Jack and Jane are married and both work. However, due to their responsibilities at home, they have decided that they do not want to work over $65$ hours per week combined. Jane is paid $\$ 14$ per hour at her job, and Jack is paid $\$ 7$ per hour at his. Neither of them are paid extra for overtime, but they are allowed to determine the number of hours per week that they wish to work. If they need to make a minimum of $\$ 770$ per week before taxes, what is the maximum amount of hours that Jack can work per week according to these limits?

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Q. Jack and Jane are married and both work. However, due to their responsibilities at home, they have decided that they do not want to work over

65

hours per week combined. Jane is paid

\$ 14

per hour at her job, and Jack is paid

\$ 7

per hour at his. Neither of them are paid extra for overtime, but they are allowed to determine the number of hours per week that they wish to work. If they need to make a minimum of

\$ 770

per week before taxes, what is the maximum amount of hours that Jack can work per week according to these limits?

Define variables: First, let's define the variables for the number of hours Jane and Jack work. Let's say Jane works $J$ hours and Jack works $K$ hours per week. We know that combined they do not want to work over $65$ hours per week. This gives us our first inequality: $\newline$ $J + K \leq 65$
Consider earnings: Next, we need to consider their earnings. Jane earns $\$14$ per hour and Jack earns $\$7$ per hour. They need to make a minimum of $\$770$ per week before taxes. This gives us our second inequality based on their earnings: $\newline$ $14J + 7K \geq 770$
Simplify inequality: We can simplify the second inequality by dividing all terms by $7$ to make the numbers smaller and easier to work with: $2J + K \geq 110$
Express maximum hours: Now, we need to express the maximum hours Jack can work in terms of the total hours they can work together and the minimum amount they need to earn. We can rearrange the first inequality to solve for $K$ : $K \leq 65 - J$
Minimize Jane's hours: We want to find the maximum hours Jack can work, so we need to minimize the number of hours Jane works while still meeting the minimum earnings requirement. To do this, we'll solve the second inequality for $J$ when $K$ is at its maximum: $\newline$ $2J + K = 110$ $\newline$ If $K$ is at its maximum, $J$ is at its minimum. Let's find the minimum $J$ by setting $K$ to its maximum from the first inequality ( $65$ hours), and then solve for $J$ : $\newline$ $2J + 65 = 110$ $\newline$ $K$ $0$ $\newline$ $K$ $1$ $\newline$ $K$ $2$ $\newline$ $J = $22$.$5$
Round up Jane's hours: Since Jane cannot work half an hour, we'll round up to the nearest whole hour for Jane's minimum working hours to ensure they meet their earnings requirement. This means Jane will work at least $23$ hours.
Find maximum hours for Jack: Now we can find the maximum hours Jack can work by substituting Jane's minimum working hours back into the inequality for $K$: $\newline$\[K \leq 65 - J\]$\newline$\[K \leq 65 - 23\]$\newline$\[K \leq 42\]
Find maximum hours for Jack: Now we can find the maximum hours Jack can work by substituting Jane's minimum working hours back into the inequality for $K$:\[K \leq 65 - J\]\[K \leq 65 - 23\]\[K \leq 42\]Therefore, the maximum amount of hours that Jack can work per week, according to these limits, is $42$ hours.