Verified answer

Answer:

[tex]\qquad \:\boxed{\begin{aligned}& \qquad \:\sf \: Number\:of\:children=40 \qquad \: \\ \\& \qquad \:\sf \:Total\:amount=Rs \: 200 \end{aligned}} \qquad \: \\ \\ [/tex]

Step-by-step explanation:

Let assume that

Number of children be x.

Share of each student be Rs y.

Total amount = Rs xy

Case :- 1 If the number of children exceeds by 10, each of them is given Rs 1 less.

So,

Number of children = x + 10

Share of each student = Rs (y - 1)

So, Total amount = Rs (x + 10)(y - 1)

Thus,

[tex]\sf \: (x + 10)(y - 1) = xy \\ \\ [/tex]

[tex]\sf \: xy - x + 10y - 10 = xy \\ \\ [/tex]

[tex]\sf \: - x + 10y - 10 = 0 \\ \\ [/tex]

[tex]\sf\implies \sf \: x = 10y - 10 \: \: - - - (1) \\ \\ [/tex]

Case :- 2 if the number of children is less by 15, each of them gets Rs 3 more.

So,

Number of children = x - 15

Share of each student = Rs (y + 3)

So, Total amount = Rs (x - 15)(y + 3)

Thus,

[tex]\sf \: (x - 15)(y + 3) = xy \\ \\ [/tex]

[tex]\sf \: xy + 3x - 15y - 45 = xy \\ \\ [/tex]

[tex]\sf \: 3x - 15y - 45 = 0 \\ \\ [/tex]

[tex]\sf \: 3(x - 5y - 15) = 0 \\ \\ [/tex]

[tex]\sf \: x - 5y - 15= 0 \\ \\ [/tex]

On substituting the value of x from equation (1), we get

[tex]\sf \: 10y - 10 - 5y - 15= 0 \\ \\ [/tex]

[tex]\sf \: 5y - 25= 0 \\ \\ [/tex]

[tex]\sf \: 5y = 25\\ \\ [/tex]

[tex]\sf\implies \sf \: y= 5 \\ \\ [/tex]

On substituting the value of y in equation (1), we get

[tex]\sf \: x = 10 \times 5 - 10 \\ \\ [/tex]

[tex]\sf \: x = 50 - 10 \\ \\ [/tex]

[tex]\sf\implies \sf \: x = 40 \\ \\ [/tex]

Hence,

[tex]\qquad \:\boxed{\begin{aligned}& \qquad \:\sf \: Number\:of\:children=40 \qquad \: \\ \\& \qquad \:\sf \:Total\:amount=Rs \: 200 \end{aligned}} \qquad \: \\ \\ [/tex]

Let assume that

Number of children be x.

Share of each student be Rs y.

Total amount = Rs xy

Case:- 1 If the number of children exceeds by 10, each of them is given Rs 1 less.

So,

Number of children = x + 10

Share of each student = Rs(y - 1)

So, Total amount = Rs(x + 10)(y - 1)

Thus,

(x + 10)(y - 1) = xy

xy - x + 10y - 10 = xy

- x + 10y - 10 = 0

Rightarrow x = 10y - 10 - - - (1)

Case :- 2 if the number of children is less by 15, each of them gets Rs 3 more.

So,

Number of children = x - 15

Share of each student = Rs(y + 3)

So, Total amount = Rs(x - 15)(y + 3)

Thus,

(x - 15)(y + 3) = xy

xy + 3x - 15y - 45 = xy

3x - 15y - 45 = 0

3(x - 5y - 15) = 0

x - 5y - 15 = 0

On substituting the value of x from equation (1), we get

10y - 10 - 5y - 15 = 0

5y - 25 = 0

5y = 25

⇒ y = 5

On substituting the value of y in equation (1), we get

x = 10 × 5 - 10

x = 50 - 10

⇒ x = 40

Hence,

Number of children = 40

Total amount = Rs 200