \]

Note:
We should remember that after calculating the new lowered entry fee, to calculate the total increased sales we must use the new lowered rates of entry fee because the sales were increased only after the rates were lowered. Do not use the old rates to calculate the gain in sales, the answer would be wrong. We assumed the original sales to be 100 in order to ease our calculations, it is recommended to take these kinds of assumptions while working on questions related to percentage.