Question

A student $X$ passes his examination with $515$ marks having scored $3\% $ above the minimum. If $Y$ had obtained $710$ marks, what percent would have been above the minimum?

Answer 1

Hint: Here we will proceed by letting the minimum marks as a variable and then by considering the first statement that the student $X$ passes his examination with $515$ marks having scored $3\% $ above the minimum, we can easily find the minimum marks. Then in the second we know the minimum marks also and the marks scored also so we can easily find the percentage increase $Y$ gets by the formula:
${\text{Percent increase}} = \dfrac{{{\text{marks scored}} - {\text{minimum marks}}}}{{{\text{minimum marks}}}} \times 100\% $

Complete step-by-step answer:
Here we are given that student $X$ passes his examination with $515$ marks having scored $3\% $ above the minimum. So according to the knowledge of this statement let us first try to find the minimum marks needed.
Let the minimum marks needed be ${m_{\min }}$
Marks scored by $X = 515$
And it is also told that he scored $3\% $ above the minimum marks.
So we can say that
${\text{Percent increase}} = \dfrac{{{\text{marks scored}} - {\text{minimum marks}}}}{{{\text{minimum marks}}}} \times 100\% $
$3 = \dfrac{{515 - {m_{\min }}}}{{{m_{\min }}}} \times 100$
$
\Rightarrow {m_{\min }} + \dfrac{3}{{100}}{m_{\min }} = 515 \\
\Rightarrow \dfrac{{103}}{{100}}{m_{\min }} = 515 \\
\Rightarrow {m_{\min }} = \dfrac{{515(100)}}{{103}} = 500 \\
 $
So we get minimum marks$ = 500$
Now we see the case (2) where it is told that $Y$ scores $710$ marks and we need to find the percentage he has scored above the minimum.
So here we know that
Marks scored$ = 710$
Minimum marks$ = 500$
So we can apply the formula as
${\text{Percent increase}} = \dfrac{{{\text{marks scored}} - {\text{minimum marks}}}}{{{\text{minimum marks}}}} \times 100\% $
${\text{Percent increase}} = \dfrac{{710 - 500}}{{500}} \times 100\% = \dfrac{{210}}{{500}} \times 100\% = 42\% $
Hence we can say that the percentage increase is $42\% $ which means that he needs the minimum marks to pass as $500$ but he scored $42\% $ above the minimum marks that were required to pass the examination. So we can say that $720$ marks are $42\% $ above the minimum marks which is $500$.

Note: In this type of question we need to avoid the grammatical mistake while solving the problem as we need to understand what the percent increase means. Here we must know that if the person has $x$ marks in exam and it is $y\% $ above the minimum marks then we can apply the formula which is
${\text{Percent increase}} = \dfrac{{{\text{marks scored}} - {\text{minimum marks}}}}{{{\text{minimum marks}}}} \times 100\% $