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Thread: Fractional Equations????

  1. #1

    Fractional Equations????

    1) An intake pipe can fill a certain tank in 6 hours when the outlet pipe is closed, but with the outlet pipe open it takes 9 hours. How long would it take the outlet pipe to fill an empty tank?

    2) Members of the ski club contributed equally to obtain $1800 for a holiday trip. When 6 members found that they could not go, their contributions were refunded and each remaining member then had to pay $10 more to raise the $1800. How many went on the trip?

    3) Because of traffic Maria could average only 40 km/h for the first 20% of her trip, but she averaged 75 km/h for the whole trip. What was her average speed for the last 80% of her trip?

    4) A number x is the harmonic mean of a and b if 1/x is the average of 1/a and 1/b. Find two positive numbers that differ by 12 and have harmonic mean 5

    I don't know how to do these please help?? I am really clueless on how to do these.....

    I don't know where to start I know I need a variable but I don't know how to make it all work

  2. #2
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    3) Because of traffic Maria could average only 40 km/h for the first 20% of her trip, but she averaged 75 km/h for the whole trip. What was her average speed for the last 80% of her trip?
    as this is my first time making a reply on this forum, someone correct me if i'm wrong please.

    We know Maria was averaging 40 km/h for the first 20% of her trip. [40 x 20%]
    We know that she averaged x km/h for the remaining 80% of her trip. [x x 80%]
    We know those two numbers together give us 75 km/h.

    so we add those figures:
    (40 x 0.2)+(x x 0.8) = 75 km/h
    8 + 0.8x = 75 km/h
    -8 + 8 + 0.8x = 75 - 8
    0.8x = 67
    - Jordan

  3. #3
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    Re: Fractional Equations????

    Quote Originally Posted by dunit0001
    1) An intake pipe can fill a certain tank in 6 hours when the outlet pipe is closed, but with the outlet pipe open it takes 9 hours. How long would it take the outlet pipe to fill an empty tank?

    2) Members of the ski club contributed equally to obtain $1800 for a holiday trip. When 6 members found that they could not go, their contributions were refunded and each remaining member then had to pay $10 more to raise the $1800. How many went on the trip?

    3) Because of traffic Maria could average only 40 km/h for the first 20% of her trip, but she averaged 75 km/h for the whole trip. What was her average speed for the last 80% of her trip?

    4) A number x is the harmonic mean of a and b if 1/x is the average of 1/a and 1/b. Find two positive numbers that differ by 12 and have harmonic mean 5

    I don't know how to do these please help?? I am really clueless on how to do these.....

    I don't know where to start I know I need a variable but I don't know how to make it all work
    3) Because of traffic Maria could average only 40 km/h for the first 20% of her trip, but she averaged 75 km/h for the whole trip. What was her average speed for the last 80% of her trip?
    Assume

    The distance = d

    speed for the last 80% = v

    then

    time for first 20% = t_1 = 0.2*d/40 hour = d/200 hr

    time for last 80% = t_2 = 0.8*d/v hour = 4d/(5v) hr

    total time = t_1 + t_2 = d(0.005 + 0.8/v)

    total average speed = d/[d(0.005 + 0.8/v)]

    Now continue...

    From the complexity (and variety) of these problems - it looks like a test or a prep for a test. If you have no clue how to do these problems - you should talk to your teacher and re-evaluate your priorities.
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  4. #4
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    Re: Fractional Equations????

    Quote Originally Posted by dunit0001
    1) An intake pipe can fill a certain tank in 6 hours when the outlet pipe is closed, but with the outlet pipe open it takes 9 hours. How long would it take the outlet pipe to fill an empty tank?



    I don't know where to start I know I need a variable but I don't know how to make it all work

    An "outlet" pipe would take water OUT of the tank, right? So, I don't see how an outlet pipe could POSSIBLY fill the tank. Please check to see if you've typed the question correctly.

  5. #5
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    Re: Fractional Equations????

    Hello, dunit0001!

    3) Because of traffic Maria could average only 40 km/h for the first 20% of her trip.
    But she averaged 75 km/h for the whole trip.
    What was her average speed for the last 80% of her trip?

    Let [tex]d[/tex] = distance (length of her trip in kilometers).

    She drove [tex]0.2d[/tex] km at 40 kph.
    . . This took her: [tex]\:\frac{0.2d}{40[/tex] hours.

    She drove [tex]0.8d[/tex] km at [tex]x[/tex] kph.
    .This took her: [tex]\:\frac{0.8d}{x}[/tex] hours.

    Hence, her total driving time was: [tex]\:\frac{0.2d}{40}\,+\,\frac{0.8d}{x}[/tex] hours.


    Her average speed is: [tex]\:\frac{\text{Total distance}}{\text{Total time}} \;=\;\L\frac{d}{\frac{0.2d}{40}\.+\,\frac{0.8d}{x} } \;=\;\frac{40x}{0.2x\,+\,32}[/tex][tex]\text{ km/h}[/tex]

    There is our equation! . . . [tex]\L\;\;\frac{40x}{0.2x\,+\,32} \:=\:75[/tex]

    Then we have: [tex]\:40x \:=\:75(0.2x\,+\,32)[/tex]

    . . [tex]40x \:=\:15x\,+\,2400[/tex]

    . . [tex]25x\:=\:2400[/tex]

    . . [tex]x \:=\:\fbox{96\text{ km/h}}[/tex]


    I'm the other of the two guys who "do" homework.

  6. #6
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    Re: Fractional Equations????

    Hello, dunit0001!

    Here's #2 . . .


    2) Members of the ski club contributed equally to obtain $1800 for a holiday trip.
    When 6 members found that they could not go, their contributions were refunded
    and each remaining member then had to pay $10 more to raise the $1800.
    How many went on the trip?

    [tex]N[/tex] people paid [tex]x[/tex] dollars each: [tex]\:Nx \:=\:1800\;\;\Rightarrow\;\;x\:=\:\frac{1800}{N}\;[/tex][1]

    When 6 members dropped out, only [tex]N-6[/tex] went on the trip.
    . . and they each paid [tex]x\,+\,10[/tex] dollars.
    . . . . [tex](N\,-\,6)(x\,+\,10) \:=\:1800\;[/tex][2]

    Substitute [1] into [2]: [tex]\:(N \,-\,6)\left(\frac{1800}{N}\,+\,10\right) \:=\:1800[/tex]

    . . and we have: [tex]\:1800\,+\,10N\,-\,\frac{10800}{N}\,-\,60\:=\:1800\;\;\Rightarrow\;\;10N\,-\,60\,-\,\frac{10800}{N} \:=\:0[/tex]

    Multiply by [tex]\frac{N}{10}:\;\;N^2\,-\,6N\,-\,1080\:=\:0[/tex]

    This factors: [tex]\:(N\,-\,36)(N\,+\,30)\:=\:0[/tex]

    . . and has roots: [tex]\:N \:=\:36,\:-30[/tex]


    Therefore, 36 members originally signed up for the trip.

    . . but only [tex]N\,-\,6\:=\:\fbox{30}[/tex] members went on the trip.

    I'm the other of the two guys who "do" homework.

  7. #7
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    Re: Fractional Equations????

    Hello again, dunit0001!

    4) A number [tex]x[/tex] is the harmonic mean of [tex]a[/tex] and [tex]b[/tex] if [tex]\frac{1}{x}[/tex] is the average of [tex]\frac{1}{a}[/tex] and [tex]\frac{1}{b}.[/tex]

    Find two positive numbers that differ by 12 and have harmonic mean 5.

    Let's translate that ugly definition . . .

    [tex]\L\frac{1}{x} \;=\;\frac{\frac{1}{a}\,+\,\frac{1}{b}}{2} \;=\;\frac{a\,+\,b}{2ab} \;\;\Rightarrow\;\;x \;=\;\frac{2ab}{a\,+\,b}[/tex]


    We have two numbers that differ by 12.
    . . Let [tex]a[/tex] = the smaller number,
    . . [tex]a\,+\,12[/tex] = the larger number.

    Their harmonic mean is: [tex]\L\:\frac{2a(a\,+\,12)}{a\,+\,(a\,+\,12)} \;=\;\frac{2a(a\,+\,12)}{2a\,+\,12} \;=\;\frac{2a(a\,+\,12)}{2(a\,+\,6)} \;=\;\frac{a(a\,+\,12)}{a\,+\,6}[/tex]

    We're told that the harmonic mean is 5: [tex]\L\:\frac{a(a\,+\,12)}{a\,+\,6} \:=\:5[/tex]

    We have: [tex]\:a(a\,+\,12)\;=\;5(a\,+\,6)\;\;\Rightarrow\;\;a^2 \,+\,12a\;=\;5a\,+\,30[/tex]

    This is a quadratic: [tex]\:a^2\,+\,7a\,-\,30\;=\;0[/tex]

    . . which factors: [tex]\:(a\,-\,3)(a\,+\,10)\;=\;0[/tex]

    . . and has roots: [tex]\:a\;=\;3,\,-10[/tex]


    Therefore, the two positive numbers are: [tex]\:\fbox{3 \text{ and }15}[/tex]
    I'm the other of the two guys who "do" homework.

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