SOLUTION: If 8x=3y=7z, find x:y:z

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Question 1196511: If 8x=3y=7z, find x:y:z
Found 3 solutions by josgarithmetic, ikleyn, greenestamps:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
8x=3y=7z
Meaning, system%288x=3y%2Cand%2C3y=7z%2Cand%2C8x=7z%29.


x%2Fy=3%2F8; y%2Fz=7%2F3;

%28x%2Fy%29%283%2F3%29=%283%2F8%29%283%2F3%29=9%2F24
and
%28y%2Fz%29%288%2F8%29=%287%2F3%29%288%2F8%29=56%2F24


x:y:z is 9:24:56

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
If 8x=3y=7z, find x:y:z
~~~~~~~~~~~~~~~


            The solution by  @josgarithmetic is incorrect,
            and his answer   " x:y:z is 9:24:56 "   is incorrect,  too.

            I  came to bring you a correct solution with full explanation. 


You are given  8x = 3y = 7z.  Let introduce special designation " N " for this number,

so we have  N = 8x = 3y = 7z.     (1)


From (1), we have 

    x = N%2F8,  y = N%2F3,  z = N%2F7.    (2)


To have x, y and z integer numbers, take  

    N = Least Common Multiple of numbers 8, 3 and 7, so  N = LCM(8,3,7) = 8*3*7 = 168.


Then from (2)

    x = 168%2F8 = 21;  y = 168%2F3 = 56;  z = 168%2F7 = 24.


Therefore,  x : y : z = 21 : 56 : 24.    ANSWER


You may check it by making multiplications  8*21 = 3*56 = 7*24 = 168.



        Instead of  168,  you can take for number N any other value - 

        you will get then other individual values  x,  y  and  z, 

        but the reduced proportion  x:y:z of three terms (in integer numbers) will not change : - it will remain the same.


Solved correctly and fully explained.



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


I have no idea what tutor @josgarithmetic was doing to solve the problem; in any case, his answer does not satisfy the given conditions.

Here are a few ways you can solve this kind of problem.

(1) Choose any value for one variable and use the given ratios to find the corresponding values of the other two. Then make a ratio of the three values, and convert the ratio to whole numbers if needed.

y = 1 --> x=3/8; z = 3/7
the ratio x:y:z is (3/8):1:(3/7) = 21:56:24

(2) Given 8x, 3y, and 7z all equal to the same number, let that number be the LCM of the three coefficients: 8*3*7 = 168. Then
x = 168/8 = 21; y = 168/3 = 56; z = 168/7 = 24
and the ratio is again x:y:z = 21:56:24

(3) Having observed how solution method (2) works, you should be able to see that the solution is simply
x:y:z = (3*7):(8*7):(8*3) = 21:56:24