Davenport Dairy Farms must lease a fleet of 21 refrigerated

Davenport Dairy Farms must lease a fleet of 21 refrigerated over-the-road tanks to transport 520,000 gallons of milk. Refrigerated tanks with three different carrying capacities are available for lease: 8,000 gallons, 16,000 gallons, and 24,000 gallons. How many of each type of tanks should be leased?

Solution

a + b + c = 24

a*8000 + b*16000 + c*24000 = 520000
8a + 16b + 24c = 520

Divide by 8 all over :
a + 2b + 3c = 65

So, we have :
a + b + c = 24
a + 2b + 3c = 65

Now, we only have two equations but 3 variables...

Subtracting :
b + 2c = 41

Let c = k
And thus, b = 41 - 2k

And using a + b + c = 24,
we get
a + 41 - 2k + k = 24
a = k - 17

So, the number of each type is :
8000 gal tanks : k - 17
16000 gal tanks : 41 - 2k
24000 gal tanks : k

where k can be any integer greater than or equal to 17 and
less than equal to 20

So, we have these possibilities :
k = 17 , 18 , 19 or 20

In these cases, the answers come out to be :
0 , 7 , 17
1 , 5 , 18
2 , 3 , 19
3 , 1 , 20