Any guidance with this logic puzzle/problem?

Question

In a literature class, 30% of students speak French, 35% speak German and 50% of students speak French or German. What is the proportion of students who a speak French and German and who speak French but not German?

Also, please note I'm not looking specifically for an answer, but some guidance on how something like this should be approached. Perhaps a Venn Diagram type of problem? Thanks! :)

Answer 1

Using the definition of Union of two events, you get

P(French) + P(German) - P(French AND German) = Union of the two events

From the data you know,

P(French) = 0.30
P(German) = 0.35
P(Union meaning French OR German) = 0.50

Solving for the interception (French AND German)

0.30 + 0.35 - P(F AND G) = 0.50

P(F AND G) = 0.15

Now that you now the interception of these two events, we can solve for the French speakers only.

From the Venn diagram = P(Only French) = P(French) - P (Interception)

= 0.30 - 0.15 = 0.15

Good luck!

Answer 2

This may be best set up as an algebra representation of a Venn.

Assume 100 students...30 speak french, 35 speak German. 50 speak French or German. Thus we know that with a total of 65 possible, only 50 are unique, and thus there must be 15 overlapping.

So for the next part, 15 of the 30 French must speak German, and 15 of the 35 German must speak French.

Answer 3

If 30% speak French and 35% speak German, and 50% speak French or German, then 15% of the total must speak both French and German (think of your Venn Diagram with circles representing the two languages - how much must be in the overlap area for the total to be 50% if the circle for French represents 30% and the circle for German represents 35%? Then if 15% is in the overlap, that leaves 15% in the French-only part of the diagram, and 20% in the German-only part. You can get your proportions from there.

Answer 4

A Venn diagram is a fine way to visualize this but really hard to draw here. Your diagram contains the whole class with circles inside for A. "Speak French" and B. "Speak German." In making this or any Venn diagram, you want all possible intersections even if their area later turns out to be zero. In this case the only intersection is between A and B. You have four areas now "Neither French or German", "French Only", "German Only". and "Both French and German". By the percentages given, you can figure the size of each area and buy simple logic, you can find the size of the "French but not German" area.

Answer 5

I would first determine my knowns % people/language(s). Then draw a graph diagram to represent the knowns. Next begin to manipulate the knowns to isolate unknowns to get the unknown %ages. Just work through it all one step at a time and it will all come together.

Answer 6

55 %