Q 4.52 - Intermediate Algebra

Question

Translate to a system of equations and solve:

Sammy has most of the ingredients he needs to make a large batch of chilli. The only items he lacks are beans

and ground beef. He needs a total of $20$ pounds combined of beans and ground beef and has a budget of $\backslash (3$ per

pound. The price of beans is $\backslash )1$ per pound and the price of ground beef is $\$5$ per pound. How many pounds of

beans and how many pounds of ground beef should he purchase?

Step-by-Step Solution

Answer

The number of pounds beans needed$=10$ pounds

The number of pounds of ground beef needed $=10$pounds

Step 1. Given information

• The weight of batch of chilli Sammy want to make is $20$ pounds.
• The cost of beans and ground beef is  $\$1$ and $\$5$ per pound respectively.

Step 2. Form the equations

Let the number pounds of beans and grounded beef used in the nuts mix is $b$ and $g$ respectively.

The weight of the batch of chilli should be $20$ pounds.

 According to the question,

$b+g=20\_\_\_\_\left(1\right)$

Now, the total cost of the batch of chilli would be

$20\times 3=\$60$.

So,

$b+5g=60\_\_\_\left(2\right)$

Step 3. Solve the equations using the elimination method.

Subtracting equation $\left(1\right)$ from equation $\left(2\right)$, we get:

$$\begin{gathered} b+5g-(b+g)=60-20 \\ 5g-g=40 \\ 4g=40 \end{gathered}$$

Dividing both sides by $4$, we get:

$$\begin{gathered} \frac{4g}{4}=\frac{40}{4} \\ g=10 \end{gathered}$$

Step 4. Find the value of b

Substituting the value $g=10$ in the equation $\left(1\right)$, we get:

$$\begin{gathered} b+g=20 \\ b+10=20 \\ b+10-10=20-10 \\ b=10 \end{gathered}$$

So, the number of pounds of beans and ground beef needed to make the batch of chilli is $10$ pounds and $10$ pounds respectively.