If the ratio of x+y to x−y is 279/262 , then the ratio of y to x can be written as a/b where a and b are coprime positive integers. Find a+b
(x+y)/(x-y) = 279/262
x+y=279
x-y=262
2x = 541
2y = 17
y/x = 17/541
...
find the ratio given 12:7
To find the ratio of y to x, we'll start by expressing the given ratio of x+y to x-y in terms of y and x.
The expression x+y can be rewritten as (x-y) + 2y. Now we have:
(x+y)/(x-y) = (x-y + 2y)/(x-y)
Simplifying further, we get:
(x+y)/(x-y) = (x-y)/(x-y) + 2y/(x-y)
= 1 + 2y/(x-y)
Given that (x+y)/(x-y) = 279/262, we can substitute the values into our equation:
279/262 = 1 + 2y/(x-y)
Subtracting 1 from both sides:
279/262 - 1 = 2y/(x-y)
Simplifying:
279/262 - 262/262 = 2y/(x-y)
17/262 = 2y/(x-y)
Cross multiplying:
17(x-y) = 2y
Expanding the left side:
17x - 17y = 2y
Bringing all the terms involving y to one side:
17x = 19y
Now, to find the ratio of y to x, divide both sides by 17:
(17x)/17 = (19y)/17
Simplifying:
x = (19/17)y
So, the required ratio of y to x is 19/17.
Since 19 and 17 are coprime positive integers, the sum of these two integers is 19 + 17 = 36.
Therefore, the final answer is 36.