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Thread: How to solve three-term ratio problem with two variables?

  1. #1

    Question [SOLVED] How to solve three-term ratio problem with two variables?

    This problem has me completely stumped:
    Code:
    (600-x)/60:(800-y)/80:(1000+x+y)/100=3:4:5
    How can I simplify this ratio in order to solve for x and y? I tried setting the first two terms equal to 3:4 or the last two terms equal to 4:5 and cross-multiplying, but that didn't work out. Is there a way to reduce the three-term ratio into two terms?

    The answer is: x=168 and y=32.

    Thoughts on solving this problem:
    • There are two variables, so I need to establish two equations in order to solve both variables.
    • If this were a two-term ratio problem, e.g., x:y=1:2, then I could setup an equation like x/y = 1/2, then cross multiply.
    • But since this is a three-term ratio problem, where x : y : z is in the same ratio as 3r : 4r : 5r, where r is a fixed factor, I cannot reduce the entire ratio to one fraction.
    • I did notice that interestingly the denominators of the three terms are in the ratio of 3:4:5. I'm not sure if that helps with anything.
    • I realize that since this is a ratio, I can perform the same operation on all three terms, e.g., multiple each term by 10.
    • I wish I could show my "attempt" in solving this problem, but I don't even know where to start!
    Last edited by supreme; 02-01-2018 at 01:35 AM.

  2. #2
    Elite Member
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    Quote Originally Posted by supreme View Post
    This problem has me completely stumped:
    (600-x)/60800-y)/801000+x+y)/1000=3:4:5

    How can I simplify this ratio in order to solve for x and y? I tried setting the first two terms equal to 3:4 or the last two terms equal to 4:5 and cross-multiplying, but that didn't work out. Is there a way to reduce the three-term ratio into two terms?

    The answer is: x=168 and y=32.
    Your problem is coming wrong on my screen - interspersed with "emoji"s.

    Most probably "(" is the offender.

    Can you please repost your problem [including your thoughts about ways to solve this problem] replacing "(" & ")" with "[" and "]".
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  3. #3
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    Also needing clarification : you have three "/" on the left hand side but two ":" on the right.

  4. #4
    I've fixed my original post. Sorry about that.

  5. #5
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    Quote Originally Posted by supreme View Post
    This problem has me completely stumped:
    Code:
    (600-x)/60:(800-y)/80:(1000+x+y)/100=3:4:5
    How can I simplify this ratio in order to solve for x and y? I tried setting the first two terms equal to 3:4 or the last two terms equal to 4:5 and cross-multiplying, but that didn't work out. Is there a way to reduce the three-term ratio into two terms?

    The answer is: x=168 and y=32.

    Thoughts on solving this problem:
    • There are two variables, so I need to establish two equations in order to solve both variables.
    • If this were a two-term ratio problem, e.g., x:y=1:2, then I could setup an equation like x/y = 1/2, then cross multiply.
    • But since this is a three-term ratio problem, where x : y : z is in the same ratio as 3r : 4r : 5r, where r is a fixed factor, I cannot reduce the entire ratio to one fraction.
    • I did notice that interestingly the denominators of the three terms are in the ratio of 3:4:5. I'm not sure if that helps with anything.
    • I realize that since this is a ratio, I can perform the same operation on all three terms, e.g., multiple each term by 10.
    • I wish I could show my "attempt" in solving this problem, but I don't even know where to start!
    Start with:

    [(600-x)/60] / [(800-y)/80] = 3/4 .... and so on
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  6. #6
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    Quote Originally Posted by supreme View Post
    This problem has me completely stumped:
    Code:
    (600-x)/60:(800-y)/80:(1000+x+y)/100=3:4:5
    How can I simplify this ratio in order to solve for x and y? I tried setting the first two terms equal to 3:4 or the last two terms equal to 4:5 and cross-multiplying, but that didn't work out. Is there a way to reduce the three-term ratio into two terms?

    The answer is: x=168 and y=32.

    Thoughts on solving this problem:
    • There are two variables, so I need to establish two equations in order to solve both variables.
    • If this were a two-term ratio problem, e.g., x:y=1:2, then I could setup an equation like x/y = 1/2, then cross multiply.
    • But since this is a three-term ratio problem, where x : y : z is in the same ratio as 3r : 4r : 5r, where r is a fixed factor, I cannot reduce the entire ratio to one fraction.
    • I did notice that interestingly the denominators of the three terms are in the ratio of 3:4:5. I'm not sure if that helps with anything.
    • I realize that since this is a ratio, I can perform the same operation on all three terms, e.g., multiple each term by 10.
    • I wish I could show my "attempt" in solving this problem, but I don't even know where to start!
    Your mention of 3r : 4r : 5r provides the key to one good method of solving this sort of problem. We introduce that r as a third variable, and write the proportion as three equations:

    (600-x)/60 = 3r
    (800-y)/80 = 4r
    (1000+x+y)/100 = 5r

    Solve this system of equations, and you'll have the answer.

  7. #7

    Red face

    Quote Originally Posted by Dr.Peterson View Post
    Your mention of 3r : 4r : 5r provides the key to one good method of solving this sort of problem. We introduce that r as a third variable, and write the proportion as three equations:

    (600-x)/60 = 3r
    (800-y)/80 = 4r
    (1000+x+y)/100 = 5r

    Solve this system of equations, and you'll have the answer.
    Thank you, Dr. Peterson. I followed your advice and solved the problem!

    600-x=180r
    800-y=320r
    1000+x+y=500r

    Add these three equations together and get:

    2400=1000r
    r=2.4
    x=168
    y=32

    Thank you so much!

  8. #8
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    Quote Originally Posted by supreme View Post
    Thank you, Dr. Peterson. I followed your advice and solved the problem!

    600-x=180r
    800-y=320r
    1000+x+y=500r

    Add these three equations together and get:

    2400=1000r
    r=2.4
    x=168
    y=32

    Thank you so much!
    Good work!

  9. #9
    Elite Member mmm4444bot's Avatar
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    Quote Originally Posted by supreme View Post
    I've fixed [the auto-emoji inserts in] my original post.
    Yes, this is one of the minor annoyances with our forum software.

    For future reference, please type spaces around colons used in such ratio expressions. You may also use the Preview Post button, to see how your posts will render on the board. Cheers
    "English is the most ambiguous language in the world." ~ Yours Truly, 1969

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