Help setting up equation

Question

A large auger can fill a grain bin in 5 h less time than a smaller auger. Together they fill the bin in 6 h. How long would it take the larger auger, by itself, to fill the bin?

I know what to do, but have trouble setting up the equation. Thanks in advance

Answer 1

The key to problems like this is that rates add.

 

Let  R = the rate for the larger auger   and r = rate for the smaller auger.  Both rates have the unit:   bins/hr

 

The fact that together they can fill the bin in 6 hr  is captured by   6 =  1/(R + r)   or  R + r = 1/6

 

The fact that larger can fill the bin in 5hr less than the smaller is captured by   5 = 1/r -  1/R

 

 This gives two equations with two unknowns.   The first equation can be solved for r and this substituted in the second to give a quadratic equation for R.    The solution is   R = 1/10,   this, in turn, gives  r = 1/15

 

Thus the time it takes the large auger alone to fill the bin is   1/R  =  10 hrs.