- The identified unit rate is \(\frac{1}{3}\) pages per minute (as indicated by the provided options \(\frac{1}{3}\) and \(\frac{40}{120}\)).
- Using this unit rate, the number of pages Dan can read in 120 minutes is calculated as \(\frac{1}{3} \times 120 = 40\).
- Final unit rate: \(\boxed{\frac{1}{3}}\) pages per minute.
- Total pages in 120 minutes: \(\boxed{40}\) pages.
Explanation
- Analyzing Reading Rates and Identifying the Unit Rate
To determine Dan's reading rate, we first analyze the two conditions provided:
- Dan can read 8 pages in 15 minutes.
- Dan can read 100 pages in \(53 \frac{1}{3}\) minutes.
First, we express the time \(53 \frac{1}{3}\) minutes as an improper fraction:
\[53 \frac{1}{3} = \frac{53 \times 3 + 1}{3} = \frac{159 + 1}{3} = \frac{160}{3} \text{ minutes.}\]
Next, we calculate the unit rate for each condition (pages per minute):
For the first condition:
\[\text{Rate}_1 = \frac{8 \text{ pages}}{15 \text{ minutes}} = \frac{8}{15} \approx 0.533 \text{ pages/min.}\]
For the second condition:
\[\text{Rate}_2 = \frac{100 \text{ pages}}{\frac{160}{3} \text{ minutes}} = 100 \times \frac{3}{160} = \frac{300}{160} = \frac{30}{16} = 1.875 \text{ pages/min.}\]
Given that the options provided for the unit rate are \(\frac{1}{3}\) or \(\frac{40}{120}\) (which is equivalent to \(\frac{1}{3}\)), we identify that the intended unit rate is \(\frac{1}{3}\) pages per minute. Note that the rates calculated from the raw numbers in the text appear inconsistent with this result, likely due to a typo in the provided data. We proceed using the intended unit rate of \(\frac{1}{3}\) pages per minute as it leads to consistent calculations for the subsequent part of the question.
- Calculating Pages Read in 120 Minutes
Using the identified unit rate of \(\frac{1}{3}\) pages per minute, we can calculate how many pages Dan can read in 120 minutes.
The calculation is as follows:
\[\text{Total pages} = \text{Unit Rate} \times \text{Time}\]
\[\text{Total pages} = \frac{1}{3} \text{ pages/min} \times 120 \text{ minutes}\]
\[\text{Total pages} = \frac{120}{3} = 40 \text{ pages.}\]
Thus, at a rate of \(\frac{1}{3}\) pages per minute, Dan can read 40 pages in 120 minutes.
Examples
Understanding unit rates is essential for planning tasks efficiently, such as estimating how long it will take to complete a reading assignment or a project. For instance, if you know your reading speed is 2 pages per minute, you can accurately predict that reading a 100-page book will take 50 minutes. This skill helps in time management and setting realistic goals for daily activities.