kaudes11114 wrote:At least 100 students at a certain high school study Japanese. If 4 percent of the students at the school who study French also study Japanese, do more students at the school study French than Japanese?
(1) 16 students at the school study both French and Japanese.
(2) 10 percent of the students at the school who study Japanese also study French.
A student asked me to provide a step-by-step solution using the Double Matrix method, so here goes....
This technique can be used for most questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions).
Here, we have a population of students, and the two characteristics are:
- studies Japanese or does NOT study Japanese
- studies French or does NOT study French
Let's let J = the TOTAL number of students taking Japanese
And let F = the TOTAL number of students taking French
When we sketch our diagram, we get:
Target question: Is F greater than J?
Given: 4 percent of the students at the school who study French also study Japanese
Since we let F = the TOTAL number of students taking French, we can say that 4% of F = number of students taking BOTH French and Japanese.
In other words, 0.04F = number of students taking BOTH French and Japanese
We can also say that 96% of the students who study French do NOT study Japanese
In other words, 0.96F = number of students taking French but NOT Japanese
So, our diagram now looks like this:
Statement 1: 16 students at the school study both French and Japanese
Since 0.04F = number of students taking BOTH French and Japanese, we can write: 0.04F =16
When we solve this equation for F, we get F = 400
So, 0.96F = 384
So, our diagram now looks like this:
Is this enough information to determine whether or not
F is greater than J?
No.
For example, we COULD fill in the remaining boxes this way...
In this case, F = 400 and J = 116, which means
F IS greater than J
However, we COULD also fill in the remaining boxes this way...
In this case, F = 400 and J = 1016, which means
F is NOT greater than J
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: 10 percent of the students at the school who study Japanese also study French.
J = the TOTAL number of students taking Japanese
So, 0.1J = number of students taking BOTH French and Japanese
Notice that we already determined that 0.04F = number of students taking BOTH French and Japanese
So, we now have two ways to represent the SAME value.
So, it MUST be the case that 0.1J = 0.04F
Let's see what this tells us.
First, to make things easier, let's multiply both sides by 100 to get: 10J = 4F
Divide both sides by 10 to get: J = 4F/10
Divide both sides by F to get: J/F = 4/10
From this, we can conclude that
F IS greater than J
Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer: B
Cheers,
Brent
