Question

Amy is counting her tips and she wants to have an average of $75 per night. The first few nights she made $90, $55, $42, and $79. What does she have to make the fifth night to meet her average she is aiming for? Write an inequality that represents her situation.

Answer 1

If Amy wants an average of $75 or more in tips, then the inequality that represents this would be:
`(90+55+42+79+x)/(5) \ge 75`

The stuff on the left is the average of {90,55,42,79,x} where x is some positive number. Adding those values up and dividing by the sample size 5 gets the average. We want the average to be 75 or larger.

Let's solve for x.

`(90+55+42+79+x)/(5) \ge 75\\\\(266+x)/(5) \ge 75\\\\266+x \ge 5*75\\\\266+x \ge 375\\\\x \ge 375-266\\\\x \ge 109\\\\`

If Amy's tips are $109 or more on the fifth night, then her average tips will be $75 or more.

Answer 2

Amy needs to make at least $109 in tips on the fifth night.

Explanation:

To find out how much Amy needs to make on the fifth night to meet her goal of having an average of $75 per night, we can set up an inequality.

Let x represent the amount Amy needs to make on the fifth night.

To calculate the average of a set of data, sum the data values and divide them by the number of data values.

Therefore, the average amount Amy makes after 5 nights is:

`(90 + 55 + 42 + 79 + x)/(5)`

She wants to have an average of $75 per night after five nights.

Therefore, the inequality that represents this is:

`(90 + 55 + 42 + 79 + x)/(5)\geq 75`

To calculate how much Amy needs to make on the fifth night to meet her average she is aiming for, solve the inequality for x:

`\begin{aligned}266 + x & \geq 375\\266+x-266&\geq375-266\\x&\geq 109\end{aligned}`

Therefore, Amy needs to make at least $109 on the fifth night to meet her goal of having an average of $75 per night.