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# KCC's Quizzes: University Elective Courses

In a university, students might (if they want to) take up to 2 optional foreign language courses: French or Spanish.

1/5 of the students who take French, also take Spanish.

1/7 of the students who take Spanish, also take French.

110 students take only one course.

We assume the two courses are not occurring on the same date.

Question: How many students take both courses?

• Here's my math...

F = # taking French

S = # taking Spanish

T = Total unique students

T = Everyone taking French + 6/7 of those taking Spanish

--> T = F + (6/7)S

-->  F = T - 6S/7

T = Everyone taking Spanish + 4/5 of those taking French

--> T = S + (4/5)F

--> T = S + (4/5)(T - 6S/7) by inserting F from above

--> T = S + 4T/5 - 24S/35 --> T/5 = 11S/35 --> T = 11S/7

Also, T = 110 + (1/7)S, which are those taking only one course plus those taking both so combining...

--> 11S/7 = 110 + S/7 --> 10S/7 = 110 --> S = 77

If S = 77, and those taking both equals S/7, then there are 11 people taking both Spanish and French!

• 11 students take both courses.

Solution:

F = total number of students who take French

S = total number of students who take Spanish

Equation 1: 1/5 F = 1/7 S -> F = 5/7 S

Equation 2: 4/5 F + 6/7 S = 110 -> 28 F + 30 S = 3850

Substitute Equation 1 to Equation 2: 28 (5/7 S) + 30 S = 3850 -> S = 77

Determining students that take both courses: 1/7 S = 1/7 (77) = 11

• F=5x

S=7x

110 = F + S - 2x

110= 5x + 7x -2x

110=10x

x=110/10

x=11

• Thanks Brian! Will comment you next week!

• Thanks Jules for your prompt feedback! Rendez-vous next week for the answer confirmation!

• It's time to give the (official) answer and reasoning on this puzzle:

Let’s call F the number of students who take French course and S the number of students who take the Spanish course.

The number of students who take French and Spanish is F/5

This means 4F/5 is the number of students for French only

The number of students who take Spanish and French) is S/7

This means 6S/7 is the number of students for Spanish only

From the 2 above relations, we can say the number of students with both French and Spanish is F/5 = S/7; thus F = 5S/7

The number of students who take only one course is 4F/5 + 6S/7 = 110

Or (5S/7)*(4/5) +6S/7 =110 or 10S/7=110 or S=77 and F=55

The number of students who take both courses are 11

Big applause to our 4 first winners:

* Jules NIKKO, Systems Integration Engineer at ADI General Trias, Philippines

* Mark CEE, Snr Systems Integration Engineer at ADI Calabarzon, Philippines

* Brian BARETT, Field Sales Engineer at ADI Columbus (OH), USA

* Jeffrey EVANKO, Key account manager at ADI Cleveland (OH), USA

• Correct