Hi Rosa,
I would set this up algebraically. We know we want to end up with 40% antifreeze, so first convert this to its decimal representation (0.4). We also have our starting materials, 20% antifreeze and 45% antifreeze. We can assume for now that we won’t be using equal amounts of these, so let’s call the amount (in gallons) we use of 20% antifreeze x, and the amount we use of 45% antifreeze y.
This means we have .2x + .45y = .4(170), where 170 is the total gallons of 40% antifreeze that we want, as given in the question. This equation ensures that the final concentration of antifreeze is correct by calculating the amount of pure antifreeze in our variable quantities of starting materials is needed to give 40% antifreeze in the product.
We also can make a second equation from the quantity constraint: x + y = 170 gallons. This means that the total volumes of our starting materials has to equal 170 gallons. This equation is easier to manipulate because it doesn’t involve decimals, so let’s solve it in terms of x (you could also choose y), and substitute that back into our first equation:
x + y = 170 —> x = 170 - y
.2x + .45y = .4(170) —> .2(170 - y) + .45y = .4(170)
This gives us an equation of one variable, which we can now solve by expanding our terms:
.2(170 - y) + .45y = .4(170) —> .2(170) - .2y + .45y = .4(170)
By rearranging and combining constants and variables:
.2(170) - .2y + .45y = .4(170) —> .25y = .2(170) —> y = 136 gallons
We can now plug y = 136 to either of our starting equations to solve for x:
x = 170 - y = 170 - 136 = 34 gallons
Therefore: we need 136 gallons of 45% pure antifreeze and 34 gallons of 20% pure antifreeze to create 170 gallons of 40% pure antifreeze.
Finally, we can check that this solution is correct by plugging these numbers back into our original equations: .2(34) + .45(136) = 68 = .4(170), where 68 represents the volume of pure ethanol in the final combined solution. We can also check that our volumes are reasonable by making sure they add to 170 gallons, which is a constraint given in the question. 34 + 136 =170, so we’re set!
This is an old question, but I hope this explanation helps others work through similar problems!
- Amelia
