Question 1196915
your are given that a/b = c/d.
multiply both sides by b and both sides by d to get:
ad = bc
(a)
start with b/a = d/c
multiply both sides by a and both sides by c to get:
bc = ad
switch sides to get:
ad = bc
this is same as you got with the given, so they're equivalent.
(b)
start with (a-b)/b = (c-d)/d
multiply both sides by b and both sides by d to get:
(a-b) * d = (c-d) * b
simplify to get:
ad - bd = cb - db
rearrange to get:
ad - bd = bc - bd
add bd to both sides to get:
ad = bc
this is the same as you got with the given so they're equivalent.
(c)
start with a/(a+b) = c/(c+d)
multiply both sides by (a+b) and both sides by (c+d) to get:
a * (c+d) = c * (a+b)
simplify to get:
ac + ad = ca + cb
rearrange to get:
ac + ad = ac + bc
subtract ac from both sides to get:
ad = bc
this is the same as you got with the given so they're equivalent.
(d)
start with (a+b)/(c+d) = b/d
multiply both sides by (c+d) and both sides by d to get:
(a+b) * d = b * (c+d)
simplify to get:
ad + bd = bc + bd
subtract bd from both sides to get:
ad = bc
this is the same as you got with the given to they're equivalent.


another way to show they're equivalent is to assign values to each variable so that a/b = c/d.
if i assumed that a = 5 and b = 25, then a/b = 1/5
i then multiply numerator and denominator by the same value to get another ratio.
i used 7 and got c=35 and d=175
when a = 5 and b = 25 and c = 35 and d = 175, a/b = c/d becomes 5/25 = 35/175 which becomes .2 = .2
cross multiply to get ad = bc which becomes 5*175 = 25*35 which becomes 875 = 875.


i used excel to do the calculations for the given and for problems (a) through (d).
the results from all options gave results from column C equal to results from column D and results from column E equal to results from column F.
this confirmed that all equations given in problems (a) through (d) are equivalent to results from the given equation.


here are the excel results.
<img src = "http://theo.x10hosting.com/2022/101301.jpg">