Solving proportion problems involving three quantities

Question

How do you solve proportion if 3 variables are given? I have looked in this site but i could not undertand it completely http://www.beatthegmat.com/ratio-proportion-3-variables-t34902.html 15 robots can make a train in 6 days working 5 hours a day. In how many days will 25 robots, working 6 hours a day complete the same work?

Answer 1

You start with $x=6 [\texttt{days}]\times \ldots $

Then you have 2 pairings of 4 numbers. hours $(5/6)$ and robots $(15/25)$.

Now, you can write a fraction of the two number of a pair. The question is, does it take longer to complete the work, if it is worked 6 hours a day instead of 5 hours a day ? The answer is no. Therefore the factor (fraction) has to be samller than 1. This is the case, if the numerator is smaller than the denominator. Therefore the fraction is $\frac{5}{6}<1$. Similar thoughts can be made with the amount of robots. You have now more robots. More robots means less time to finish the work. Thus the fraction is $\frac{15}{25}<1$.

In total you get $x=6 [\texttt{days}]\times \frac{5}{6} \times \frac{15}{25}$

Answer 2

15 robots -> 6 days -> 5 hours

25 robots -> x days -> 6 hours

The first step, you need to find out how many days will be took to create the robot in 6 days again, but in 6 hours, such that:

15 robots -> 6 days -> 5 hours

X robots -> 6 days -> 6 hours

By using proportion, we got 15/X = 6/5, which is X= 12.5 robots

Now, we got two equations involving 6 hours, such that:

25 robots -> x days -> 6 hours

12.5 robots -> 6 days -> 6 hours

By using proportion again, we got

25/12.5 = 6/x

x = 3 days