Combined Work Formula - How to derive total T (Time)?

Question

Can someone please explain how the formula \(\frac{xy}{x+y}\) is derived?

It is shown here: the-discreet-charm-of-the-ds-126962-20.html#p1039633

The solution given states "sum of the rates equal to the combined rate or reciprocal of total time": \(\frac{1}{x}+\frac{1}{y}=\frac{1}{T}\) --> \(T=\frac{xy}{x+y}\)).

I know the basic combined worked formula, but when you solve for T, shouldn't it be \(T=\frac{x+y}{xy}\)?

Thanks for all help in advance in understanding this. I tried to search around, but couldn't find any place that derived this formula.

~ Im2bz2p345

argha · Answer

Hi,

Happy to help

If A can do a job in x hours,
then in 1 hour, he does 1/x part of the job

If B can do the same job in y hours,
then in 1 hour, he does 1/y part of the job

So if A & B jointly work, 
then in 1 hour, they will do (1/x+1/y) part of the job ie. (x+y)/xy part of the job

Therefore, they jointly will do the whole work in 1/((x+y)/xy)=(xy/(x+y) hours= T hours

Regards

Argha

Im2bz2p345 · Answer

As you correctly wrote, the original equation to solve combined work problems is: \frac{1}{x}+\frac{1}{y}=\frac{1}{T} I understand your response up until the part the red part above. Shouldn't it be \frac{1}{(x+y/xy)}=\frac{1}{T} ? Why does it equal just T? You can't just remove the "1" that is numerator of \frac{1}{T} fraction. In order to solve for T, you need to take the reciprocal of the left hand side equal, so the final equal should be: \frac{xy}{x+y}=T Can someone please explain?? ~ Im2bz2p345

argha · Answer

Hi,

If you have to go by equation route, then your equation is correct:

(1/x+1/y)=1/T

Solving this, T= xy/(x+y)

Alternatively, by my explanation:

Jointly, they do (1/x+1/y) in 1 hour
Therefore, they will do the whole(1) in 1/(1/x+1/y)=xy/(x+y) which can be called T hours.

Suggest, try plugging in numbers in place of x,y and then it will be clear.

Regards

Argha

KarishmaB · Answer

Rate of work is additive, time is not. This means that if a machine makes 2 nuts in a minute and another makes 5 nuts in a minute, together, they make 2+5 = 7 nuts in a minute. So you added the two rates to get the combined rate. On the other hand, time is not. If a machine takes 3 hrs to complete a work and another takes 4 hrs, total time taken by both of them together to complete the work is not 3+4 = 7 hrs. They will take less than 3 hrs to complete the work together.

Usually you are given the time taken by the two machines independently and you need to find the time taken by them together.

How do you find rate when time is given e.g. a machine takes 3 hrs to complete a particular work. Rate of work is work done per hour i.e. (1/3)rd of the work per hour. 
Rate = Work/Time
So if there is 1 work, Rate = 1/TIme

That is why, given one machine takes x hrs and another takes y hrs, rate of one machine is 1/x and another is 1/y
Total rate = 1/x + 1/y = 1/Total time

(x+y)/xy = 1/T
T = xy/(x+y) (Cross multiply)

Im2bz2p345 · Answer

Thank you so much for your followups Argha & Karishma! I think that I was just confusing myself when I looked at the equation for the first time. Your explanations helped me better understand the concept. ~ Im2bz2p345

Combined Work Formula - How to derive total T (Time)?

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