Bunuel wrote:

Every student at Darcy School is in at least one of three clubs: horseback riding, embroidery, and country dancing, which are the only clubs in existence at the school. The ratio of the number of students in exactly two clubs to the number of students in exactly one club is 4:3, while the ratio of the number of students in exactly two clubs to the number of students in at least two clubs is 5:7. Which of the following could be the total number of students at Darcy School?

A. 63

B. 69

C. 74

D. 82

E. 86

Kudos for a correct solution.

This is a good question to test the concept of fractions and ratios.

It took me a while to solve this but once you know the trick, it is quite easy.

From the question, we know that \(\frac{Exactly 2 Clubs}{Exactly 1 Club}= \frac{4}{3}\) and \(\frac{Exactly 2 Clubs}{At least 2 Clubs}\) \(= \frac{5}{7}\)

The number of students in exactly 2 clubs is our link between 2 information here, so we have to make them equal by multiplying \(\frac{4}{3}\) by \(\frac{5}{5}\) and \(\frac{5}{7}\) by \(\frac{4}{4}\)

You will get the ratios of Exactly 1 Club : Exactly 2 Clubs : At least 2 Clubs (or Exactly 2 clubs + Exactly 3 clubs) \(= 15 : 20 : 28\)

You now know that the ratio of students in exactly 2 clubs is \(20\), therefore the ratio of the students in exactly 3 clubs is \(8\).

You will get the final ratio of Exactly 1 Club : Exactly 2 Clubs : Exactly 3 clubs \(= 15 : 20 : 8\)

So, total student must be multiple of \(15+20+8 = 43\) and the only answer that is multiple of \(43\) is \(86\)