Last Updated on May 10, 2023
A school administrator will assign each student in a…
Solution:
The first task in solving this question is to determine exactly what is being given and what is being asked. We are given that each student in a group of n students is going to be assigned to one of m classrooms; however, we are being asked whether it is possible to assign each of the n students to one of the m classrooms so that each classroom has the same number of students. In order for this to be possible, the number of classrooms (or m) must evenly divide into the number of students (or n). Thus we can rewrite the question as:
Is n/m = integer?
Statement One Alone:
It is possible to assign each of 3n students to one of m classrooms so that each classroom has the same number of students assigned to it.
Statement one is telling us that 3n is evenly divisible by m. Writing this mathematically we can say:
3n/m = integer
Based on this information, we can verify that this statement could be true when n IS divisible by m and when n IS NOT divisible by m. We know this because it’s possible for m to divide evenly into 3n (thus holding the statement true) but not into n. It’s also possible for m to divide evenly into both n and 3n (thus holding the statement true).
Let’s plug in some real number to test this theory. Remember, we know that m is between 3 and 13 and that n is greater than 13.
Case #1: When n is divisible by m
n = 16
m = 4
We see that 3n/m = (3 x 16)/4 = 3 x 4 = 12, so we know that 3n/m = integer. We also see that n is evenly divisible by m, so we answer YES to the question “is n/m = integer?”
Case #2: When n is NOT divisible by m:
n = 20
m = 6
We see that 3n/m = (3 x 20)/6 = 60/6 = 10, so we know that 3n/m = integer. We also see that n is NOT evenly divisible by m, so we answer NO to the question “is n/m = integer?”
Because we had a “yes” answer for case 1 and a “no” answer for case 2, statement one is not sufficient. We can eliminate answers A and D.
Statement Two Alone:
It is possible to assign each of 13n students to one of m classrooms so that each classroom has the same number of students assigned to it.
This statement is telling us that 13n is divisible by m. Writing this mathematically we cansay:
13n/m = integer
What is interesting about this statement is that we know that n is greater than 13 and that m is less than 13 and greater than 3. Thus, we know that m could equal any of the following: 4, 5, 6, 7, 8, 9, 10, 11, or 12. We see that none of those values (4 through 12) will divide evenly into 13.
Knowing this, we can say conclusively that m will never divide evenly into 13. Thus, in order for m to divide into 13n, m must divide evenly into n. Therefore, without having to plug in any values, we can say that statement two is sufficient to answer the question “is n/m = integer?”.
Answer: B
