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# A Tricky Problem

By Neel Shah

A simple math problem that has a tricky solution .
Question:-
Alice and Bob can complete a job in 2 hours. Alice And Charlie can complete the same job in 3 hours. Bob and Charlie can complete the same job in 4 hours. (They work at a constant rate)
So how long it will take if three work together??
Seems simple right?
One college professor explained, nearly all seniors were unable to solve the problem,because they set up incorrect equations.
Normally people make equations assuming Alice as a, Bob as b and Charlie as c.
Equations that are formed are:-
a+b=2
b+c=3
a+c=4
By adding all equations we get:-
a+b+b+c+c+a=2+3+4
2a+2b+2c=9
2(a+b+c)=9
a+b+c=4.5
According to this solution if three of them worked together to complete the job then it will be done in 4.5 hours which is wrong because even though all three are working together then also the time taken ie. 4.5 hours, is more than the time taken if any combination of two people work together.
So what went wrong?
The equations that are made does not make any sense. Common mistake people make is that they consider names of people as variables (in this case Alice as a,Bob as b, and Charlie as c)
So what should be the correct approach?
We know that Alice and Bob will take two hours to complete a job. This means if we add up the percentage of the job Alice did and the percentage of the job done by Bob, the result will be 100% or 1.
(%job Alice does in 2 hours) + (% job Bob does in 2 hours) = 1
Since each works at a constant rate, the amount of work each does in two hours is twice the work done by each of them in 1 hour.
So we have equation:-
2(%job Alice does in 1 hours) + 2(% job Bob does in 1 hours) = 1
Now let’s set up the variables:
a= % job Alice does in 1 hour
b= % job Bob does in 1 hour
We then have the equation:-
2a+2b=1
We can similarly write:
c=% job Charlie does in 1 hour.

Now we can set up the other equations.
For Alice and Charlie complete the job in three hours we have:-
3a+3c=1

For Bob and Charlie complete the job in four hours we have:-
4b+4c=1
So our three equations are:-
2a+2b=1
3a+3c=1
4b+4c=1
We could solve the equations, but we are interested in how long it takes all 3 people to do the job working together. So we need to solve for t in the equation:
t(a+b+c)=1
For the first three equations we take L.C.M to get same coefficient for all variables by multiplying equation 1 by 6, equation 2 by 4 and equation 3 by 3, we get equations:-
12a+12b=6
12a+12c=4
12b+12c=3
Now we add up all equations
We get:-
24(a+b+c)=13
In 24 hours all three of them complete the job 13 times. So we divide by 13 to get the number of hours taken to complete the job 1 time.
(24/13)(a+b+c)=1
The correct answer is all three working together would take 24/13 hours, about 111minutes or 1 hour 51 mins, to complete the job.