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Aptitude - Problems on H.C.F and L.C.M - Discussion

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"We cannot solve our problems with the same thinking we used when we created them."
- Albert Einstein
3. 

Six bells commence tolling together and toll at intervals of 2, 4, 6, 8 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together ?

[A]. 4[B]. 10
[C]. 15[D]. 16

Answer: Option

Explanation:

L.C.M. of 2, 4, 6, 8, 10, 12 is 120.

So, the bells will toll together after every 120 seconds(2 minutes).

In 30 minutes, they will toll together 30 + 1 = 16 times.
2


Punk said: (Sun, Jul 11, 2010 12:44:22 PM)    
 
Why 1 is added? please explain.

Sundar said: (Wed, Dec 29, 2010 07:05:24 AM)    
 
If two bells toll after every 3 secs and 4 secs respectively and if they commence tolling at the same time.

then the first bell tolls after every 3, 6, 9, 12 secs...
the second bell tolls after every 4, 8, 12, ....
So they toll together again after 12 secs, which is the LCM

Henceforth they toll after every 12 seconds, i.e whenever the time is a common multiple of both 3 and 4.

Since the bells start tolling together, the first toll also needs to be counted. Therefore we need to add 1.

Kushal said: (Sun, Jan 23, 2011 05:55:33 AM)    
 
Super explanation sundar.

Vishnu said: (Wed, Jan 26, 2011 01:52:08 AM)    
 
So We need to count that first commence according to your answer.

But according to the question we need to find the time after first tolling together na ?

Mehar said: (Mon, Jan 31, 2011 02:02:34 AM)    
 
WoW...super explanation .thank you:)

Sravanreddypailla said: (Thu, Apr 28, 2011 08:55:47 AM)    
 
Commense is nothing but start tolling. here six bells are there and toll together at intervals 2,4,6,8,10,12

first bell tolls at every :2secs,4secs,6sec......120sec....

second bell tolls at every :4secs,8secs,12secs.....120secs.....

third bell tolls together at every:6sec,12sec,18sec..120sec...

fourth bell tolls together at every:8sec,16sec,24sec...120sec..

fifth bell tolls together at every:10sec,20sec,30sec ,....120sec...

sixth ell tolls together at every:12sec,24sec,36sec....120sec...

NOW OBSERVE ALL TOLLS the common sec in *All* bell tolls is 120sec

thats why we need to find l.c.m in shortcut
come to question:
1 toll altogether takes time=120sec
? tolls altogether takes time=1800sec(30min*60sec)

The number of tolls are 1800*1/120 = 15 tolls(times)


But in question:
Six bells commence tolling together(first toll) and also toll at intervals of 2, 4, 6, 8 10 and 12 beginning itself started from tollinf so we should consider that toll also.

Hence answer is 15 Tolls + First Toll == 16 dat's it.

Hope you understand.

Sathya said: (Thu, Apr 28, 2011 10:07:57 AM)    
 
Excellent sravan now I understand everything in the question n wt you said. Its really very helpful to me. , AND YOUR EXPLANATION IS VERY VERY CLEAR.

THAN YOU VERY MUCH sravan!

Revathy said: (Fri, Jun 17, 2011 04:13:04 AM)    
 
Why do you divide 30 by 2?

Ashish said: (Fri, Jun 17, 2011 11:38:26 AM)    
 
Forget about 30/2, look at the answer given by Sravan. 1800/120=15

Thanks Sravan.

Sudha said: (Sat, Jun 18, 2011 01:10:52 AM)    
 
Excellent sravan.

Manju said: (Wed, Jul 6, 2011 12:42:25 PM)    
 
Awesome explaination sravan.

Kundharapu Shravan Reddy said: (Thu, Jul 28, 2011 12:56:35 AM)    
 
Excellent explanation sravanreddy.

Nagaratna said: (Mon, Aug 1, 2011 09:19:07 PM)    
 
@Revathy

At first time all bells toll together at 120sec (i.e, 2 minutes).

But not only at 2 mins they toll together at multiples of 2mins they toll together. That means at 2min, 4min, 6min...........30min, 32,34,..........That is they continue the same.

But they have asked to find tolling only upto 30mins. So now if you divide 30min by 2 you will get 15.

Neem said: (Thu, Aug 11, 2011 12:36:51 AM)    
 
Can you please tell me how to calculate LCM of above series no. ? Please tell me.

Deepak said: (Sat, Aug 13, 2011 02:37:08 PM)    
 
@neem
2=2
4=2*2
6=2*3
8=2*2*2
10=5*2
12=2*2*3

nw select d max count for 2 3 5
for 2. Its in 8. I.e. 2*2*2.
Then 5 n 3 are only once in any combination.

So lcm=2*2*2*3*5=120

Krishna said: (Sun, Aug 14, 2011 12:53:56 AM)    
 
Thank you Sravan I got it.

Gayatri said: (Sat, Aug 27, 2011 11:17:03 AM)    
 
Thank u deepak...

Ali said: (Mon, Aug 29, 2011 06:52:09 AM)    
 
Excellent explanation sravanreddy

Nimmi said: (Wed, Sep 28, 2011 05:39:56 PM)    
 
But why have counted till 120. ?

Rahul said: (Fri, Sep 30, 2011 09:34:23 PM)    
 
Thanks Sravanreddypailla. Its truly good.

Vairam said: (Sat, Nov 5, 2011 09:55:06 AM)    
 
Why 1 is added?

Naveen said: (Sat, Nov 12, 2011 11:17:59 PM)    
 
Really good explnation.

Lakshmisha said: (Thu, Nov 24, 2011 09:19:03 PM)    
 
@Sravanreddypailla, excellent dude...!!!

Sangeeta said: (Fri, Dec 2, 2011 08:42:04 AM)    
 
I didn't understand why did we add 1?

Please explain.

Malatha said: (Thu, Dec 8, 2011 12:17:10 PM)    
 
I also didn't understand why did we add '1'? please tell me.

Balaraju Hari said: (Sun, Dec 18, 2011 04:10:39 PM)    
 
Its simple, according to the shortcut formula everyone understood why the tolling together takes place at every 120sec(2min) period from the l.c.m concept.

But why 1 is added means, it is clearly mentioned in the question that the 6 bells starts tolling together at the starting stage(first tolling is done by together) and tolls at the intervals 2, 4, 6 ,8, 10, 12. so, before tolling together at 120sec(2min), they already tolls together at the 1st(zeroeth sec i.e starting) itself. so that 1 is added.

Aashish said: (Wed, Dec 28, 2011 09:28:10 PM)    
 
Thanks sundar. Amazing explanation!

Vasu said: (Fri, Dec 30, 2011 08:59:07 PM)    
 
Thanks to both Sravan and Sundar for your explanation.

Fozia Sheikh said: (Wed, Jan 11, 2012 02:34:01 AM)    
 
Four bells toll after intervals of 8, 9, 12and 15minutes, respectively. If they toll together at 3pm. When will they toll together next?

Ramesh said: (Wed, Jan 11, 2012 12:51:55 PM)    
 
Awesome sravan!!

Suparna said: (Thu, Jan 12, 2012 08:53:55 AM)    
 
Why 30 divided by 2 and 1 added?

Divya said: (Fri, Feb 10, 2012 11:20:03 PM)    
 
Please present answer to fozia sheihk.

Manasa said: (Sat, Jun 2, 2012 02:29:36 PM)    
 
120seconds is 2 min so divided by 2.

Saj said: (Fri, Jun 15, 2012 04:38:03 PM)    
 
@Fozia.

I guess the answer should be 9pm, since the LCM of 8, 9, 12, 15 is 360. i.e 360 minutes i.e. after 6 hours.

Yunus said: (Wed, Jul 4, 2012 08:40:12 PM)    
 
One is add because it is clearly given in the question that.

"Six bells commence tolling together" that means all bells rang once than after that rang in intervals of 2 4 6 8 10 and 12.

So add 1 to answer.

Rasheed Tolulope said: (Wed, Sep 12, 2012 05:36:24 AM)    
 
Assuming the tolling follows Arithmetic Progression (AP).
2s, 4s, 6s, 8s, 10s, 12s.
a = 2 and d = 2
In 30 minutes i.e 1800s
Tn = 1800, n = ?
1800 = 2 + (n - 1) 2 = 2 ( 1 + n - 1)
1800 = 2n

n (minutes) = 900s ~ 15 (minutes)

n = 15 times
But note that the six bells started tolling together. Then, plus the first tolling, the number of times they will tolling together in 30 minutes is 16 times

ANSWER = 16 times

Rasheed Tolulope said: (Sun, Oct 14, 2012 02:30:48 PM)    
 
@ Neem.
To find the LCM of 2,4,6,8,10, and 12 using Rasabtol Cross Method (RCM).

2 x 4 x 6 x 8 x 10 x 12 = 2 [1 x 2 x 3 x 4 x 5 x 6]

2 [1 x 3 x 5 x {2 ( 1 x 2 x 3)}]

Apply BODMAS rule,

= 2 [1 x3 x 5 x 2 x 6]

= 2 [1 x 2 x 5 x 3 (1x2)]

= 2 [1 x 2 x 5 x 3 (2)]

= 2 [60]

= 120.
Hence, the LCM = 120

Rahul Vasu said: (Sun, Nov 25, 2012 09:57:32 PM)    
 
LCM method
2 sec, 4 sec, 6 sec, 8 sec, 10 sec, 12 sec = all number is divisible by 2
i.e 1,2,3,4,5,6
= 1x2x3x4x5x6=720
=720/6(bells) = 120 seconds i.e 120/60(seconds) =2 minutes

In 30 minutes, they will toll together = 30/2=15

Since the bells start tolling together, the first toll also needs to be counted. Therefore we need to add 1.

Therefore 15+1=16.

Niharika said: (Fri, Jan 25, 2013 08:00:15 PM)    
 
30/2 in order to calculate time for in fractions to each minute, as the LCM gave 2 minutes as the time.

Kp Mukesh Kumar Singh said: (Sun, Apr 28, 2013 01:37:35 AM)    
 
Superb, it is indispensable for general competition.

Sruthi said: (Thu, Sep 19, 2013 02:55:10 PM)    
 
Why at last 1 has been added?

Mr Perfect said: (Fri, Oct 18, 2013 03:07:13 PM)    
 
Since the bells start tolling together, the first toll also needs to be counted. Therefore we need to add 1.

Kumar said: (Fri, Oct 18, 2013 03:37:31 PM)    
 
It's asked in the question that Six bells commence tolling together and toll at intervals of 2, 4, 6, 8 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together?

Here as per the question the 1st bell can toll 15 times in 30 min i.e (30/2), 2nd bell can toll 7.5 times in 30 min i.e 30/4, 3rd bell 5 times, 4th bell 3.7 times, 5th bell 3 times and 6th one 2.5 times in 30 min...together they tolled 36.7 times in 30 min..

Why this approach is wrong?

Lal Chand said: (Sun, Nov 24, 2013 10:47:37 PM)    
 
Why 30 is divide by 2, please explain?

Veda said: (Wed, Jan 29, 2014 08:04:05 PM)    
 
Well, I will break it into English but no maths.

Generally all the bells rang (toll) at its first encounter, that means all 6 bells started its journey by joining all their hands at once, i.e., at 0th sec<< remember this started journey>>.

Then they lost their hands and joined again at 120th second, in the mean-while they are tolling individually and not combinedly.

1. It is nothing but at 14th 19th. 25th 38th 57th 69th. Until 119th sec they didn't rang combinedly then at *120th* sec they combined their hands and rung. This is the least number that those 6 bells have waited for their combined toll (nothing but LCM in maths criteria), shortcut used.

2. Then they again rung combinedly after *120*+120th=240th sec+. 120+120+..until 1800.

3. They had given us 30min =1800 sec) , so if we divide that 1800 sec by 120sec (intervals) 1800/120=15 we will get.

Actually this (15) should be the answer if all the bells did not start combinedly, but our bad luck, they mentioned in the question that ""Six bells commence tolling together. "" that means they have started its journey itself by ringing combinedly.

4. The extra +1 is nothing but this started journey.

Now go through what @Saravan and all others said you will get more detailed idea.

Hope this understands. Thank you.

Dushanthi said: (Mon, Feb 24, 2014 11:15:33 AM)    
 
Why we find the L.C.M.?

I can't understand the L.C.M. method. What is the difference between L.C.M and H.C.M. Why didn't use the H.C.M. method in here. Please anyone can explain me.

Prem said: (Thu, May 15, 2014 05:56:58 PM)    
 
I am unable to understand why we are using lcm method for this problem. Can anyone explain?

Atishay said: (Sat, Jul 19, 2014 02:41:30 AM)    
 
Why not using HCM:

Suppose 3 bells are tolling at 2, 4, 6 sec,

HCM is 2, so now think, is it possible that they will toll at every 2 sec. NO.

We will have to find a common number (multiple) which is occurring to every bell and that is called LCM.

So LCM here is 12.

That means every bell will toll at 12th second.

Nakum Paras said: (Thu, Oct 16, 2014 07:43:12 PM)    
 
Actually got solution.

All bells starting from 1 second at a time 1 time bells tolling at a time.

1st bell tolling every 2 sec means 2, 4, 6, 8,...120...240... 360....480

2nd bell tolling every 4 sec means 4, 8, 12, 16...120...240...360...480

3rd bell tolling every 6 sec means 6, 12, 18, 24...120...240...360...480

4th bell tolling every 8 sec means 8, 16, 24...120...240...360...480

5th bell tolling every 10 sec means 10, 20, 30...120...240...360...480

6th bell tolling every 6 sec means 12, 24, 36...120...240...360...480

So 1 sec =1 time tolling at a time.

120 sec = 2 time tolling at a time.

240 sec = 3 time tolling at a time.

360 sec = 4 time tolling at a time.

4800 sec = 5 time tolling.

600 sec = 6 time tolling at a time.
.
.
Last 1800 sec = 16 time tolling at a time.

Total time bells tolling at a time =16 times.

Rohan Bhoir said: (Mon, Oct 20, 2014 12:35:37 PM)    
 
Can you tel me how is the HCM can find? Explain with any example.

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