Question

in 22 and 23, use the diagram at the right.

1. at what times between 10:00 a.m. and 5:00 p.m. do the chemistry presentation and the recycling presentation start simultaneously?
2. the museum performs shows in schools every monday and in public libraries every fifth day (on both weekdays and weekends). if the museum did both a school show and a library show on monday, how many days will it be until it does both shows on the same day again?

Explanation:

22.

Step1: Find the least common multiple (LCM) of 10 and 6

Prime - factorize the numbers: \(10 = 2\times5\) and \(6=2\times3\).
The LCM is \(2\times3\times5=30\). This means the chemistry and recycling presentations start simultaneously every 30 minutes.

Step2: List the times

Starting from 10:00 A.M., the times are 10:00 A.M., 10:30 A.M., 11:00 A.M., 11:30 A.M., 12:00 P.M., 12:30 P.M., 1:00 P.M., 1:30 P.M., 2:00 P.M., 2:30 P.M., 3:00 P.M., 3:30 P.M., 4:00 P.M., 4:30 P.M.

Step1: Find the LCM of 7 (days in a week) and 5

Since 7 and 5 are co - prime (their greatest common divisor is 1), the LCM of 7 and 5 is \(7\times5 = 35\)

Answer:

10:00 A.M., 10:30 A.M., 11:00 A.M., 11:30 A.M., 12:00 P.M., 12:30 P.M., 1:00 P.M., 1:30 P.M., 2:00 P.M., 2:30 P.M., 3:00 P.M., 3:30 P.M., 4:00 P.M., 4:30 P.M.

23.