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Question 480615: Suppose z varies directly as x and y and inversely as w, and z =8 when x =6, y =5 and w =10. Find z when x =3, y =4 and w = 5.
Found 2 solutions by Edwin McCravy, Theo:
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website! Suppose z varies directly as x and y and inversely as w, and z =8 when x =6, y =5 and w =10. Find z when x =3, y =4 and w = 5.
FORMULA:
product of "directly" or "jointly" variables
"Varies as" variable = k* --------------------------------------------
product of "inversely" variables
The "varies as" variable is z, so write:
z = k*-----
The "directly variables" are x and y, so write them on the top:
xy
z = k*-----
There is just one "inversely variable", w, so write it on the bottom:
xy
z = k*-----
w
Now we're ready to substitute the numbers for all the letters except the
constant k from this: "z =8 when x =6, y =5 and w =10". Substituting:
6*5
8 = k*-----
10
Solve for k:
30
8 = k*-----
10
8 = k*3
8
--- = k
3
Now go back to this equation:
xy
z = k*-----
w
and substitute 3/8 for the constant k.
3 xy
z = ---*----
8 w
or
3xy
z = -----
8w
Now we substitute in x =3, y =4 and w = 5, to find z:
3(3)(4)
z = --------
8(5)
Then finish it up:
36
z = ----
40
Reduce the fraction:
9
z = ----
10
The answer is 9/10.
[In many problems one of the types of variables "directly", "jointly",
or "inversely" variables are missing, in which case you put 1 for them.]
Edwin
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! this reference might be helpful.
http://www.purplemath.com/modules/variatn.htm
the wording is important.
i interpreted your problem as:
z varies directly as x and y and inversely as w meaning:
z = kxy/w
k is the constant of proportionality.
you are given:
z = 8
x = 6
y = 5
w = 10
plug that into the above equation to get:
8 = (6*5*k)/10
solve for k to get:
k = (8*10)/(6*5) = 80/30 = 8/3
now that you have solved for k, you can go on to solve your problem.
you use the same formula of:
z = kxy/w
the values you are given are:
x = 3
y = 4
w = 5
the value of k that you previously solved for is:
k = 8/3
you want to find z.
plug those values into your equation to get:
z = ((8/3) * 3 * 4)/5
this results in:
z = (32/5) = 6.4
you can also do this using ratios.
this way does not involve the constant of proportionality (k).
if z is directly proportional to y, this means that:
z1/z2 = y1/y2
if z is directly proportional to x, this means that:
z1/z2 = x1/x2
if z is inversely proportional to w, this means that:
z1/z2 = w2/w1
notice the reverse proportionality when dealing with z and w.
when you now say that z is directly proportional to x and y and inversely proportional to w, this leads to the equation:
z1/z2 = (x1*y1*w2)/(x2*y2*w1)
plug your known values into this equation and solve for the unknown.
your equation becomes:
8/z2 = (6*5*5)/(3*4*10)
simplify this to get:
8/z2 = 150/120
multiply both sides of this equation by z2 to get:
8 = (150/120)*z2
multiply both sides of this equation by (120/150) to get:
z2 = 8 * (120/150) = 8 * (12/15) = 8 * (4/5) = (32/5) = 6.4
you get the same answer.
the reference gives you good tips on how to translate the words into equation form.
it also gives you good tips on how to solve these types of equations using the constant of proportionality method.
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