Question 1013841
your solution is (a+c)/(b+d) = 5/7.


you can find it easily by just assuming some values for a and b and c and d and solving.


you can also find it in terms of a and b and c and d much more arduously by performing algebra on the variables directly.


in that case you will get (a+c)/(b+d) = (25*(b+d))/(49*(a+c)).


when you replace a and b and c and d with qualifying numbers, the ratio will wind up being the same, i.e. it will be 5/7.


first i'll do it using numbers.


then i'll do it using variables.


your ratio is a:b = c:d = 5:7.


this can also be written as a/b = c/d = 5/7


this means that a/b = 5/7 and that c/d = 5/7.


if we assume that a is 5 and b is 7, then the ratio is 5/7 for them.


if we assume that c is 5 and d is 7, then the ratio is 5/7 for them.


since a = 5 and b = 7 and c = 5 and d = 7, then the ratio of (a+c)/(b+d) becomes (5+5)/(7+7) which becomes 10/14 which simplifies to 5/7.


you can assume that c = 10 and d = 14, or c = 15 and d = 21.
as long as the ratio of c/d simplifies to 5/7, all those values are good and will lead to the same conclusion.


for example:


assume a = 5 and b = 7 and c = 15 and d = 21.


(a+c) = 20
(b + d) = 28


(a+c)/(b+d) = 20/28 which simplifies to 5/7.


with letters it becomes a little more complex, but  you'll get an answer that will be equivalent to the answer that you got with numbers.


start with a/b = 5/7 and c/d = 5/7


cross multiply to get 7a = 5b and 7c = 5d


solve for a to get a = 5b/7
solve for b to get b = 7a/5
solve for c to get c = 5d/7
solve for d to get d = 7c/5


a+c is equal to 5b/7 + 5d/7 = (5b+5d)/7 = 5(b+d)/7.


b+d is equal to 7a/5 + 7c/5 = (7a+7c)/5 = 7(a+c)/5.


(a+c)/(b+d) = 5(b+d)/7 divided by 7(a+c)/5.


this is equivalent to 5(b+d)/7 times 5/(7(a+c))


this is equivalent to 5(b+d)*5/ (7*7(a+c))


simplify further to get {{{25(b+d)/(49(a+c))}}}


to confirm, we replace a,b,c,d with qualifying values.


we'll assume a = 5, b = 7, c = 15, d = 21


the equation of {{{25(b+d)/(49(a+c))}}} becomes {{{25(7+21)/(49(5+15))}}}


simplify this to get {{{(25*28)/(49*20)}}}


perform the multiplication to get {{{700/980}}}


when you simplify this, you will find that it is equal to 5/7.