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Roberto plans to start a new job. In preparation, he decides that he should spend no more than 30 hours per week on the job and homework combined. If Roberto wants to have at least 2 homework hours for every 1 hour at his job, what is the maximum number of hours that he should spend at his job each week?
Choose 1 answer:
(A) 9 hours
(B) 10 hours
(c) 20 hours
(D) 21 hours

Roberto plans to start a new job. In preparation, he decides that he should spend no more than $30$ hours per week on the job and homework combined. If Roberto wants to have at least $2$ homework hours for every $1$ hour at his job, what is the maximum number of hours that he should spend at his job each week? $\newline$ Choose $1$ answer: $\newline$ (A) $9$ hours $\newline$ (B) $10$ hours $\newline$ (C) $20$ hours $\newline$ (D) $21$ hours

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Counting principle

Full solution

Q. Roberto plans to start a new job. In preparation, he decides that he should spend no more than

30

hours per week on the job and homework combined. If Roberto wants to have at least

2

homework hours for every

1

hour at his job, what is the maximum number of hours that he should spend at his job each week?

\newline

Choose

1

answer:

\newline

(A)

9

hours

\newline

(B)

10

hours

\newline

(C)

20

hours

\newline

(D)

21

hours

Denoting hours for job and homework: Let's denote the number of hours Roberto spends at his job as $J$ and the number of hours he spends on homework as $H$ . According to the problem, Roberto wants to spend at least $2$ hours on homework for every hour at his job. This can be written as: $\newline$ $H = 2J$
Total hours constraint: We also know that the total number of hours spent on the job and homework combined should not exceed $30$ hours per week. This can be expressed as: $\newline$ $J + H \leq 30$
Substituting equations: Substituting the first equation $H = 2J$ into the second equation $J + H \leq 30$ , we get: $J + 2J \leq 30$
Combining like terms: Combining like terms, we have: $3J \leq 30$
Dividing the inequality: To find the maximum number of hours Roberto can spend at his job, we divide both sides of the inequality by $3$ : $\newline$ $J \leq \frac{30}{3}$
Calculating the maximum hours: Calculating the division, we get: $\newline$ $J \leq 10$
Calculating the maximum hours: Calculating the division, we get: $\newline$ $J \leq 10$ Therefore, the maximum number of hours Roberto should spend at his job each week is $10$ hours.

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