Q 4.51 - Intermediate Algebra

Question

Translate to a system of equations and solve:

Greta wants to make $5$ pounds of a nut mix using peanuts and cashews. Her budget requires the mixture to cost

her $\backslash (6$ per pound. Peanuts are $\backslash )4$ per pound and cashews are $\$9$ per pound. How many pounds of peanuts and

how many pounds of cashews should she use?

Step-by-Step Solution

Answer

The number of pounds of peanuts used$=3$

The number of pounds of cashews used$=2$

Step 1. Given information

• The weight of the nut mix is $5$ pounds.
• The cost of peanuts and cashews is $\$4$ and $\$9$ per pound respectively.

Step 2. Form the equations

Let the number pounds of peanuts and cashews used in the nuts mix is $p$ and $c$ respectively.

The total amount of nut mix is  $5$pounds.

According to the question, we get:

$p+c=5\_\_\_\left(1\right)$.

Now, the total cost of nut mix would be 

$6\times 5=\$30$

So, 

$4p+9c=30\_\_\_\left(2\right)$

Step 3. Solve the equations using the elimination method.

Multiplying equation $\left(1\right)$ by $4$, we get:

$$\begin{gathered} 4p+4c=5\times 4 \\ 4p+4c=20\_\_\_\_\left(3\right) \end{gathered}$$

Subtracting equation $\left(2\right)$ from equation $\left(3\right)$, we get:

$$\begin{gathered} 4p+4c-(4p+9c)=20-30 \\ 4c-9c=-10 \\ -5c=-10 \\ 5c=10 \end{gathered}$$

Dividing both sides by $5$, we get:

$$\begin{gathered} \frac{5c}{5}=\frac{10}{5} \\ c=2 \end{gathered}$$

Step 4. Find the value of p

Substituting the value of $c=2$ in the equation $\left(1\right)$, we get:

$$\begin{gathered} p+c=5 \\ p+2=5 \\ p=5-2 \\ p=3 \end{gathered}$$

So, Greta uses $3$ pounds of peanuts and $2$ pounds of cashews.