Question

hussain has a total of RS.590 of currency notes in the denominations of Rs.50,Rs.20,and Rs.10. The ratio of the number of Rs.50 and Rs.20 notes is 3:5. If he has a total of 25 notes, how many 10-rupee notes does he have?

Answer 1

Solution.

Let $x$ be the number of 50-rupee notes, $y$ be the number of 20-rupee notes, $z$ be the number of 10-rupee notes.

Solve this problem by using a system of equations. So

$$\left\{ \begin{array}{c} 50x + 20y + 10z = 590 \\ \frac{x}{y} = \frac{3}{5} \\ x + y + z = 25 \end{array} \right.$$

Solve the following system:

$$\left\{ \begin{array}{c} 50x + 20y + 10z = 590 \\ 5x - 3y = 0 \\ x + y + z = 25 \end{array} \right.$$

Subtract $\frac{1}{10} \times$ (equation 1) from equation 2:

$$\left\{ \begin{array}{c} 50x + 20y + 10z = 590 \\ -5y - z = -59 \\ x + y + z = 25 \end{array} \right.$$

Divide equation 1 by 10:

$$\left\{ \begin{array}{l} 5x + 2y + z = 59 \\ -5y - z = -59 \\ x + y + z = 25 \end{array} \right.$$

Multiply equation 2 by $-1$:

$$\left\{ \begin{array}{c} 5x + 2y + z = 59 \\ 5y + z = 59 \\ x + y + z = 25 \end{array} \right.$$

Subtract $\frac{1}{5} \times$ (equation 1) from equation 3:

$$\left\{ \begin{array}{c} 5x + 2y + z = 59 \\ 5y + z = 59 \\ \frac{3y}{5} + \frac{4z}{5} = \frac{66}{5} \end{array} \right.$$

Multiply equation 3 by 5:

$$\left\{ \begin{array}{c} 5x + 2y + z = 59 \\ 5y + z = 59 \\ 3y + 4z = 66 \end{array} \right.$$

Subtract $\frac{3}{5} \times$ (equation 2) from equation 3:

$$\left\{ \begin{array}{c} 5x + 2y + z = 59 \\ 5y + z = 59 \\ \frac{17z}{5} = \frac{153}{5} \end{array} \right.$$

Multiply equation 3 by $\frac{5}{17}$:

$$\left\{ \begin{array}{c} 5x + 2y + z = 59 \\ 5y + z = 59 \\ z = 9 \end{array} \right.$$

Substitute $z$ into equation 1 and equation 2:

$$\left\{ \begin{array}{c} 5x + 2y = 50 \\ 5y = 50 \\ z = 9 \end{array} \right.$$ $$\left\{ \begin{array}{c} 5x + 2y = 50 \\ y = 10 \\ z = 9 \end{array} \right.$$

Substitute $y$ into equation 1:

$$\left\{ \begin{array}{l} 5x = 30 \\ y = 10 \\ z = 9 \end{array} \right.$$ $$\left\{ \begin{array}{l} x = 6 \\ y = 10 \\ z = 9 \end{array} \right.$$

**Answer**: he has 9 currency notes in the denomination of Rs. 10.