Montgomery College has a 64-piece band and a 35-piece orchestra. If 14 people are members of both the band and the orchestra, can the band and orchestra travel in two 45-passenger buses?
Let U = {6, 7, 8, 9, 10, 11, 12}, A = {6, 7, 8, 9}, B = {6, 7, 10, 11}, and C = {8, 10, 12}. List all the members of the given set. (Enter your answers as a comma-separated list.)
(A ∪ B) ∩ C
64+35-14 = 86
So, two buses can hold the lot. Even chaperones!
(A ∪ B) = {6,7,8,9,10,11}
So,
(A ∪ B) ∩ C = {8,10}
To determine if the band and orchestra can travel in two 45-passenger buses, we need to calculate the total number of people in both groups.
Step 1: Calculate the total number of members in the band:
Total number of band members = Number of band members - Number of people in the intersection of band and orchestra
Total number of band members = 64 - 14 = 50
Step 2: Calculate the total number of members in the orchestra:
Total number of orchestra members = Number of orchestra members - Number of people in the intersection of band and orchestra
Total number of orchestra members = 35 - 14 = 21
Step 3: Calculate the total number of people in the band and orchestra:
Total number of people = Total number of band members + Total number of orchestra members + Number of people in the intersection of band and orchestra
Total number of people = 50 + 21 + 14 = 85
Step 4: Check if two 45-passenger buses are enough to accommodate the total number of people:
45-passenger bus capacity = 2 * 45 = 90
Since the total number of people (85) is less than the combined capacity of two 45-passenger buses (90), the band and orchestra can travel in two 45-passenger buses.
To determine if the band and orchestra can travel in two 45-passenger buses, we need to find out the total number of people in the band and orchestra.
First, we can calculate the total number of people in the band and orchestra separately. The band has 64 members, and the orchestra has 35 members. However, since 14 people are members of both the band and the orchestra, we need to subtract this overlapping count once to avoid double counting.
So, let's calculate:
Total number of people in the band = 64
Total number of people in the orchestra = 35
Overlapping count (members in both band and orchestra) = 14
Total number of people in the band = 64 - 14 = 50
Total number of people in the orchestra = 35 - 14 = 21
Now that we have the number of people in the band and orchestra respectively, we can check if they can travel in two 45-passenger buses.
Since each bus has a capacity of 45 passengers, we need to divide the total number of people in each group by the bus capacity to see if it fits.
Number of buses needed for the band = Total number of people in the band / Bus capacity
Number of buses needed for the orchestra = Total number of people in the orchestra / Bus capacity
Number of buses needed for the band = 50 / 45 ≈ 1.11
Number of buses needed for the orchestra = 21 / 45 ≈ 0.47
As the number of buses must be a whole number, we need to round up the fractional numbers to the nearest whole number.
Number of buses needed for the band = 2 (rounded up from 1.11)
Number of buses needed for the orchestra = 1 (rounded up from 0.47)
So, the band needs 2 buses, and the orchestra needs 1 bus. In total, 3 buses would be required. Therefore, two 45-passenger buses would not be enough to accommodate both the band and the orchestra at the same time.