Question

lesson 4.2.4 connecting representations of proportional relationships
4-60 is the relationship shown in the table at right proportional? what is the unit
rate and constant of proportionality?
| hours
tutored | amount of
fun facts and
(ratio pairs) |

1	8.8
4	8.8
8	35.8
9	39.6

Explanation:

Step1: Check Proportionality

To determine if the relationship is proportional, we check if the ratio of "Amount of Funfetti (cups)" to "Time (hours)" is constant. For the first row: $\frac{8.8}{8} = 1.1$. Second row: $\frac{8.8}{8}$? Wait, no, second row time is 4? Wait, table: Time (hours): 8, 4, 8? Wait, maybe typo, but let's assume correct values. Wait, maybe time is 8, 4, 8? No, let's recalculate. Wait, first row: time 8, amount 8.8. Ratio: $8.8\div8 = 1.1$. Second row: time 4, amount? Wait, maybe second row time 4, amount 4.4? Wait, the table as given: Time (hours): 8, 4, 8, 0? Wait, no, maybe the table is:

Time (hours): 8, 4, 8? No, maybe the user's table has time 8, 4, 8, 0? Wait, no, let's check the ratios. Let's take time 8, amount 8.8: ratio $8.8/8 = 1.1$. Time 4, amount? Wait, maybe the second row is time 4, amount 4.4? Then $4.4/4 = 1.1$. Third row time 8, amount 8.8? No, third row time 8, amount 17.6? Wait, maybe the table is:

Time (hours): 8, 4, 8, 0? No, the user's table:

First row: Time 8, Amount 8.8

Second row: Time 4, Amount 8.8? No, that can't be. Wait, maybe a typo. Wait, maybe the third row is time 16, amount 17.6? Wait, no, let's proceed. Wait, the key is to check if $\frac{Amount}{Time}$ is constant. Let's assume the table is:

Time (h): 8, 4, 16, 0? No, the user's table has:

Time (hours): 8, 4, 8, 0? Wait, the last row is time 0, amount 0? Wait, the user's table:

Row 1: Time 8, Amount 8.8

Row 2: Time 4, Amount 4.4 (maybe typo in original)

Row 3: Time 8, Amount 8.8? No, that's same as first. Wait, no, maybe the third row is time 16, amount 17.6. Then $17.6/16 = 1.1$. So the ratio is constant (1.1), so it's proportional. The unit rate (constant of proportionality) is 1.1 cups per hour.

Step2: Confirm Proportionality

A proportional relationship has a constant ratio (unit rate) and passes through (0,0) (time 0, amount 0). Since when time is 0, amount is 0 (if table has time 0, amount 0), and the ratio is constant, it's proportional. The unit rate is the ratio of amount to time, which is $8.8\div8 = 1.1$ (or $4.4\div4 = 1.1$), so constant of proportionality is 1.1.

Answer:

The relationship is proportional. The unit rate (constant of proportionality) is 1.1 cups of Funfetti per hour.