 QUESTION

# One gallon of fuel mixture contains antifreeze in the ratio of five parts fuel to one part antifreeze. To this is added half a gallon of mixture which is one third antifreeze and two thirds fuel. What is the ratio of fuel to antifreeze in the final mixture? (Grid your answer as a fraction: fuel/antifreeze)(a). $\dfrac{3}{2}$(b). $\dfrac{5}{2}$(c). $\dfrac{7}{2}$(d). $\dfrac{9}{2}$

Hint: Find the amount of fuel and antifreeze in the one gallon of fuel mixture using the ratio given. Then find the amount of antifreeze and fuel in the half a gallon of the mixture which is added. Then find the ratio of the final mixture.

In one gallon of fuel mixture, there is one part of the antifreeze for every five parts of fuel. Then the amount of fuel is given as follows:
Amount of fuel in one gallon of mixture = $\dfrac{5}{{5 + 1}} \times 1$
Amount of fuel in one gallon of mixture = $\dfrac{5}{6}...............(1)$
Amount of antifreeze in one gallon of mixture = $1 - \dfrac{5}{6}$
Amount of antifreeze in one gallon of mixture = $\dfrac{1}{6}.................(2)$
Then a half a gallon of the mixture is added which contains one-third of antifreeze and two-thirds of fuel.
Amount of antifreeze in half a gallon that is added = $\dfrac{1}{3} \times \dfrac{1}{2}$
Amount of antifreeze in half a gallon that is added = $\dfrac{1}{6}............(3)$
Amount of fuel in half a gallon that is added = $\dfrac{2}{3} \times \dfrac{1}{2}$
Amount of fuel in half a gallon that is added = $\dfrac{1}{3}.............(4)$
The total antifreeze in the final mixture is the sum of equations (2) and (3).
Total antifreeze = $\dfrac{1}{6} + \dfrac{1}{6}$
Total antifreeze = $\dfrac{2}{6}$
Total antifreeze = $\dfrac{1}{3}..............(5)$
The total fuel in the final mixture is the sum of equations (1) and (4).
Total fuel = $\dfrac{5}{6} + \dfrac{1}{3}$
Total fuel = $\dfrac{{5 + 2}}{6}$
Total fuel = $\dfrac{7}{6}................(6)$
The ratio of fuel to antifreeze using equations (5) and (6) is given as follows:
Ratio of fuel to antifreeze = $\dfrac{{\dfrac{7}{6}}}{{\dfrac{1}{3}}}$
Ratio of fuel to antifreeze = $\dfrac{7}{2}$
Hence, the correct answer is option (c).

Note: You can not directly add the ratios to find the final ratio since the quantities to which the ratios are assigned are different, that is, one gallon and half a gallon. Hence, find the amount of fuel and the amount of antifreeze and find the ratio.