alltimeacheiver wrote:John, Karen, and Luke collected cans of vegetables for a food drive. The number of cans
that John collected was 1/2 the number of cans that Karen collected and 1/3 the number
of cans that Luke collected. The number of cans that Luke collected was what fraction of
the total number of cans that John, Karen, and Luke collected?
2
A. 1/5
B. 1/3
C. 2/5
D. 1/2
E. 2/3

I like taneja's answer.

Here's the algebraic approach:

First, we're told that the number of cans that John collected was 1/2 the number of cans that Karen collected

Let's assign one variable here.

When using only one variable, I prefer to assign the variable to whatever appears to be the smallest value. That way we can avoid using fractions. Here, it appears that John collected the fewest cans.

So, if we let J = the number of cans that John collected, then it must be true that:
2J = the number of cans that Karen collected (if John collected 1/2 as many cans, then Karen collected twice as many cans)

We're also told that the number of cans that John collected was 1/3 the number of cans that Luke collected

This means that Luke collected three times as many cans as John.

So, 3J = the number of cans that Luke collected

This means the total number of cans collected was J + 2J + 3J = 6J

Of those 6J cans, Luke collected 3J cans, so Luke collected 1/2 of all the cans (3J/6J = 1/2).

So, the answer is D

alltimeacheiver wrote:John, Karen, and Luke collected cans of vegetables for a food drive. The number of cans
that John collected was 1/2 the number of cans that Karen collected and 1/3 the number
of cans that Luke collected. The number of cans that Luke collected was what fraction of
the total number of cans that John, Karen, and Luke collected?
2
A. 1/5
B. 1/3
C. 2/5
D. 1/2
E. 2/3

We can create the equations:

J = K/2

2J = K

And

J = L/3

3J = L

So the fraction that Luke collected was:

L/(J + K + L) = 3J/(J + 2J + 3J) = 3J/6J = 1/2

Alternate Solution:

Let's assume that John collected 5 cans. Since he collected half as many cans as Karen collected, then she collected 10 cans. Since John collected 1/3 the number of cans that Luke collected, then Luke collected 15 cans. Thus, the total number of cans collected was 5 + 10 + 15 = 30. So Luke collected 15/30 = ½ of all the cans collected.

Answer: D