Answer :
To find the equivalent rates at which Michael read, we need to determine if the rates in the options match with the rate at which Michael originally read the pages. Here's how you can do it step-by-step:
1. Determine Michael's Reading Rate:
- Michael read 135 pages in [tex]\(1 \frac{1}{2}\)[/tex] hours, which is 1.5 hours.
- To find the rate, divide the number of pages by the time:
[tex]\[
\text{Rate} = \frac{135 \text{ pages}}{1.5 \text{ hours}} = 90 \text{ pages per hour}
\][/tex]
2. Check Each Option:
- Option A: 180 pages in 2 hours
[tex]\[
\text{Rate} = \frac{180 \text{ pages}}{2 \text{ hours}} = 90 \text{ pages per hour}
\][/tex]
This matches Michael's rate.
- Option B: 60 pages in 30 minutes
- 30 minutes is 0.5 hours.
[tex]\[
\text{Rate} = \frac{60 \text{ pages}}{0.5 \text{ hours}} = 120 \text{ pages per hour}
\][/tex]
This does not match Michael's rate.
- Option C: 108 pages in [tex]\(2 \frac{1}{2}\)[/tex] hours
- [tex]\(2 \frac{1}{2}\)[/tex] hours is 2.5 hours.
[tex]\[
\text{Rate} = \frac{108 \text{ pages}}{2.5 \text{ hours}} = 43.2 \text{ pages per hour}
\][/tex]
This does not match Michael's rate.
- Option D: 150 pages in 1 hour
[tex]\[
\text{Rate} = \frac{150 \text{ pages}}{1 \text{ hour}} = 150 \text{ pages per hour}
\][/tex]
This does not match Michael's rate.
- Option E: 225 pages in 150 minutes
- 150 minutes is 2.5 hours.
[tex]\[
\text{Rate} = \frac{225 \text{ pages}}{2.5 \text{ hours}} = 90 \text{ pages per hour}
\][/tex]
This matches Michael's rate.
- Option F: 210 pages in 2 hours 20 minutes
- 2 hours 20 minutes is [tex]\(2 \frac{1}{3}\)[/tex] hours, approximately 2.333 hours.
[tex]\[
\text{Rate} = \frac{210 \text{ pages}}{2.333 \text{ hours}} \approx 90.01 \text{ pages per hour}
\][/tex]
This does not exactly match Michael's rate due to rounding.
3. Select Matching Rates:
- Options that have the same rate as Michael's are A and E.
So, the rates that are equivalent to Michael's reading rate are A and E.
1. Determine Michael's Reading Rate:
- Michael read 135 pages in [tex]\(1 \frac{1}{2}\)[/tex] hours, which is 1.5 hours.
- To find the rate, divide the number of pages by the time:
[tex]\[
\text{Rate} = \frac{135 \text{ pages}}{1.5 \text{ hours}} = 90 \text{ pages per hour}
\][/tex]
2. Check Each Option:
- Option A: 180 pages in 2 hours
[tex]\[
\text{Rate} = \frac{180 \text{ pages}}{2 \text{ hours}} = 90 \text{ pages per hour}
\][/tex]
This matches Michael's rate.
- Option B: 60 pages in 30 minutes
- 30 minutes is 0.5 hours.
[tex]\[
\text{Rate} = \frac{60 \text{ pages}}{0.5 \text{ hours}} = 120 \text{ pages per hour}
\][/tex]
This does not match Michael's rate.
- Option C: 108 pages in [tex]\(2 \frac{1}{2}\)[/tex] hours
- [tex]\(2 \frac{1}{2}\)[/tex] hours is 2.5 hours.
[tex]\[
\text{Rate} = \frac{108 \text{ pages}}{2.5 \text{ hours}} = 43.2 \text{ pages per hour}
\][/tex]
This does not match Michael's rate.
- Option D: 150 pages in 1 hour
[tex]\[
\text{Rate} = \frac{150 \text{ pages}}{1 \text{ hour}} = 150 \text{ pages per hour}
\][/tex]
This does not match Michael's rate.
- Option E: 225 pages in 150 minutes
- 150 minutes is 2.5 hours.
[tex]\[
\text{Rate} = \frac{225 \text{ pages}}{2.5 \text{ hours}} = 90 \text{ pages per hour}
\][/tex]
This matches Michael's rate.
- Option F: 210 pages in 2 hours 20 minutes
- 2 hours 20 minutes is [tex]\(2 \frac{1}{3}\)[/tex] hours, approximately 2.333 hours.
[tex]\[
\text{Rate} = \frac{210 \text{ pages}}{2.333 \text{ hours}} \approx 90.01 \text{ pages per hour}
\][/tex]
This does not exactly match Michael's rate due to rounding.
3. Select Matching Rates:
- Options that have the same rate as Michael's are A and E.
So, the rates that are equivalent to Michael's reading rate are A and E.
