please understand my doubt

Question

the difference between the numerator and the denominator of fraction is 5. If 5 is added to its denominator, the fraction is decreased by 5/4 find the value of the fraction

 

denominator x
numerator is x+5

x=x+5                                                         1

(x+5)/(x+5)=(x+5)/(x)+5/4                          2

i have four doubts
1 is my second equation correct i tried but answer came wrong so
2 explain me why my second equation is wrong
3 i am using one variable but i get two equation and how we ignore any first or second equation. Becuse every question's word get information
4 how to determine which equation will evaluate answer

Answer 1

Assuming that the original fraction is "proper," i.e., numerator smaller than denominator:

Original fraction x / (x + 5)

Altered fraction x / (x + 10)

Altered = Original - 5/4

x / (x + 10) = 1 / (x + 5) - 5/4

4(x + 5)x = 4(x + 10) = 5(x + 10)(x + 5)      multiply by LCD

 

Can you solve for x?

Answer 2

Let's call the numerator x, and the denominator can be y. 

Then we know that x - y = 5

or x = y + 5

 

The problem also tells us that when 5 is added to the denominator y, then the original fraction x/y is decreased by 5/4:

 

x/(y + 5) = x/y - 5/4

 

The common denominator her is 4y(y + 5), so let's multiply both sides of the equation by this and cancel terms where possible:

 

4xy = 4x(y + 5) - 5y(y + 5)

4xy = 4xy + 20x - 5y² - 25y   Let's subtract 4xy from both sides. That leaves us with:

0 = 20x - 5y² - 25y

 

However, we know that x = y + 5, so we can re-write this as:

 

0 = 20(y + 5) - 5y² - 25y

0 = 20y + 100 - 5y² - 25y

0 = 100 - 5y - 5y²     when we combine like terms. Let's divide thru by -5 to get:

0 = -20 + y + y²       or:

y² + y - 20 = 0         This can be factored as:

(y - 4)(y + 5) = 0

y = 4 and y = -5

 

However, we can eliminate y = -5. Why, because the problem states that eventually 5 is added to the denominator. If we add 5 + (-5), we get 0, and division by 0 is undefined. So, our only solution is y = 4.

 

If y = 4, then x = y + 5 = 4 + 5 = 9

 

The original fraction is 9/4. We can check this. Add 5 to this denominator. 

5 + 4 = 9, so the fraction becomes:

9/9 = 1

 

If we subtract 9/4 - 1, that is the same as:

9/4 - 4/4 = 5/4, which is what the problem stated, that the fraction would be decreased by 5/4 if we add 5 to the denominator.