- The pay Eva receives is proportional to the time worked, so we find her hourly rate by dividing her total pay by the hours worked: \(\frac{\$72}{8} = \$9/\text{hour}\).
- Multiplying this hourly rate by 5 hours gives her total pay for 5 hours: \(5 \times \$9 = \$45\).
- To check if 8 can be divided by a whole number to equal 5, we solve \(\frac{8}{n} = 5\), which gives \(n = 1.6\); since 1.6 is not an integer, the answer is "No".
- Final Answers: Can 8 be divided by a whole number to get 5? \(\boxed{\text{No}}\). Total pay for 5 hours: \(\boxed{\$45}\).
Explanation
- Analyzing the Problem Data
We are given that Eva's pay is directly proportional to the number of hours she works. She worked 8 hours and earned \(\$72\). We need to determine two things:
- Whether 8 can be divided by a whole number to equal 5.
- How much Eva will make for 5 hours of work.
The problem states that the pay is proportional to the time worked, which means we can express the pay \(P\) as a function of the hours worked \(h\) using the formula \(P = k \times h\), where \(k\) is the constant hourly rate.
- Evaluating the Sub-question
To answer the sub-question, we need to check if there exists a whole number (an integer) \(n\) such that \(8 / n = 5\).
We can set up the equation:
\[8 / n = 5\]
Multiplying both sides by \(n\):
\[8 = 5n\]
Dividing by 5:
\[n = \frac{8}{5} = 1.6\]
Since 1.6 is not a whole number, there is no whole number that divides 8 to result in 5. Therefore, the answer is "No".
- Calculating Pay for 5 Hours
Since the hourly rate is constant, we first determine the rate \(k\) by dividing the total pay for 8 hours by the number of hours:
\[k = \frac{\$72}{8 \text{ hours}} = \$9 \text{ per hour}\]
Now that we know the hourly rate is \(\$9\), we can calculate the pay for 5 hours by multiplying the hourly rate by 5:
\[\text{Pay} = 5 \text{ hours} \times \$9/\text{hour} = \$45\]
Thus, for 5 hours of work, Eva will make \(\$45\).
Examples
Understanding proportional relationships is essential for everyday financial tasks, such as calculating earnings based on time or estimating costs when shopping. For instance, if you are paid a fixed hourly wage, you can easily project your total earnings for different shifts using the constant hourly rate. This same concept applies to scaling recipes for cooking, calculating distances on a map using a scale, or even budgeting for recurring expenses. By identifying the unit rate, you can quickly solve for any unknown quantity in a proportional relationship.