A denominator of a fraction exceeds the numerator by 1 if 2 is taken from each the sum of the reciprocal of the new fraction and 4 times the original fraction is 5. Find the original fraction

Why did the denominator cross the road?

To exceed the numerator by 1, of course!

Let's assume that the numerator of the original fraction is represented by 'x' and the denominator is represented by 'x + 1'.

According to the given information, if 2 is subtracted from the sum of the reciprocal of the new fraction (1/(x + 1)) and 4 times the original fraction (4x/(x + 1)), the result is 5.

So we can write the equation as:
1/(x + 1) + 4x/(x + 1) - 2 = 5

To solve this equation, we need to simplify it:

First, let's combine the fractions on the left side:
(1 + 4x)/(x + 1) - 2 = 5

Next, let's multiply both sides of the equation by (x + 1) to eliminate the fraction:
(1 + 4x) - 2(x + 1) = 5(x + 1)

Now, let's distribute and simplify further:
1 + 4x - 2x - 2 = 5x + 5

Combining like terms:
2x - 1 = 5x + 5

Let's simplify and isolate the variable:
2x - 5x = 5 + 1
-3x = 6

Now, let's solve for 'x':
x = -6/3
x = -2

Since the numerator cannot be negative, the original fraction does not exist.

To solve this problem, let's start by representing the original fraction as x/y, where x is the numerator and y is the denominator.

According to the problem, the denominator exceeds the numerator by 1, so we can write the equation:

y = x + 1

Next, we need to translate the second part of the problem into an equation. It states that if 2 is taken from each the sum of the reciprocal of the new fraction (1/(x-2)/(y-2)) and 4 times the original fraction (4(x/y)), the result is 5.

Mathematically, we have:

1/(x-2)/(y-2) + 4(x/y) = 5

Now, let's simplify this equation. First, let's simplify 4(x/y):

4(x/y) = 4x/y

Next, let's simplify 1/(x-2)/(y-2) by multiplying the numerator and denominator by the reciprocal:

1/(x-2)/(y-2) = (y-2)/(x-2)

Substituting these simplified expressions back into the equation, we get:

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

Now, let's multiply both sides of the equation by y(x-2) to eliminate the denominators:

y(x-2) * [(y-2)/(x-2)] + y(x-2) * [4x/y] = 5 * y(x-2)

Simplifying the left side:

(y-2)(x-2) + 4x(x-2) = 5y(x-2)

Expanding the products:

xy - 2y - 2x + 4 + 4x^2 - 8x = 5xy - 10y

Rearranging the terms:

xy - 5xy + 4x^2 - 2x - 8x + 2y - 10y + 4 = 0

Combining like terms:

4x^2 - 15xy - 8x - 8y + 4 = 0

We now have a quadratic equation in terms of x and y. However, we need one more equation to solve for the values of x and y. Let's use the equation y = x + 1, which we obtained earlier.

Substituting y = x + 1 into the quadratic equation:

4x^2 - 15x(x + 1) - 8x - 8(x + 1) + 4 = 0

Simplifying this equation:

4x^2 - 15x^2 - 15x - 8x - 8x - 8 + 4 = 0

Combining like terms:

-11x^2 - 31x - 4 = 0

Now, we can use the quadratic formula to solve for x:

x = [-(-31) +/- sqrt((-31)^2 - 4(-11)(-4))]/(2(-11))

Simplifying further:

x = [31 +/- sqrt(961 - 176)]/(-22)

x = [31 +/- sqrt(785)]/(-22)

x = (31 +/- sqrt(785))/(-22)

Therefore, the possible values for x are:

x = (31 + sqrt(785))/(-22) or x = (31 - sqrt(785))/(-22)

We can then substitute these values into the equation y = x + 1 to find the corresponding values for y.

Finally, we will have the original fraction x/y where x and y are the solutions to the equations.

original fraction ---- x/(x+1)

new fraction = (x-2)/(x-1)

(x-1)/(x-2) + 4x/(x+1) = 5
times (x-2)(x+1), the LCD

(x-1)(x+1) + 4x(x-2) = 5(x-2)(x+1)
x^2 - 1 + 4x^2 - 8x = 5x^2 - 5x - 10
3x - 9 = 0
x = 3

original fraction is 3/4

check:
new fraction = 1/2
1/(1/2) + 4(3/4)
= 2 + 3 = 5 , as needed