At least 100 students at a certain high school study japanes

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Is it B?
1. 16 students study both F and J, so there are 400 students studying F: insuff
2. 10% J study F while 4%F study J, so the number of students studying F is greater than that of studens studying J

We have J >= 100, and 0.04F also study Japanese.

1. 16 students study both French and Japanese. This means 400 students study French, since 4% of 400 is 16, however we don't know how many students study Japanese. INSUFF

2. This tells us 0.1J also study French.
This number is actually identical to 0.04F because it represents the same quantity, which is the number of students who study both French and Japanese!

=> 0.1J = 0.04F
=> J = 0.4F
Thus more students study Japanese than French! Sufficient

Pick B

IMO, it cannot be B:

The 4% is for J&F not F alone (Thus it comprises students who study both Japanese and French). Hence, the statement 10% of J = 4 % of F is not correct. If we combine the two statements, we will get a definite answer:

From (1), we get F=400
From (2), we get 10% of J = 4% of J&F. From (1), J&F=16, therefore, if 10% of J = 16, J=160. Thus J<F. The two statements combined are sufficient.

Correct me if I am wrong...

Thus, my ans. is (C).

OA for this question is B.

Let # of students who study Japanese be \(J\), the # of of students who study French be \(F\) and # of students who study Japanese and French be \(F&J\).

From stem # of students who study Japanese and French is 4% of the # of students who study French --> \(F&J=4%F\);

From (2) # of students who study Japanese and French is 10% of the # of students who study Japanese (so MORE share of the same group) --> \(F&J=10%J\).

So, \(10%J=4%F\) --> \(\frac{F}{J}=2.5\) --> \(F>J\).

Hi,

Statement 1:

16 students study F and J. There should be 400 students studying F. The number of J students could be any of 300 or 400 or 500 (lesser or greater? cannot tell).

Statement 2:

Let x be the number of students who study F and J.

x is 4% of F
x is 10% of J. This implies that J is definitely lesser than F.

Hence, sufficient.

Hope this helps. Thanks.

Let J represent the set of students studying Japanese and F represent the set of students studying French
F^J is the set of students studying both Japanese and French
From the question J>=100 , .04F= F^J , is F>J ?

1) F^J=16 => .04F =16 => F=400 . We still don't have any information about J. It can >=100. If 100<= j<400 , F>J. Otherwise F<=J. So not sufficient
2) .1J = F^J => .1J=.04F => F = 2.5 J . Clearly F is bigger than J. So sufficient

Answer B

This question can also be solved by using venn diagram.

Interpreting the Given Info
The question tells us about students in a school studying Japanese and French. Let's represent them using a venn diagram.



a - Number of students who study only Japanese
b - Students who study both Japanese and French
c- Students who study only French
d- Students who study neither Japanese nor French

We are given that atleast 100 students study Japanese i.e. a + b => 100

Also we are told that 4% of the students who study French also study Japanese i.e. 4%(b + c) = b. It can be simplified to c = 24b

We are asked to find if there are more students at the school who study French than Japanese i.e. if b + c > b + a, which simplifies to c > a?

We can either
i. find the values of c and a to answer the question or
ii. express a in terms of b and then compare it with c to answer the question.

Let's see if the statements provide us with the required information.

Statement-I
St-I tells us that 16 students study both French and Japanese i.e. b = 16. This would give us a => 84 and c = 384. As a => 84, a > 384 or a < 384. Since we do not have a definite value of a we can't say for sure if c > a.

Hence st-I is insufficient to answer the question.

Statement-II
St-II tells us that 10% of the students who study French also study Japanese i.e. 10%(a + b) = b i.e. a = 9b.
Now, we have a and c both in terms of b. We see that c =24b and a = 9b i.e. c > a .

Hence st-II is sufficient to answer the question.

Hope this helps

Regards
Harsh

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Hi Bunuel,

So, "4 percent of the students at the school who study french also study japanese" and "10 percent of the students at the school who study japanese also study french" - these 2 are the same group of people?

Yes. Students who study french and also study japanese are those who study both languages the same way students who study japanese and also study french are also those are those who study both languages.

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

RELATED VIDEO

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.

Hence B.

A more thorough video explanation can be found here:
https://www.youtube.com/watch?v=fmam6p_Ello&t=2s

We need a mathematical relationship between F and J. Statement (1) doesn not provide one, but (2) does:

.10(J) = .04(F)

proportion F/J = 4/10

For every 4 students taking Japanese, 10 take French. Sufficient information to answer the question. Answer B.

Note: Instead of writing a Venn diagram, its often easier to visualize things by writing a matrix with two columns and two rows

Columns: study Japanese (J), don't study J
Rows: study French (F), don't study F

We can let f = the number of students who study French and j = number of who study Japanese, and we need to determine whether f is greater than j. Furthermore, we are given that j ≥ 100 and 4 percent of the students at the school who study French also study Japanese; in other words, 4 percent of the students at the school who study French study both languages.

Statement One Alone:

16 students at the school study both French and Japanese.

We can create the equation:

0.04f = 16

4f = 1600

f = 400

However, since we don’t know the value of j (except that it’s at least 100), statement one alone is not sufficient.

Statement Two Alone:

10 percent of the students at the school who study Japanese also study French.

In other words, 10 percent of the students at the school who study Japanese study both languages. Even though we don’t know the exact number of students who study both languages, we know that it’s 0.04f. Therefore,

0.1j = 0.04f

10j = 4f

2.5j = f

We see that the number of students who study French is 2.5 times those who study Japanese. So f is indeed greater than j. Statement two alone is sufficient.

Answer: B

This question made me chuckle. While we spend days and months memorizing formulas and rules, it is actually questions as simple as this one that decide the fate of GMAT takers.
4% of X = 10% Y. A primary school kid can tell you that X > Y. But as the statistics suggest, 49% of us over thought it.

IS it just me who read the question wrong as 4% OF French speak Jap and 10% of Jap Speak French so common = 0.04F +0.1F

Can you assume in this question that only two languages are taught in this school? Won't the possibility of the third language change the answer?