Logical Reasoning Discussion

**Common Information**

In a survey conducted at a University, it was found that 51% of the students wanted to learn French as a foreign language, 48% wanted to learn German and 52% wanted to learn Russian.

Of the surveyed students, 21% wanted to learn both French and German, 23% wanted German and Russian and 24% wanted French and Russian. Only 12% wanted to learn all three languages. A total of 500 students were surveyed.

Q. |
What is the ratio of the number of students interested in exactly two languages to those interested in only one language? |

✖ A. |
30/51 |

✔ B. |
32/51 |

✖ C. |
40/51 |

✖ D. |
42/51 |

**Solution:**

Option(**B**) is correct

To solve this question set, it is absolutely critical to mark the different sections of the Venn diagram appropriately or you may stand to get confused.

Let *F* denote French, *G* denote German and *R* denote Russian.

Total number of students surveyed = n(*U*) = 100% = 500

So, it follows that n(*F*) = 51%, n(*G*) = 48% and n(*R*) = 52%

To represent the fact that there are students who wanted to learn more than one language, we should use the intersection notation as it represents elements common to two or more sets. As per the information given in the question statement, we have:

n(*F * ∩ *G*) = 21%, n(*G *∩ *R*) = 23%, n(*R *∩ *F*)

= 24%, n(*F *∩ *G *∩ *R*) = 12%

As per the diagram, it is seen that:

The percentage of students who wanted to learn *F* and *G* only = the percentage of students who wanted to learn both *F* and *G* – the percentage of students who wanted to learn all 3 languages

= n(*F *∩ *G*) - n(*F *∩ *G *∩ *R*)

= (21 - 12)%

= 9%

Similarly,

The percentage of students who wanted to learn *G* and *R* only

= n(*G* ∩ *R*) - n(*F *∩ *G *∩ *R*)

= (23 - 12) %

= 11%

The percentage of students who wanted to learn *R* and *F* only

= n(*R *∩ *F*) - n(*F *∩ *G *∩ *R*)

= 12%

Now that we have all this information from the common data, we can go ahead and answer the question.

The required ratio is given by:

$=\dfrac{\text{% of students interested in any two languages only}}{\text{% of students interested in any one language only}}$

$=\dfrac{(9+12+11)\%}{(18+17+16)\%}$

$=\dfrac{32}{51}$