Question

José has a summer job as the batboy for a baseball team. He earns $5 per hour for the practice or game time he works. His pay also includes two season tickets worth a total of $80. Charlie works at the concession stand. He earns $6 per hour plus a $50 bonus if he works at least 20 games. Let h represent the number of hours worked. Let I represent the income received.

a. What equation shows how José’s total income I depends on the

number of hours h he works?


b. Write an inequality that describes the number of hours that José must

work to earn a total income of less than $260. Then determine how

many hours he needs to work to meet this condition.


c. Suppose Charlie works at least 20 games. Write an equation that

describes his total income.


d. Suppose Charlie’s total income is greater than José’s total income,

after working at least 20 games. Write an inequality that describes this

condition. Then dete

Answer 1

a. We know form our problem that Jose makes 5$ per hour. If we let $h$ be the number of ours worked, the rate per our of Jose will be $5h$. We also know that his pay also includes two season tickets worth a total of $80, so his total income will be given by the expression $l=5h+80$.  

We can conclude that the equation that shows  Jose's total income after working $h$ hours is: $l=5h+80$

b. We know that the total income after $h$ hours is given by the right hand side of our equation: $5h+80$. Now, to set up our inequality we are going to set that right

 hand side less than $260:
$5h+80\ \textless \ 260$
To determine the number of hours he should work, we just need to solve for $h$:
$5h+80\ \textless \ 260$
$5h\ \textless \ 180$
$h\ \textless \ \frac{180}{5}$
$h\ \textless \ 36$

We can conclude that Jose needs to work less than 36 hours to make less than $260.

c. Just like before, we know form our problem that Charlie makes 6$ per hour; letting $h$ be the number of worked hours, we get that Charlie's hourly rate is $6h$. Since we are assuming that he is working at least 20 games, he is going to get the $50 bonus, so his total income woll be given by the expression: $l=6h+50$

We can conclude that the equation that describes Charlie's total income is $l=6h+50$.

d. We know form our problem that after working at least 20 games Charlie's total income is grater to Jose's income. We also know from our previous calculation that Charlie's total income is $l=6h+50$ and Jose's total income is $l=5h+80$. Since Charlies total income is greater than Jose's total income, we should set the right hand side of Charlie's equation greater than the right hand side of Jose's equation to reflect that fact:
$6h+50\ \textgreater \ 5h+80$
To find the number of hours we just need to solve for $h$:
$6h+50\ \textgreater \ 5h+80$
$6h-5h\ \textgreater \ 80-50$
$h\ \textgreater \ 30$

We can conclude that they need to work more than 30 hours for Charlie to make more money than Jose.