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Math Help - Fraction Word Problems

  1. #1
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    Fraction Word Problems

    I have two fraction homework questions i need help on..I am totally stuck. Please help me if you know how to solve it. Thank you, your help is greatly appreciated!!


    1.) A water is purification plant works by passing the dirty water through three filters in turn. The first takes out all but 1/4 of the dirt in the water, the second leaves 1/5 of the dirt reaching it and the last takes out 2/3 of what's left.
    (a) What fraction of the dirt is left in the water after it has gone through the plant?
    (b) What fraction of the dirt has the plant removed?

    2) A large pizza costs $12.75, three friends decide to split one. Jack takes a third, and then Jill takes a third of the remainder and then Tom takes half of what's left.
    (a) How should they split the total cost so that each pays an amount proportional to what he has eaten so far?
    (b) Can they split the remaining pizza so that everyone ends up having eaten the same total amount? If so, how?

    PLEASE HELP ME!!...It would AT least help if someone taught me how to start it, or teach me how to do it step by step, so i tackle similar problems on my test ><..Thank you
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  2. #2
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    1.) A water is purification plant works by passing the dirty water through three filters in turn. The first takes out all but 1/4 of the dirt in the water, the second leaves 1/5 of the dirt reaching it and the last takes out 2/3 of what's left.
    (a) What fraction of the dirt is left in the water after it has gone through the plant?
    (b) What fraction of the dirt has the plant removed?

    Let x = total dirt in the water before it passed any filter.

    After passing the first filter, (1/4)x or x/4 is left.

    After passing the second filter, (1/5) of (x/4) is left
    That is (1/5)*(x/4) = (1*x)/(5*4) = x/20 is left.

    Finally, after the 3rd filter, 2/3 of x/20 is removed.
    So 1/3 of x/20 is left, or (1/3)*(x/20) = (1*x)/(3*20) = x/60 is left.

    (a) What fraction of the dirt is left in the water after it has gone through the plant?
    x/60 or 1/60 of the dirt. --------------answer.

    (b) What fraction of the dirt has the plant removed?
    x -x/60
    = x/1 -x/60
    = (60*x -x)60
    = (59x)/60
    Therefore, 59/60 of the dirt is removed.
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  3. #3
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    Hello, lemontea!

    You just have to "talk" your way through these problems.
    There are no neat formulas for filtering dirt or sharing a pizza.


    1) A water is purification plant works by passing the dirty water through three filters in turn.
    The first takes out all but 1/4 of the dirt in the water,
    the second leaves 1/5 of the dirt reaching it
    and the last takes out 2/3 of what's left.

    (a) What fraction of the dirt is left in the water after it has gone through the plant?
    Let D be the amount of dirt originally in the water.

    Filter #1 leaves 1/4 of the dirt in the water.
    . . There is now \frac{1}{4}D in the water.

    Filter #2 leaves 1/5 of that in the water.
    . . There is now: . \frac{1}{5}\cdot\frac{1}{4}D \:=\:\frac{1}{20}D in the water.

    Filter #3 removes 2/3 of the dirt; it leaves 1/3 of the dirt.
    . . There is now: . \frac{1}{3}\cdot\frac{1}{20}D \:=\:\frac{1}{60}D in the water.

    Therefore, one-sixtieth \left(\frac{1}{60}\right) of the dirt is still in the water.


    (b) What fraction of the dirt has the plant removed?
    The plant removed: . 1 -\frac{1}{60} \:=\:\frac{59}{60} of the dirt.



    2) A large pizza costs $12.75, three friends decide to split one.
    Jack takes a third, and then Jill takes a third of the remainder
    and then Tom takes half of what's left.

    (a) How should they split the total cost so that each pays
    an amount proportional to what he has eaten so far?
    Jack takes \frac{1}{3} of the pizza; there is \frac{2}{3} left.

    Jill takes \frac{1}{3} of that: . \frac{1}{3}\cdot\frac{2}{3} \:=\:\frac{2}{9}
    . . She takes \frac{2}{9} of the pizza. .There is: . \frac{2}{3} - \frac{2}{9} \:=\:\frac{4}{9} of the pizza left.

    Tom takes \frac{1}{2} of that: . \frac{1}{2}\cdot\frac{4}{9}\:=\:\frac{2}{9}
    . . He takes \frac{2}{9} of the pizza. .There is: . \frac{4}{9} - \frac{2}{9} of the pizza left.


    Their shares are in the ratio: . \frac{1}{3}:\frac{2}{9}:\frac{2}{9} \;=\;\frac{3}{9}:\frac{2}{9}:\frac{2}{9} \;=\;3:2:2

    So far they have eaten: . \frac{3}{7}:\frac{2}{7}:\frac{2}{7} of the consumed pizza, respectively.

    So Jack pays: . \frac{3}{7}\times12.75 \:=\:5.4642... \:\approx\:\$5.46
    And Jill and Tom each pay: . \frac{2}{7}\times12.75 \:=\:3.6428... \:\approx\:\$3.64


    (b) Can they split the remaining pizza so that everyone
    ends up having eaten the same total amount? If so, how?
    Each must have eaten \frac{1}{3} of the pizza.

    Jack has already had his share.
    Jill and Tom will split the remaining \frac{2}{9} of the pizza.
    . . Then both will have: . \frac{2}{9} + \frac{1}{9} \:=\:\frac{3}{9} \:=\:\frac{1}{3} of the pizza.

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  4. #4
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    2) A large pizza costs $12.75, three friends decide to split one. Jack takes a third, and then Jill takes a third of the remainder and then Tom takes half of what's left.
    (a) How should they split the total cost so that each pays an amount proportional to what he has eaten so far?
    (b) Can they split the remaining pizza so that everyone ends up having eaten the same total amount? If so, how?

    Jack takes 1/3 of the whole pizza. So 2/3 is left.

    Jill takes 1/3 of the 2/3 of the whole.
    So what is left is 2/3 of 2/3 of the whole pizza.
    That is (2/3)(2/3) = (2*2)/(3*3) = 4/9 of the whole left.

    Tom takes 1/2 of 4/9 of the whole.
    So what is left is 1/2 of 4/9 also.
    That is (1/2)(4/9) = 4/18 = 2/9 of the whole left.

    (a) How should they split the total cost so that each pays an amount proportional to what he has eaten so far?

    Jack got and ate 1/3 of the whole. That is 3/9 of the whole.
    Jill took and ate 1/3 of 2/3. That is 2/9 of the whole.
    Tom took and ate 1/2 of 4/9, which is 2/9 of the whole also.
    (And what is left or not eaten, is the last 2/9 of the whole.)

    That means 3/9 +2/9 +2/9 = 7/9 is eaten.

    Therefore, by proportion,
    Jack should pay 3/7 of the total cost.
    Jill should pay 2/7 of the total cost
    Tom should pay 2/7 of the total cost also.

    (b) Can they split the remaining pizza so that everyone ends up having eaten the same total amount? If so, how?

    Yes. So that Jack, Jill and Tom end up eating the same amount, then each should eat 1/3 of the whole pizza.
    So,
    Jack will not get anymore.
    Jill will get additional 1/3 -2/9 = 3/9 -2/9 = 1/9 of the whole.
    Tom, likewise, will get additional 1/9 of the whole also.
    Or, Jill and Tom split equally the remaining, uneaten 2/9 of the whole.
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  5. #5
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    thank you for helping me^^..but i got a question regarding the second question 2a

    you said:

    Therefore, by proportion,
    Jack should pay 3/7 of the total cost.
    Jill should pay 2/7 of the total cost
    Tom should pay 2/7 of the total cost also.

    but i want to know where does the denominator of 7 come from?
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  6. #6
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    Quote Originally Posted by lemontea View Post
    thank you for helping me^^..but i got a question regarding the second question 2a

    you said:

    Therefore, by proportion,
    Jack should pay 3/7 of the total cost.
    Jill should pay 2/7 of the total cost
    Tom should pay 2/7 of the total cost also.

    but i want to know where does the denominator of 7 come from?
    Let's show the complete answer:

    (a) How should they split the total cost so that each pays an amount proportional to what he has eaten so far?

    Jack got and ate 1/3 of the whole. That is 3/9 of the whole.
    Jill took and ate 1/3 of 2/3. That is 2/9 of the whole.
    Tom took and ate 1/2 of 4/9, which is 2/9 of the whole also.

    (And what is left or not eaten, is the last 2/9 of the whole.)

    That means 3/9 +2/9 +2/9 = 7/9 is eaten.

    Therefore, by proportion,
    Jack should pay 3/7 of the total cost.
    Jill should pay 2/7 of the total cost
    Tom should pay 2/7 of the total cost also.


    Would the highlights answer your question?

    There are 9/9, or nine of 1/9, parts of the whole pizza.
    Only 7/9 are eaten.
    The 3 guys are to pay the whole 9/9 parts proportional to each has eaten.
    If the 3 guys pay only for the 7/9 parts they have eaten, then who will pay for the 2/9 parts not eaten?
    So, since only 7/9, or seven 1/9, parts are eaten, then the basis for proportion should be the seven 1/9 parts eaten.

    Meaning, the cost based on the seven 1/9 parts eaten should pay for the nine 1/9 parts whole pizza.
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