Question

Ruth can read 15 pages in 30 minutes. She decides to create a table to keep track of her progress. From the data provided, at what ratio is the constant of proportionality?
Time (Hours) | Pages read
1/2 | 15
1 | 30
1 1/2 | 45
2 | 60
Answers:
A. (1 1/2, 30)
B. ( 1, 30)
C. (2, 60)
D. (15, 1/2)

Answer 1

To find the constant of proportionality in this context, we need to identify the ratio of pages read to time in hours.

We can see that Ruth reads a constant number of pages in every half hour. According to the data provided:

- In 1/2 hour, she reads 15 pages.
- In 1 hour, she reads 30 pages.
- In 1 1/2 hours, she reads 45 pages.
- In 2 hours, she reads 60 pages.

To find the constant of proportionality (k), we can use the first time and page count provided:

**For 1/2 hour:**
\[ k = \frac{\text{Pages}}{\text{Time}} = \frac{15 \text{ pages}}{1/2 \text{ hour}} = 15 \div (1/2) = 15 \times 2 = 30 \text{ pages per hour} \]

**For 1 hour:**
\[ k = \frac{30 \text{ pages}}{1 \text{ hour}} = 30 \text{ pages per hour} \]

**For 1 1/2 hours:**
\[ k = \frac{45 \text{ pages}}{1.5 \text{ hours}} = 45 \div 1.5 = 30 \text{ pages per hour} \]

**For 2 hours:**
\[ k = \frac{60 \text{ pages}}{2 \text{ hours}} = 30 \text{ pages per hour} \]

In every case, the value of k is consistent, confirming that the constant ratio of pages read per hour is 30.

Now identifying the pairs from the answers provided, all options have the form (Pages, Time). We note that 30 pages per hour corresponds to the pair (30, 1) or (k, 1) since we set k to correspond to 1 hour.

Thus, the correct answer that corresponds with the ratio we computed is:

**B: (1, 30)**

This maintains the relationship of pages per hour and matches our computations as k = 30 pages/hour.