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PCAT Quantitative Reasoning – Languages Word Problem

A classroom at a Swiss university contains 300 students. 68 of these students only speak English and 76 only speak French. Among these students, 52 speak more than one language. 18 of the students speak both English and German but not French, and 146 speak at least German. There are twice as many students who speak both French and German, but not English, as there are students who speak all three languages. How many students are trilingual in this classroom?

  1. 4
  2. 6
  3. 8
  4. 10

C is correct. This word problem is best solved using a Venn diagram, where English, German, and French speakers are each represented by a circle. 

We know that the total number of students is 300. Then we can start filling in each compartment of the Venn diagram. The number of English-only speakers is 68, and the number of French-only speakers is 76. However, we are not told how many only speak German. Then we are told that 52 students speak more than one language. This must mean that all of the remaining students only speak one language:

300 – 52 = 248.

Now we can solve for the number of German-only speakers (G) because the number of English-only, German-only, and French-only speakers must add to 248:

68 + 76 + G = 248

G = 104

Then we are told that 18 English speakers speak both English and German. We are also told that 146 speak German, which is equal to the total number of enclosed by the red circle. Twice as many speak German and French only (A in the diagram below) as those who speak all three languages (B in the diagram below).

Therefore 104 + 18 + A + B = 146 and A = 2B. Now we can solve for B, the number of trilingual students:

A = 2B

104 + 18 + A + B = 146

104 + 18 + 2B + B = 146

3B = 24

B = 8

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