Question

A report of 20 sheets of 56 lines and each such line consist of 65 characters. The report is reduced into sheets each of 65 lines such that the line consists of 70 characters. What is the percentage reduction in the number of sheets?
(a) 20%
(b) 30%
(c) 15%
(d) 25%

Answer 1

Hint: Assume the number of the second type of sheet used equal to ‘n’. Equate the total number of characters written on both types of sheets and find the value of n. To find the total number of characters written, multiply the number of sheets with the number of lines and number of characters in each line. Now, use the formula: - percentage reduction = \[\left( \dfrac{20-n}{20} \right)\times 100%\] to get the answer.

Complete step by step solution:
Here, we have been provided with two types of sheets with different number of lines and number of characters written in each line. We are asked to find the percentage reduction in the number of sheets. But first we need to find the number of second type sheets used.
Now, consider the first type sheet which contains 56 lines and each line consists of 65 characters. So, we have,
\[\Rightarrow \] Number of sheets = 20
\[\Rightarrow \] Total number of lines in these 20 sheets = 20 \[\times \] 56
\[\Rightarrow \] Total number of characters used = 20 \[\times \] 56 \[\times \] 65
Now, let us consider the second type sheet which contains 65 lines and each line consists of 70 characters. Here, we are assuming that the total number of second type sheets used is ‘n’. So, we have,
\[\Rightarrow \] Number of sheets = n
\[\Rightarrow \] Total number of lines in these ‘n’ sheets = n \[\times \] 56
\[\Rightarrow \] Total number of characters used = n \[\times \] 65 \[\times \] 70
Since, it is given to us that the report in the two types of sheet is the same, that means the total number of characters used in the two types of sheet will be the same. So, equating the total number of characters used in these sheets, we get,
\[\Rightarrow n\times 65\times 70=20\times 56\times 65\]
\[\Rightarrow n=\dfrac{20\times 56\times 65}{65\times 70}\]
Cancelling the common factors, we get,
\[\Rightarrow n=16\]
Therefore, the number of the second type of sheet used will be 16.
\[\Rightarrow \] Reduction in the number of sheets = 20 – 16
\[\Rightarrow \] Reduction in the number of sheets = 4
Applying the formula: - (Reduction / Initial value) \[\times 100%\], we get,
\[\Rightarrow \] Percentage reduction = \[\dfrac{4}{20}\times 100%\]
On simplifying, we get,
\[\Rightarrow \] Percentage reduction = 20%
Hence, option (a) is the correct answer.

Note: One may note that here we have to equate the total number of characters used in the two types of sheets because the report in them is the same. You must not equate the total number of lines or sheets otherwise we will not get the value of n. Note that here the value of n is less than 20 and that is why the reduction takes place. If the value of n would have been greater than 20 then we would have found the percentage increment. Remember that while calculating the percentage we always consider initial value in the denominator.