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At least 100 students at a certain high school study Japanese. If 4 percent of the studetns 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

At least 100 students at a certain high school study Japanese. If 4 percent of the studetns 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.

We are asked whether we can find if more students study French than Japanese. The concept used here is , we can compare similar expressions of measuring in finding out which one of the measured is greater, say we can compare percentages or compare absolute numbers .

In statement 1 absolute number is given, and in the stem percentage is given. Also the first is about percentage of French studying Japanese and the second is the number who study both. We can intuitively understand there is insufficient information in this.

In statement 2 , percentage is given and we can now compare the percentage who study French and the percentage who study Japanese in the school. So sufficient.