The denominator of a certain fraction exceeds the numerator by 1. The reciprocal of the fraction exceeds 3/4 by 7/12. Find the fraction.
If the fraction is x/(x+1), then we are told
(x+1)/x = 3/4 + 7/12
To solve this problem, we can start by representing the fraction in terms of its numerator and denominator.
Let's define the fraction as "x/y", where x is the numerator and y is the denominator. According to the problem statement, we know that the denominator exceeds the numerator by 1. We can express this as:
y = x + 1 ...(Equation 1)
Now, let's find the reciprocal of the fraction. The reciprocal of a fraction x/y is y/x. According to the problem statement, the reciprocal of the fraction exceeds 3/4 by 7/12. We can express this as an equation:
y/x = 3/4 + 7/12 ...(Equation 2)
To simplify the equation, let's find a common denominator for the fractions on the right side. The common denominator for 4 and 12 is 12. Now, we can rewrite Equation 2:
y/x = (9/12) + (7/12)
Combining the fractions gives us:
y/x = 16/12
Next, let's simplify the fraction on the right side by dividing both the numerator and denominator by their greatest common divisor, which is 4:
y/x = (16/4) / (12/4)
= 4/3
Now we can substitute Equation 1 into Equation 2 to solve for the value of x:
x + 1 = 4y/3
To remove the fraction, we can multiply both sides of the equation by 3:
3(x + 1) = 4y
Expanding the left side of the equation gives us:
3x + 3 = 4y
Now let's substitute the value of y from Equation 1 into the above equation:
3x + 3 = 4(x + 1)
3x + 3 = 4x + 4
Simplifying the equation gives us:
3x - 4x = 4 - 3
-x = 1
Multiplying both sides by -1, we have:
x = -1
Now we can find the value of y by substituting the value of x back into Equation 1:
y = x + 1
y = -1 + 1
y = 0
Therefore, the fraction is -1/0.