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# Anchal collected a sum of Rs. 150 in a piggy bank. She has put only Rs. 2, Rs. 5, and Rs. 10 coins. If Rs. 10 and Rs. 5 coins are in ratio 3:7 and there is a total of 30 coins, find the number of coins of each type.

Last updated date: 13th Jun 2024
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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.

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.