Horace is a professional hair stylist. $\newline$ Let $C$ represent the number of child haircuts and $A$ represent the number of adult haircuts that Horace can give within $7$ hours. $\newline$ $0.75C + 1.25A \leq 7$ $\newline$ Horace gave $5$ child haircuts. How many adult haircuts at most can he give with the remaining time? $\newline$ Choose $1$ answer: $\newline$ (A) Horace can give at most $1$ adult haircut. $\newline$ (B) Horace can give at most $2$ adult haircuts. $\newline$ (C) Horace can give at most $3$ adult haircuts. $\newline$ (D) Horace can give at most $5$ adult haircuts.

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Math Problems

Algebra 1

Modeling with linear inequalities

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Q. Horace is a professional hair stylist.

\newline

Let

C

represent the number of child haircuts and

A

represent the number of adult haircuts that Horace can give within

7

hours.

\newline

0.75C + 1.25A \leq 7

\newline

Horace gave

5

child haircuts. How many adult haircuts at most can he give with the remaining time?

\newline

Choose

1

answer:

\newline

(A) Horace can give at most

1

adult haircut.

\newline

(B) Horace can give at most

2

adult haircuts.

\newline

3

adult haircuts.

\newline

(D) Horace can give at most

5

adult haircuts.

Understanding the inequality: Understand the inequality that represents the time Horace spends on haircuts. $\newline$ The inequality $0.75C + 1.25A \leq 7$ represents the maximum time Horace can spend on child haircuts $(C)$ and adult haircuts $(A)$ within $7$ hours. Here, $0.75$ is the time in hours Horace takes for one child haircut, and $1.25$ is the time in hours for one adult haircut.
Substituting child haircuts: Substitute the number of child haircuts Horace gave into the inequality. $\newline$ Horace gave $5$ child haircuts, so we substitute $C$ with $5$ in the inequality: $\newline$ $0.75 \times 5 + 1.25A \leq 7$
Calculating time spent on child haircuts: Calculate the total time spent on $5$ child haircuts. $0.75 \times 5 = 3.75$ So, the inequality becomes: $3.75 + 1.25A \leq 7$
Finding time left for adult haircuts: Subtract the time spent on child haircuts from the total available time to find the time left for adult haircuts. $\newline$ $7 - 3.75 = 3.25$ $\newline$ Now the inequality is: $\newline$ $1.25A \leq 3.25$
Determining maximum number of adult haircuts: Divide both sides of the inequality by the time it takes for one adult haircut to find the maximum number of adult haircuts Horace can give. $\newline$ $\frac{1.25A}{1.25} \leq \frac{3.25}{1.25}$ $\newline$ $A \leq \frac{3.25}{1.25}$
Calculating maximum number of adult haircuts: Calculate the maximum number of adult haircuts Horace can give. $\newline$ $A \leq \frac{3.25}{1.25}$ $\newline$ $A \leq 2.6$ $\newline$ Since Horace cannot give a fraction of a haircut, he can give at most $2$ adult haircuts.