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Math Help - two percentage word problems!!! No clue!

  1. #1
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    Unhappy two percentage word problems!!! No clue!

    1) In the chemistry lab I had to mix some 33and1/3 alcohol with some 12and1/2 alcohol to get a 50 litre jug of 25% alcohol. How many litres of the 33and1/3% and of the 12and1/2% alcohol did I use?

    2) Harry is trying to quit smoking. He reduced his habit to 1/8 of a carton per week except when he's studying for exams and, being all stressed, then he smokes 1/3 of a carton per week. Over the last 50 weeks, Harry has smoked 1/4 of a carton per week on average. During those 50 weeks, how many weeks did Harry study for exams, and how many weeks did he not study for exams?
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  2. #2
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    For problem #1, you know you need to end up with 50 liters of 25% alcohol. In other words, you need to end up with 12.5 liters of total, pure alcohol (50 * .25).

    Let's say you use x liters of the 33 1/3% alcohol. This gives you x/3 liters of pure alcohol.

    Then, since you need 50 total liters, you must use 50 - x liters of the 12.5% alcohol mixture. This gives you (50 - x)*.125 = 6.25 - .125x liters of pure alcohol.

    Combine the two amounts you get from the mixtures and set it equal to the final total.

    In other words, x/3 + 6.25 - .125x = 12.5. Now, solve for x - first, I'd multiply everything by 3 to get rid of the fraction:

    x + 18.75 - .375x = 37.5

    Move the 18.75 to the other side and combine the two x terms on the left:

    .625x = 18.75.

    Divide by .625:

    x = 30.

    Remember what you chose x to represent: the 33 1/3% alcohol mixture. So you need 30 liters of that and 20 of the other.
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  3. #3
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    2 is similar. We know he has 50 total weeks and his average over that time was 1/4 carton per week. (You can think of these weeks as being a "mixture" of the 1/3 pack per week and 1/8 pack per week types.) So, in the 50 weeks, he smoked 12.5 packs.

    Now, let x be the number of weeks he studied for tests and thus smoked 1/3 of a pack. That means, in all those weeks, he smoked x/3 packs.

    In the remaining weeks, of which there are 50 - x, he smoked 1/8 of a pack per week, for a total of (50 - x) / 8 packs. Add these together, and we should get 12.5, the total number of packs he smoked.

    \frac{x}{3} + \frac{50-x}{8} = 12.5

    Multiply everything by 24 to eliminate the fractions:

    8x + 3(50 - x) = 300

    8x + 150 - 3x = 300

    5x = 150

    x = 30.

    Just remember what x stood for and you should be in good shape.

    Really, both these problems are mixture problems - in fact, they are the SAME PROBLEM.

    In the first, we have a solution that is 33 1/3% alcohol, which is really just 1/3 alcohol, which corresponds to the 1/3 pack per week.

    The 12 1/2% alcohol is the same as the 1/8 pack per week (1/8 = .125 = 12.5%).

    The 25% final mixture is the same as the 1/4 per pack final average, with the 50 liters corresponding to the 50 weeks.
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