Answer

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**Hint:**First convert the ratio in equation form. Then, multiply the equation of the total number of coins by 2 and subtract from the equation of the total amount of the coin. After that substitute the value of the number of coins of Rs. 5 in terms of Rs. 10 coins and solve it. It will give the number of Rs. 10 coins. Then find the number of Rs. 5 coins from it. After that substitute the number of coins of Rs. 5 and Rs. 10 in the total number of coins equation to find the number of Rs. 2 coins.

**Complete step by step answer:**

Given: Total amount in piggy bank $ = Rs. 150$

Total number of coins $= 30$

Rs. 10 and Rs. 5 coins are in ratio $3:7$

Let the number of Rs. 2 coins be $x$, Rs. 5 coins be $y$ and Rs. 10 coins be $z$.

Since the ratio of Rs. 10 and Rs. 5 coins is $3:7$

$\Rightarrow \dfrac{z}{y}=\dfrac{3}{7}$

Cross multiply the terms,

$\Rightarrow 3y=7z$ ……………...….. (1)

Now the total number of coins is 30. Then,

$\Rightarrow x+y+z=30$ ……………..….. (2)

Also, the total amount is Rs. 150,

$\Rightarrow 2x+5y+10z=150$ …………...….. (3)

Multiply equation (2) by 2 and subtract from equation (3),

$\begin{align}

& 2x+5y+10z=150 \\

& \underline{2x+2y+2z\,\,\,=60} \\

& \,\,\,\,\,\,\,\,\,\,\,3y+8z\,\,\,=90 \\

\end{align}$

Substituting the value of $3y$ from equation (1),

$\Rightarrow 7z+8z=90$

Add the terms on the left side,

$\Rightarrow 15z=90$

Divide both sides by $15$,

$\Rightarrow z=6$

Now substituting the value of $z$ in equation (1),

$\Rightarrow 3y=7\times 6$

Multiply the term on the right side,

$\Rightarrow 3y=42$

Dividing both sides by 3,

$\Rightarrow y=14$

Substituting the value of $y$ and $z$ in equation (2),

$\Rightarrow x+14+6=30$

Adding the term on the left side,

$\Rightarrow x+20=30$

Move $20$ on the right side and subtract from $30$,

$\Rightarrow x=10$

**Therefore, the number of coins of Rs. 2 is 10, Rs. 5 is 14, and Rs. 10 is 6.**

**Note:**

If $a, b, c$, and $r$ are the real numbers and $a, b, c$ are not all equal to $0$ then \[ax+by+cz=r\] is called a linear equation in three variables. The three variables are the $x, y$, and $z$. The numbers $a, b$, and $c$ are called the coefficients of the equation. The number r is called the constant of the equation.

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