Question

If four bells toll together at 9:00am they toll after 7,8,11,12 seconds respectively then how many times will they toll together again in the next 3 hours​

Answer 1

Gɪᴠᴇɴ :-

• Four bells toll together at 9:00AM.
• They toll after 7,8,11,12 seconds respectively..

Tᴏ Fɪɴᴅ :-

• How many times will they toll together again in the next 3 hours ?

ᴄᴏɴᴄᴇᴘᴛ ᴜsᴇᴅ :-

• If some bells toll After a,b,c,d Seconds , Than, They will toll together again after LCM of (a,b,c,d).
• 1 Minute = 60 Seconds .

Sᴏʟᴜᴛɪᴏɴ :-

Four Bells toll together at 9:00AM. .

Than, After 9AM first Time They will bell After :- LCM of (7,8,11,12)

Prime Factorization :-

→ 7 = 1 * 7

→ 8 = 2*2*2 = 2³

→ 11 = 1 * 11

→ 12 = 2 * 2 * 3 = 2² * 3

→ LCM = 7 * 8 * 11 * 3 = 1848 Seconds. = (1848/60) = 30 Min + 48 Seconds .

Hence, After 9AM all 4 Bells toll after 30 Min + 48 Seconds . .

_________________

Now, in Next 3 Hours They will Toll :-

→ 3 Hours = 3 * 60 = 180 Minutes.

→ 30 Min + 48 Seconds = 30 (48/60) = 30(4/5) = (154/5) Min.

So,

→ In (154/5) Min. Bells Toll = 1 Time

→ in 1 Min. Bells Toll = (1) / (154/5) = (5/154) Time.

→ in 180 Min. Bells Toll = (5/154) * 180 = 5 Times.

Therefore, They toll together again 5 Times in the next 3 hours .

Answer 2

Answer:

Bell1 : tolls at  t = 0seconds, 7s, 14s, 21s....

     2"                  0s, 8s, 16s, 24s,...

    3:  0,11,22, 33s,...

    4:  0, 12, 24,36,48s,...

LCM of 7s , 8s ,11s ,12 is :  7 * 2 *2*2 *11 * 3 = 1848 sec = 30 min and 48 seconds.

so we add 30 m 48 sec to the previous time instant that they tolled together.

next time all the bells will toll together will be at  9 h 30min 48 sec

next time it will be  10 h 1 min 36 sec

next it will be at  10 h 32 min 24 sec

next it will be  at  11 h  3 min 12 sec

next it will be at    11 h  34 min 0 sec

next it will be at  12 h  04 min 48 sec..  this is after 3 hrs from 9 am.

===>  to do it simply:  in 180 minutes = 3 hrs,  they toll together

  =  180 min / 30min 48sec  times

    =  5.84  times  that is  5 times.