Question

1. gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. how long would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time? (2002 amc 8 problem, question #18)

a. 1 hr b. 1 hr 10 min c. 1 hr 20 min d. 1 hr 40 min e. 2 hr
can you use the method of calculating the average to find an equal relationship?
be careful with the units here!
make them the same first!

Explanation:

Step1: Convert time to minutes

1 hour 15 minutes = \(60 + 15=75\) minutes.
He skated 75 minutes each day for 5 days, so the total time for these 5 days is \(75\times5 = 375\) minutes.
1 hour 30 minutes = \(60+30 = 90\) minutes.
He skated 90 minutes each day for 3 days, so the total time for these 3 days is \(90\times3=270\) minutes.
Let the time he skates on the 9th day be \(x\) minutes.

Step2: Use the average formula

The average formula is \(\text{Average}=\frac{\text{Sum of all values}}{\text{Number of values}}\).
Here, the number of days \(n = 9\) and the average \(\bar{x}=85\) minutes.
The sum of all skating times is \(375 + 270+x\).
We know that \(85=\frac{375 + 270+x}{9}\).
Multiply both sides by 9: \(85\times9=375 + 270+x\).
Calculate \(85\times9=765\), and \(375 + 270=645\).
So, \(765=645+x\).
Subtract 645 from both sides: \(x=765 - 645=120\) minutes.

Answer:

Since \(120\) minutes \( = 2\) hours, the answer is E. 2 hr.