Question

A woman has twice as many dimes as quarters in her purse. If the dimes were quarters and the quarters were dimes, she would have$\$1.20$ more than she now has. How many of each does she have now?

Answer 1

Step 1: Calculate the number of dimes and quarters she has in her purse.

Let $x$ denote the number of quarters and $2x$ denote the number of dimes.
The total value is $0.25x+\left(0.10\right)\left(2x\right)$.
When the numbers of quarters and dimes are interchanged and she would have $\$1.20$ more than the original.
The formula to calculate the new value is $\mathrm{New}\mathrm{Value}=\mathrm{Original}\mathrm{Value}+1.20$.
Substitute $0.25\left(2x\right)+0.10x$ for new value and $0.25x+\left(0.10\right)\left(2x\right)$ for original value in the equation,

$\mathrm{New}\mathrm{Value}=\mathrm{Original}\mathrm{Value}+1.20$.

$$\begin{gathered} 0.25\left(2x\right)+0.10x=0.25x+\left(0.10\right)\left(2x\right)+\$1.20 \\ \Rightarrow 0.6x=0.45x+\$1.20 \\ \therefore 0.15x=\$1.20 \end{gathered}$$

Step 2: Solve for $x$.

Divide $1.20$ by $0.15.$

$$\begin{gathered} 0.15x=\$1.20 \\ \Rightarrow x=\frac{\$1.20}{0.15} \\ \Rightarrow x=8\text{quarters} \\ \therefore 2x=16\text{dimes} \end{gathered}$$

Hence, she has $16$ dimes and $8$ quarters.