Question

There are 80 students, every third athlete is a musician, and every second musician is an athlete. How many are both athletes and musicians?

Answer 1

SOLUTION

GIVEN

There are 80 students, every third athlete is a musician, and every second musician is an athlete.

TO DETERMINE

The number of students both athletes and musicians

EVALUATION

Here it is given that there are 80 students, every third athlete is a musician, and every second musician is an athlete.

Now the LCM of 2 & 3 is 6

So every 6th, 12th,....., 78th is both athletes and musicians

Here the arithmetic progression is

6, 12,......., 78

First term = a = 6

Common Difference = d = 12 - 6 = 6

Let 78 be the nth term

So by the given condition

$\sf{a + (n - 1)d = 78}$

$\sf{ \implies \: 6 + 6(n - 1) = 78}$

$\sf{ \implies \: 6(n - 1) = 72}$

$\sf{ \implies \: (n - 1) = 12}$

$\sf{ \implies \: n = 13}$

Hence the number of students both athletes and musicians is 13

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