Question

Tionna has saved dimes and quarters in her piggy bank. Define the variables, and write equations to find the number of dimes and quarters. Then solve using the best method for the situation.

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25 coins

Short answer

The number of dimes and quarters are 15 and 10 respectively.

Step-by-step solution

Step 1. State the concept of elimination method.

The elimination method is the method in which addition or subtraction method is used to get an equation in the one variable.

If the coefficients are opposite, use the addition and when the coefficients are same, use the subtraction.

Step 2. Add or subtract the equations to eliminate one variable. 

Let the number of dimes be $x$(1 dime = $\$0.1$) and the number of quarters be $y$(1 quarter = $\$0.25$).

As per the conditions the system of equations thus formed will be:

$x+y=25$ … (1)

$0.1x+0.25y=4$ … (2)

Multiply equation (1) by 0.1.

$\begin{array}{l}0.1(x+y)=0.1\times \left(25\right)\\ 0.1x+0.1y=2.5\text{\hspace{0.17em}.....\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(3\right)\end{array}$

Note that both the equations (2) and (3) have $0.1x$in same sign, therefore, variable $x$can be eliminated if both equations are subtracted.

$\begin{array}{l}0.1x+0.1y=2.5\\ \underset{¯}{(-)0.1x+0.25y=4}\\ -0.15y=-1.5\text{Dividebothsidesby-0.15}\\ y=10\end{array}$

Step 3. Solve for x.

Substitute 10 for $y$into the equation (1) and solve for $x$.

$\begin{array}{c}x+y=25\\ x+\left(10\right)-10=25-10\\ x=15\end{array}$

Therefore, the number of dimes and quarters are 15 and 10 respectively.