Question

Jenny and Shally bought a special 360 days joint membership of a tennis club. Jenny will use the club every alternate day and Shally will use the club every third day. They both use the club on the first day. How many days will neither person use the club in the 360 days?

Answer 1

Solution:-
Jenny will use the club every alternate day
so, the series of the days Jenny uses the club = 1, 3, 5, 7, 9 .... 359
Let the number of days Jenny used the club be 'x'
⇒ an = a +(n-1) d
⇒ 359 = 1 +(x-1) 2
⇒ 359 - 1 = 2x - 2
⇒ 358 + 2 = 2x 
⇒ 2x = 360
⇒ x = 180 days
Sally will use the club every third day.
So, the series of days Sally uses the club = 1, 4, 7, 10, 13 .....358.
Let the number of days Sally used the club be 'y'.
⇒ an = a + (n-1) d
⇒ 358 = 1 + (y-1) 3
⇒ 358 - 1 = 3y - 3
⇒ 357 + 3 = 3y
⇒ 360 = 3y
⇒ y = 120.
Every sixth day both Jenny and Sally use the club together,
So, the series of days when both Jenny and Sally use together = 1, 7, 13, 19.... 355.
Let the total number of days when both Jenny and Sally will go to the club together be 'L'.
⇒ an = a + (n-1) d
⇒ 355 = 1 + (L-1) 6
⇒ 355 - 1 = 6L - 6
⇒ 354 + 6 = 6L
⇒ 360 = 6L
⇒ L = 60
So, total number of days they both went to the club = x + y - L
⇒ 180 + 120 - 60  
⇒ 300 - 60 = 240
So, the number of days when none of them went to the club = 360 - 240 = 120 days.
Answer.