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). \[\dfrac{9}{2}\]
ANSWER
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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.
Complete step-by-step answer: 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.