The students of a school are made to stand in rows. If the number of students in each row is increased by ^@6^@, there would be ^@4 ^@ row less. If the number of students in each rows is reduced by ^@6^@, there would be ^@5^@ rows more. Find the number of students in the school.


Answer:

^@2160^@

Step by Step Explanation:
  1. Let us assume that there are ^@ x ^@ students in each row and ^@ y ^@ rows in total.

    Thus, the total number of students in the school is ^@ xy^@.
  2. If the number of students in each row is increased by ^@6^@, then the number of rows becomes ^@(y - 4)^@.

    As the total number of students remain same, we have @^ \begin{aligned} & xy = (x + 6)(y - 4) \\ {\implies} &xy = xy - 4 x + 6 y - 24 &&[\text{ Cancelling } xy {\space} ]\\ {\implies} &4 x - 6 y = -24 &&\ldots{\text{(i)}} \\ \end{aligned} @^
  3. If the number of students in each row is reduced by ^@6^@, then the number of rows becomes ^@(y + 5)^@. @^ \begin{aligned} &\therefore &&xy = (x - 6)(y + 5) \\ &{\implies} &&xy = xy + 5 x - 6 y - 30 &&[\text{ Cancelling } xy {\space} ] \\ &{\implies} &&-5 x + 6 y = -30 &&\ldots{\text{(ii)}} \\ \end{aligned} @^
  4. Adding ^@\text{eq(i)}^@ and ^@\text{eq(ii)}^@, we get @^ \begin{aligned} {\implies}& (4 - 5) x = (-24 -30) \\ {\implies}& -x = -54 \\ {\implies}& x = 54 \end{aligned} @^
  5. Substituting the value of ^@x ^@ in ^@\text{eq(i)}^@, we get @^ \begin{aligned} &4 \times 54 - 6 y = -24 \\ {\implies} & - 6 y = -24 - 216 \\ {\implies} & - 6 y = -240 \\ {\implies} & y = 40 \end{aligned} @^
  6. Therefore, the total number of students in the school = ^@ xy^@ = 54 ^@{\times} ^@ 40 = ^@2160^@.

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