The numerator of a fraction is 7 less than the denominator. If the numerator is increased by 2 and the denominator by 9,we again get the same fraction. Find the fraction.
n = d-7
(n+2)/(d+9) = n/d
(d-7+2)/(d+9) = (d-7)/d
d(d-5) = (d-7)(d+9)
d^2-5d = d^2+2d-63
7d = 63
d=9
check:
original fraction: 2/9
modified fraction: 4/18 = 2/9
To solve this problem, we can set up equations based on the given information.
Let the denominator of the fraction be represented by the variable "x".
Then, the numerator of the fraction is "x - 7" (since it is 7 less than the denominator).
According to the problem, if we increase the numerator by 2 and the denominator by 9, we still get the same fraction. This can be expressed as an equation:
(x - 7 + 2) / (x + 9) = (x - 7) / x
Now, we can cross-multiply to solve for x:
(x - 5)(x) = (x - 7)(x + 9)
Expanding both sides of the equation yields:
x^2 - 5x = x^2 + 2x - 63
Next, simplify the equation:
x^2 - x^2 - 5x - 2x = -63
Combine like terms:
-7x = -63
Divide both sides by -7:
x = (-63) / (-7) = 9
So, the denominator of the fraction is 9.
Now, substitute this value back into the numerator formula to find the numerator:
Numerator = 9 - 7 = 2
Thus, the fraction is 2/9.