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A clothier who only makes shirts and pants can make a shirt in $4$ hours and a pair of pants in $6$ hours. Which of the following equations shows the number of shirts, $s$ , and the number of pants, $p$ , that the clothier can make in a $40$ hour work week?

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Write a linear function: word problems

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Q. A clothier who only makes shirts and pants can make a shirt in

4

hours and a pair of pants in

6

hours. Which of the following equations shows the number of shirts,

s

, and the number of pants,

p

, that the clothier can make in a

40

hour work week?

Calculate Rates: Determine the rate at which the clothier can make shirts and pants. $\newline$ The clothier can make a shirt in $4$ hours and a pair of pants in $6$ hours. This means in one hour, the clothier can make $\frac{1}{4}$ of a shirt and $\frac{1}{6}$ of a pair of pants.
Total Production in $40$ Hours: Calculate the total number of shirts and pants that can be made in $40$ hours. $\newline$ Since the clothier works for $40$ hours, we multiply the hourly rate of making shirts and pants by $40$ to find the total number of shirts and pants that can be made. $\newline$ For shirts: $\frac{1}{4} \text{ shirt/hour} \times 40 \text{ hours} = 10 \text{ shirts}$ $\newline$ For pants: $\frac{1}{6} \text{ pair of pants/hour} \times 40 \text{ hours} = \frac{40}{6} \text{ pairs of pants}$
Equation for Shirts and Pants: Write the equation that represents the relationship between the number of shirts, s, and the number of pants, p, that the clothier can make in $40$ hours. $\newline$ Let s be the number of shirts and p be the number of pants. The clothier's work for shirts and pants can be represented by the equations: $\newline$ $s = \frac{1}{4} \times 40$ $\newline$ $p = \frac{1}{6} \times 40$ $\newline$ However, these equations give us the exact number of shirts and pants, not a relationship between s and p. We need to find an equation that relates s and p to the total time worked, which is $40$ hours.
Combine Rates for Equation: Combine the rates to form a single equation. $\newline$ The total time spent making shirts and pants should add up to $40$ hours. The time spent on shirts is $4$ hours per shirt, and the time spent on pants is $6$ hours per pair of pants. The equation is: $\newline$ $4s + 6p = 40$ $\newline$ This equation shows the relationship between the number of shirts, s, and the number of pants, p, that the clothier can make in a $40$ -hour work week.

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