Question

the students of a school contributed as many paisa each for a charity show as the number of students in the school .if the total collection amounts to Rs.506.25.find the number of students in the school and the amount paid by each​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

↝ The students of a school contributed as many paisa each for a charity show as the number of students in the school.

Let assume that

↝ The number of students in the school be x.

So,

↝ Amount paid by each student be 'x' paisa.

Thus, we have

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{Number \: of \: students = x} \\ &\sf{Amount \: by \: each \: student = x \: paisa} \end{cases}\end{gathered}\end{gathered}$

So,

$\begin{gathered}\begin{gathered}\bf \: \rm :\longmapsto\:\begin{cases} &\sf{Total \: Amount \: collected = x \times x = {x}^{2} }\end{cases}\end{gathered}\end{gathered}$

Now, it is given that,

↝ Total amount collected = Rs 506.25 = 50625 paisa.

Thus, we have now

$\rm :\longmapsto\: {x}^{2} = 50625$

$\rm :\longmapsto\:x = \sqrt{50625}$

So, let find the prime factorization of 50625

$\purple{\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{5}}}&{\underline{\sf{\:\:50625 \:\:}}}\\ {\underline{\sf{5}}}& \underline{\sf{\:\:10125 \:\:}} \\\underline{\sf{5}}&\underline{\sf{\:\:2025\:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:405 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:81 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:27 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:9 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3 \:\:}} \\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}}$

Therefore,

$\rm :\longmapsto\: \sqrt{50625} = \sqrt{5.5.5.5.3.3.3.3}$

$\bf\implies \: \sqrt{50625} = 5 \times 5 \times 3 \times 3 = 225$

Hence,

$\red{\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{Number \: of \: students = 225} \\ &\sf{Amount \: by \: each \: student = 225 \: paisa} \end{cases}\end{gathered}\end{gathered}}$