• # General Mathematics puzzles-1: Fun for all

Mathematics is an interesting subject. It has many branches and students have to learn different subjects some of which are very specific ones. Here we are talking of general mathematics with which most of the people are acquainted and understand it. I am giving one such mathematical puzzle here which not only the students but all other people will also enjoy and for some of them it will definitely be a fun activity.
Puzzle:
A and B complete a work in 2 hours. A and C complete it in 3 hours and B and C complete it in 4 hours. If they all work together in how much time the work will be completed?

After getting a few responses, I would post the answer.
• A few days ago I posted a thread My thread in which the content related to the number of people working and the duration of the work was shared in the context of real life.
Regarding this thread of the author, I am a little confused whether it is actually related to the correct answer to a maths question or as the author has called it a puzzle, then the answer may be slightly different. According to me, I understood only two answers to this, one which I have already provided in my thread and the second is its mathematical solution, and the answer, based on the formula will be 24 / 13 hours.

Now I am waiting for the response of the author.

• After a long time, going back to equation solving. Even though my answer might be wrong. But I am going to try.
Question can be depicted like this
A+B=2
A+C=3
B+C=4
A+B+C=?

Solution

A=2-B
2-B+C=3
-B+C=1
B+C=4
2C=5
A+2B+C=6
3+2B=6
2B=3
2A+B+C=5
2A=1
2A+2B+2C=1+3+5=9

A+B+C=9/2

So I think, the answer will be 4.5 hours. But I wonder that it will not be logical.

“It takes a great deal of bravery to stand up to our enemies, but just as much to stand up to our friends."
– Albus Dumbledore, Harry Potter and the Sorcerer's Stone, Chapter 17

• Swati,
it is a simple mathematical problem and because most of the people can try to attempt it and as it is not related to any higher mathematical concept it may provide some fun also that's why I have called it a puzzle. Your answer is correct.

Ann Wilson,
Good that you tried it. Now I give you some hints as how this type of problem is to be solved. You have to assume that let A alone can do p1 part of the job in one hour, B alone can do p2 part of the job in one hour, and C alone can do p3 part of the job in one hour.

When A and B complete it in 2 hours then it is obvious that -
2p1 + 2 p2 = 1
So you will get two more equations like that and solve them to get value of p1 + p2 + p3 which will lead you to find the required time.

Knowledge is power.

• Mr, Wilson.
Your approach was alright. There are different ways to arrive at the final answer.
First Method -
Given that
A+ B = 2 hrs
A+C = 3 hrs
If you add these two equations, you will get
2A + B+ C = 5 hrs
Also taking into consideration of B+C =4 hrs, put the value of B+C in the given equation 2A + B+C =5 hrs, we get 2A +4 =5 hrs or 2A = 1 Therefore A = 1/2 hrs
From A+B = 2 hrs, put the value of A obtained above B = 3/2 hrs
Similarly from B+C =4 hrs and B= 3/2 hrs, we get C = 5/2 hrs
Therefore A +B +C = 1/2 +3/2 +5/2 = 9/2 hrs
Alternatively adding A+B = 2 hrs, A+C = 3 hrs and B+C = 4 hrs, we get 2 ( A +B + C ) = 9
Therefore A + B + C = 9/2 hrs. The later process saves time in calculation.

• Until we find the individual rate of working, these problems cannot be solved. A and B complete the work in 2 hours does not mean A + B = 2.
That is why we have to derive equation like 2p1 + 2p2 = 1
I will put it like -
Work done by A in 2 hours + work done by B during the same time = 1
Here 1 denotes complete job.

Knowledge is power.

• The time required for A and B and c to complete the work by working together is about 1 hour and 48 minutes.
We have to find out the speed at which each worker works in rate per hour. Once we get that we can add up and see how much work they can complete in one hour and then the total time required for completing the work.,
A and B together can complete half the work in one hour
A and C together can complete one-third of the work in one hour
B and C together can complete one-fourth of the work in one hour
Two As and 2 Bs and 2 Cs can complete 1/2 + 1/3+ 1/4 work in one hour. That means they can complete 13/12 th work in one hour. So A and B and C together can do 13/24th work in one hour. They will take 24/13 hours to complete the work That is almost equal to one hour and 48 minutes.

drrao
always confident

• Yes my earlier calculations were wrong, it may please be seen in this way - A+B is dong the job in 2 hrs, hence 1/2 job has been done in an hour. Similarly A+C completes 1/3 job in hour. B+C completes 1/4 job in an hour.
Hence job done by 2( A+B +C ) = 1/2 part +1/3 part + 1/4 part = 13/12 part in hour
Hence ( A +B + C ) carried out 13/ 24 part in hour hence to complete the full part of the job it will take 24/13 hour.
Hence it is perfectly matching with Swati's answer.

• Dr Rao has explained the solution very nicely. Thanks.