The denominator of a fraction exceeds the numerator by 1.if 2 be taken from each ,the sum of the reciprocal of the new fraction and 4 times the original fraction is 5 .find the original fraction?
To solve this problem, let's consider the given information and break it down step by step.
Let's say the numerator of the original fraction is represented by 'x'. Since the denominator exceeds the numerator by 1, we can represent the denominator as 'x + 1'.
The original fraction can be expressed as x/(x + 1).
Now, according to the problem, if we subtract 2 from both the numerator and denominator, the new fraction becomes (x - 2)/((x + 1) - 2), which simplifies to (x - 2)/x.
The problem states that the sum of the reciprocal of the new fraction and 4 times the original fraction is 5. Mathematically, this can be written as:
1/((x - 2)/x) + 4 * (x/(x + 1)) = 5
To solve this equation, let's find a common denominator and simplify:
1/((x - 2)/x) + 4 * (x/(x + 1)) = 5
Multiply the first fraction by x/x:
1 * (x/(x - 2)) + 4 * (x/(x + 1)) = 5
Multiply each fraction by its respective denominator:
x/(x - 2) + 4x/(x + 1) = 5
To get rid of the denominators, let's find a common denominator:
Common denominator = (x - 2)(x + 1)
Multiply each term by the common denominator:
x(x + 1) + 4x(x - 2) = 5(x - 2)(x + 1)
Simplify the equation:
x^2 + x + 4x^2 - 8x = 5(x^2 - x - 2)
Combine like terms:
5x^2 - 7x = 5x^2 - 5x - 10
Now, subtract 5x^2 from both sides:
-7x = -5x - 10
Next, add 5x to both sides:
-2x = -10
Finally, divide both sides by -2 to solve for 'x':
x = -10 / -2
x = 5
So the numerator of the original fraction is 5. Using this value, we can determine the denominator:
Denominator = x + 1 = 5 + 1 = 6
Therefore, the original fraction is 5/6.