The numerator of a fraction is two less than its denominator. When both numerator and denominator are increased by 3, the fraction is incre
ased by 3/20. Find the original fraction.
(n-2)/n
(n+1)/(n+3) = (n-2)/n + 3/20
multiply both sides by 20 n(n+3)
20 n(n+1) = 20(n+3)(n-2) + 3n(n+3)
20n^2 + 20n =20(n^2 +n-6) +3n^2 + 9 n
20n^2 + 20n = 20n^2 +20n -120 + 3n^2 + 9n
solve quadratic for n
then do
(n-2)/n
To solve this problem, we need to come up with equations based on the given information and then solve them to find the original fraction.
Let's start by representing the numerator as "n" and the denominator as "d".
According to the problem, the numerator is two less than the denominator, so we can write the first equation as:
n = d - 2 --- Equation 1
Next, it says that when both the numerator and denominator are increased by 3, the fraction is increased by 3/20. This means the new fraction can be represented as:
(n + 3)/(d + 3) = (n/d) + 3/20 --- Equation 2
Now we have two equations, and we can substitute the value of "n" from Equation 1 into Equation 2 to solve for "d".
Substituting the value of n from Equation 1 into Equation 2, we have:
(d - 2 + 3)/(d + 3) = ((d - 2)/d) + 3/20
Simplifying this equation, we get:
(d + 1)/(d + 3) = (d - 2)/d + 3/20
To eliminate the fractions, we can cross-multiply:
(d + 1)(d) = (d - 2)(d + 3) + (3/20)(d)(d + 3)
Expanding on both sides and simplifying, we have:
d^2 + d = d^2 + d - 6 + 3d + (3/20)d^2 + 9/20d
Combining like terms, the equation becomes:
0 = (3/20)d^2 + (9/20 - 1)d - 6
Multiplying both sides of the equation by 20 to remove fractions, we get:
0 = 3d^2 + 9d - 20
Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, the quadratic equation factors as:
0 = (3d - 4)(d + 5)
Setting each factor equal to zero, we have two possible solutions:
3d - 4 = 0 --> d = 4/3
d + 5 = 0 --> d = -5
Since the denominator of a fraction cannot be negative, we discard the solution d = -5.
Therefore, the original denominator is d = 4/3.
Now we can substitute this value back into Equation 1 to find the numerator:
n = d - 2
n = (4/3) - 2/1
n = (4 - 6)/3
n = -2/3
So, the original fraction is -2/3.