Question

A man has $14$ coins in his pocket, all of which are dimes and quarters. If the total value of his change is $\$2.75$, how many dimes and how many quarters does he have?

Answer 1

Determining the number of dimes and quarters.

If $d$ is the number of dimes the man has and $q$ is the number of quarters.

The fact that he has $14$ coins mean that

$d+q=14...\left(1\right)$

We know that

$$\begin{gathered} \Rightarrow 1\text{dimes=10 cents} \\ \Rightarrow d\text{dimes=}\left(10\times d\right)\text{cents} \\ \Rightarrow 1\text{quarter=25 cents} \\ \Rightarrow q\text{quarters=}\left(25\times q\right)\text{cents} \end{gathered}$$

We know that, he has $\$2.75or275\text{cents}$ means that

$10d+25q=275...\left(2\right)$

Rewrite the equation (1) to get the value of $d$ in terms of $q$

$d=14-q$

Substitute the above value into equation $\left(2\right)$, we get

$$\begin{gathered} \Rightarrow 10\left(14-q\right)+25q=275 \\ \Rightarrow 140-10q+25q=275 \\ \Rightarrow 140+15q=275 \\ \Rightarrow 15q=275-140 \\ \Rightarrow 15q=135 \\ \Rightarrow q=\frac{135}{15} \\ \Rightarrow q=9 \end{gathered}$$

Substitute the above value into equation $\left(2\right)$, we get

$$\begin{gathered} \Rightarrow 10d+25\left(9\right)=275 \\ \Rightarrow 10d+225=275 \\ \Rightarrow 10d=275-225 \\ \Rightarrow 10d=50 \\ \Rightarrow d=\frac{50}{10} \\ \Rightarrow d=5 \end{gathered}$$

Therefore, the man has $5\mathrm{dimes}$ and $9\mathrm{quarters}$.