Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Two workers in a holiday boutique are filling stockings with small gifts and candy. Beth has already filled $4$ stockings and will continue to fill them at a rate of $2$ stockings per hour. Aaron, who just arrived to help, can fill $4$ stockings per hour. At some point, Aaron will catch up with Beth and they will have completed the same number of stockings. How many stockings will each worker have filled by then? How long will that take? $\newline$ The workers will each have filled $\_$ stockings in $\_$ hours.

View step-by-step help

Home

Math Problems

Algebra 2

Solve a system of equations using any method: word problems

Full solution

Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Two workers in a holiday boutique are filling stockings with small gifts and candy. Beth has already filled

4

stockings and will continue to fill them at a rate of

2

stockings per hour. Aaron, who just arrived to help, can fill

4

stockings per hour. At some point, Aaron will catch up with Beth and they will have completed the same number of stockings. How many stockings will each worker have filled by then? How long will that take?

\newline

The workers will each have filled

\_

stockings in

\_

hours.

Define variables: Let's define the variables. Let $x$ be the number of hours Aaron works until he catches up with Beth. Since Beth has already filled $4$ stockings and continues at a rate of $2$ per hour, the total number of stockings she will have filled by the time Aaron catches up is $4 + 2x$ . Aaron fills stockings at a rate of $4$ per hour, so he will have filled $4x$ stockings in $x$ hours.
Set up equation: Set up the equation based on the information that Aaron will catch up with Beth. At that point, they will have filled the same number of stockings, so we can equate Beth's total to Aaron's total: $\newline$ $4 + 2x = 4x$
Solve for x: Solve the equation for x. Subtract $2x$ from both sides to isolate the variable on one side: $\newline$ $4 + 2x - 2x = 4x - 2x$ $\newline$ $4 = 2x$ $\newline$ Divide both sides by $2$ to solve for x: $\newline$ $\frac{4}{2} = \frac{2x}{2}$ $\newline$ $x = 2$
Calculate total stockings: Now that we know $x = 2$ , we can find out how many stockings each worker will have filled. For Beth: $\newline$ Total stockings filled by Beth = $4 + 2x = 4 + 2(2) = 4 + 4 = 8$ $\newline$ For Aaron: $\newline$ Total stockings filled by Aaron = $4x = 4(2) = 8$
Check solution: Check the solution by substituting $x$ back into the original equation to ensure both sides are equal: $\newline$ $4 + 2(2) = 4(2)$ $\newline$ $4 + 4 = 8$ $\newline$ $8 = 8$ $\newline$ Since both sides are equal, the solution is correct.