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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 'B'.

an = a + (n–1) d

355 = 1 + (B–1) 6

355 – 1 = 6B – 6

354 + 6 = 6B

360 = 6B

B = 60

So, total number of days they both went to the club = x + y – B

180 + 120 – 60

300 – 60 = 240

So, the number of days when none of them went to the club = 360 – 240 = 120 days.

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RS Aggarwal - Mathematics