Hi s91arvindh,
The shortcut that you have mentioned in this problem is based on a simple fact - when we multiply a number with an improper fraction its value incrases and when we multiply a number with proper fraction its value decreases.
In this case the number of days is found out in three steps, but the reason behind these steps is the normal method by which we find out time required to get the work done.
Firstly as you constructed the table,
Patients Pints Days
Case 1: 50 2000 30
Case 2: 60 1680 X
Step 1: firstly we consider the ratio of the number of days now and the number of days earlier that is X/30
Step 2: now, in case 2 the number of patients have increased, so it will take less days to finish the amount of glucose and the ratio X/30 will decrease. Since the ratio is decreasing we make a proper fraction out of the the number of patients in case 1 and 2 that is 50/60. The ratio x/30 will decrease and will be equal to 50/60 (given the amount of glucose remains same) X/30=50/60 [ had the ratio been increasing, we would have meade an improper fraction out of the two figures].
Step 3: now we consider a decrease in the amount of glucose available. Pints of glucose decrease so it will take less number of days for glucose to finish, thus ratio X/30 will decrease. Again, we make a proper fraction out of the pints in both cases, that will be 1680/2000. But since the number of patients have alreafy increased, the ratio X/30 will be
X/30 = 50/60 x 1680/2000
X= 30 x 50/60 x 1680/2000
When encountered with such a problem you can create this table quicky and depending on the effect the change in other values have on X you can create proper or improper fractions and multiply them.
For example you can consider another problem,
Men Days Work
20 10 2
10 12 X
X= 2 x 10/12 x 20/10
(Amount of work earlier) x (days increased so work decreased - proper fration) x men decreased so work increased - improper fraction)
Cheers
Sukriti
