A college class with 30 sophomores, 18 juniors, and 12 seniors is
divided into project groups where each group has the same number
of sophomores, juniors, and seniors. What is the greatest number of
groups that can be formed? How many sophomores, juniors, and
seniors are in each project group?
5,3 and 2 respectively
5, 3, 2
To find the greatest number of groups that can be formed, we need to find the greatest common divisor (GCD) of the numbers 30, 18, and 12. The GCD will determine the maximum number of groups.
1. Find the GCD of 30, 18, and 12:
- The prime factorization of 30 is 2 * 3 * 5.
- The prime factorization of 18 is 2 * 3^2.
- The prime factorization of 12 is 2^2 * 3.
- The common factors are 2 * 3 = 6.
- Therefore, the GCD of 30, 18, and 12 is 6.
So, the greatest number of groups that can be formed is 6.
To determine the number of sophomores, juniors, and seniors in each group, divide the total number of each class by the number of groups:
- Number of sophomores per group = 30 / 6 = 5.
- Number of juniors per group = 18 / 6 = 3.
- Number of seniors per group = 12 / 6 = 2.
Therefore, each project group will have 5 sophomores, 3 juniors, and 2 seniors.
To find the greatest number of project groups that can be formed, we need to determine the common number of sophomores, juniors, and seniors in each group.
First, let's find the greatest common factor (GCF) of 30, 18, and 12. The GCF is the largest number that divides evenly into all three numbers.
To find the GCF, we can list the factors of each number and see which one is common to all three:
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 12: 1, 2, 3, 4, 6, 12
Looking at the factors, we see that 6 is the largest number that divides evenly into 30, 18, and 12.
Therefore, the greatest number of project groups that can be formed is 6.
Now, let's determine the number of sophomores, juniors, and seniors in each group.
To find the number of sophomores in each group, we divide the total number of sophomores (30) by the number of groups (6).
30 sophomores / 6 groups = 5 sophomores per group.
To find the number of juniors and seniors in each group, we divide the total number of juniors (18) and total number of seniors (12) by the number of groups (6).
18 juniors / 6 groups = 3 juniors per group.
12 seniors / 6 groups = 2 seniors per group.
Therefore, each project group would consist of 5 sophomores, 3 juniors, and 2 seniors.