Achemical company makes two brands of antifreeze. the first brand is 70% pure antifreeze, and the second brand is 95% pure antifreeze. in order to obtain 180 gallons of a mixture that contains 90% pure antifreeze, how many gallons of each brand of antifreeze must be used?

Let a be the amount needed of 70% pure antifreeze, and let b be the amount needed of 95% pure antifreeze.
0.7a + 0.95b = 180 * 0.9 (1)
a + b = 180 (2)
b = 180 - a (3)
Plugging the expression for b in equation (3) into equation (1), we get:
0.7a + 0.95(180 - a) = 162 (4)
Solving equation (4) gives the value a = 36, and plugging this value for a into equation (2) we get b = 144.
Amounts needed are therefore 36 gallons of 70% pure antifreeze and 144 gallons of 95% pure antifreeze.