Question

Translate to a system of equations and solve:

Juan has a pocket full of nickel sand dimes. The total value of the coins is $8.10. The number of dimes is 9 less than twice the number of nickels. How many nickels and how many dimes does Juan have?

Short answer

The number of nickels and number of dimes are 36 and 63 respectively.

Step-by-step solution

Step 1. Given Information  

The given data is that Juan has a pocket full of nickel sand dimes. The total value of the coins is $8.10. The number of dimes is 9 less than twice the number of nickels.

Step 2. Calculation  

Let the number of nickels be x and the number of dimes be y.

Nickel is one-twenty of the value and one dime has one tenth value.

The total number of nickel and dimes is $\$8.10$that is localid="1647678017292" $\frac{1}{20}x+\frac{1}{10}y=8.10--\left(1\right)$

The number of dimes is 9 less than twice the number of nickels that is $y=2x-9--\left(2\right)$

Substitute equation (2) in equation (1) and solve for x.

localid="1656376984781" $\begin{array}{rcl}\frac{1}{20}x+\frac{1}{10}(2x-9)& =& 8.10\\ \frac{x}{20}+\frac{x}{5}-\frac{9}{10}& =& 8.10\\ \frac{x+4x}{20}& =& 8.10+0.9\\ \frac{5x}{20}& =& 9\\ \frac{x}{4}& =& 9\\ x& =& 36\end{array}$

Substitute the value of x in equation (2) to find the value of y.

localid="1648709393124" $$\begin{gathered} y=2\left(36\right)-9 \\ y=72-9 \\ y=63 \end{gathered}$$