Question

(i) The ages of Sahil and Nikhil are in the ratio \[4:3\]. Five years hence, the ratio of their ages will be \[5:4\]. Find their present ages.
(ii) A student has obtained \[35\% \] marks in a test. If the student scored \[24.5\] marks, find the maximum marks.

Answer 1

Hint: As there are ratios present in (i), we are going to assume a common factor between the ratios of their ages to find their present ages and solve further. In part (ii), we are given a percentage of the maximum marks scored by the student and his actual marks, so we will assume the maximum marks.

Complete step-by-step answer:
(i) Let the common factor between the ages of Sahil and Nikhil be \[x\].
So, the present age of Sahil \[ = 4x\]
and the present age of Nikhil\[ = 3x\]
Five years hence, the ratio of their ages changes from \[4:3\] to\[5:4\].
So, the age of Sahil after five years\[ = 4x + 5\]
And the age of Nikhil after 5 years\[ = 3x + 5\]
According to the question, we know that the ratio of their ages after five years is\[5:4\].
Hence, we get
\[
  4x + 5:3x + 5 = 5:4 \\
   \Rightarrow \dfrac{{4x + 5}}{{3x + 5}} = \dfrac{5}{4} \\
   \Rightarrow 4(4x + 5) = 5(3x + 5) \\
   \Rightarrow 16x + 20 = 15x + 25 \\
   \Rightarrow 16x - 15x = 25 - 20 \\
   \Rightarrow x = 5 \\
\]
Therefore, Sahil’s present age\[ = 4x = 4 \times 5 = 20\]years
Nikhil’s present age\[ = 3x = 3 \times 5 = 15\]years
(ii) Here, we are given the percentage of the marks obtained and the actual marks obtained. So, we will consider the maximum marks to be\[m\].
Since, the student scored \[35\% \] marks in the test,
\[\therefore \]the student’s marks\[ = 35\% \]of \[m\]\[ = \dfrac{{35}}{{100}}m\]
But according to the question, he scored \[24.5\]marks in the test.
\[
  \therefore \dfrac{{35}}{{100}}m = 24.5 \\
   \Rightarrow 35m = 24.5 \times 100 \\
   \Rightarrow 35m = 2450 \\
   \Rightarrow m = \dfrac{{2450}}{{35}} \\
   \Rightarrow m = 70 \\
\]
Therefore, the maximum marks are \[70\]marks.

Note: When there are ratios present in the problem, our first thought should be to assume a common factor and proceed further according to the question. As it is said that the ratio changes five years hence, thus we will add \[5\] to their present ages to find their ages five years hence. The second part of the question is a simple percentage problem. We assume the maximum marks to be \[m\] because we are given a percentage of the maximum marks that the student is scoring and also his actual marks. When these two things are given to us, we have to equate them in order to find the missing variable.