Q. 149 - Intermediate Algebra

Question

A $90\%$ antifreeze solution is to be mixed with a $75\%$ antifreeze solution to get $360$ liters of an $85\%$ solution. How many liters of the $90\%$ and how many liters of the $75\%$ will be used?

Step-by-Step Solution

Answer

$240$ liters of the $90\%$ solution and $120$ liters of  the $75\%$ solution.

Step 1. Given information.

Consider the given question,

$90\%$solution is to be mixed with the $75\%$ solution to get $360$ liters of the $85\%$.

Assume $x$ to be the number of liters of $90\%$ solution and $y$ to be the number of liters of $75\%$ solution.

Step 2. Form the equations.

Consider the given question,

$x+y=360 ...... \left(i\right)$

Also, $90\% of \left(x\right)+75\% of \left(y\right)=85\% of \left(360\right)$

Then,

$$\begin{gathered} 0.90x+0.75y=0.85\left(360\right) \\ 0.90x+0.75y=306 ...... \left(ii\right) \end{gathered}$$

From equation (i),

$$\begin{gathered} x+y=360 \\ x=360-y ...... \left(iii\right) \end{gathered}$$

Step 3. Substitute the values in the equations and solve.

Substitute the values of $x$ in equation (ii),

$$\begin{gathered} 0.9\left(360-y\right)+0.75y=306 \\ 324-0.9y+0.75y=306 \\ -0.15y=-18 \\ y=\frac{-18}{-0.15} \\ y=120 \end{gathered}$$

Substitute the value of $y$ in equation (iii),

$$\begin{gathered} x+120=360 \\ x=240 \end{gathered}$$

Step 4. Check the answers.

Substitute the values in equation (i),

$$\begin{gathered} 240+120=360 \\ 360=360 \end{gathered}$$

This is true.

Substitute the values in equation (ii),

$$\begin{gathered} 0.9\left(240\right)+0.75\left(120\right)=306 \\ 216+90=306 \\ 306=306 \end{gathered}$$

This is also true.

Therefore, the number of liters of $90\%$ solution is about $240$ liters and the number of liters of $75\%$ solution is about $120$ liters.