An experiment calls for a 5% salt solution. Only a 3% salt solution and an 8% salt solution are available. How many gallons of each must be combined to obtain 30 gallons of a 5% salt solution?

Answer 1

If we want to have a 5 percent solution, we want 0.05 (30 gallons) of salt, so we want 1.5 gallons of salt.

Since we have a 3 percent and 5 percent solution, and we want there to be 1.5 gallons of salt we know that

0.03x + 0.08y = 1.5, where x is the number of gallons of the 3% solution and y is the number of gallons of the 8% solution. The reason for this is there are 0.03x gallons of salt for x gallons of liquid in the 3 percent concentration and 0.08y gallons of salt in y gallons of the concentrate. A more convenient way of writing this is to multiply both sides of this equation by 100 (thus not changing it), so we get all integers. If we do that we get an equivalent, nicer-looking equation: 3x + 8y = 150 (1)

Since we want there to be 30 gallons of solution, we also want x + y = 30.

In this case, we can solve our second equation for y and get y = 30 - x.

Since we have y = 30 - x, we can plug that in for y in equation (1) and get:

3x + 8(30 - x) = 150.

3x + 240 - 8x = 150.

-5x + 240 = 150

-5x = -90

x = 18.

Since y = 30 - x, y = 30 - 18 = 12.

So we need 18 gallons of the 3 percent solution and 12 gallons of the 8 percent solution.