If the ratio of x+y to x−y is 114, then the ratio of y to x can be written as ab where a and b are coprime positive integers. Find a+b.
I think you mean 11/4 right...
Here, I'll help you, but I want you to solve the problem in the end. The key is simplification.
You have x+y = 11 ... and x-y = 4 If you put the two equations together you'll get Y. If you submit y into the original equation you'll get x and the summation of that will be your final answer. Sorry I know the answer but I don't want to answer it for you. Have a go at it at least... I know you can do it! ;)
So David, you mean,
x + y = 11
x - y = 4
x = 7.5
y = 3.5
Then we need to substitute y into the original equation?
x + 3.5 = 11
x = 7.5
x - 3.5 = 4
x = 7.5
Adding the values lead x = 15?
Nope... I had it wrong... And I know I was a hypocrite there for a sec! :)
Solution 1: By componendo and dividendo, we get that (x+y)−(x−y)/(x+y)+(x−y)=11−4/11+4=715. Hence, yx=715, so a+b=22.
Solution 2: Since the ratio of x+y to x−y is 114, we can write
x+y/x−y=114.
We cross-multiply to get rid of the fractions and get 4(x+y)=11(x−y). Distributing the constants into the parentheses gives us 4x+4y=11x−11y. Adding 11y to both sides and subtracting 4x from both sides of this equation gives us 15y=7x. Now in order to find the ratio of y to x, we divide both sides by 15 and by x to obtain yx=7/15. (Note that we can divide by x, since we know x≠0 from the given x+y/x−y=11/4.)
Therefore, the ratio of y to x is 7/15. So a=7 and b=15, hence a+b=22.
The Answer is 22
I guess then you had part of it right...
To find the ratio of y to x, we can start by setting up the equation with the given information.
Let's represent the ratio of x+y to x-y as (x+y)/(x-y). According to the problem, this ratio is equal to 114:
(x+y)/(x-y) = 114
Now, we can solve this equation to find the value of x+y in terms of x-y:
(x+y) = 114 * (x-y)
Expanding the right side of the equation:
x+y = 114x - 114y
Rearranging the equation to isolate y:
y + 114y = 114x - x
115y = 113x
Now, we can express the ratio of y to x as y/x:
y/x = (115y)/(113x)
Simplifying the ratio by dividing both the numerator and denominator by the greatest common divisor (GCD) of 115 and 113, which is 1 since 115 and 113 are coprime:
y/x = (115/113) * (y/x)
The ratio of y to x can be written as 115/113. Therefore, a = 115 and b = 113, which are coprime positive integers.
Finally, the sum of a and b is:
a + b = 115 + 113 = 228
So, a+b = 228.