Solve this plxxx
The numerator of a fraction is 2less than it's denominator.when both numerator and denominator are increased by 3,the fraction is increased by 3/20.find the original fraction.
first denominator -- x
first numerator ---- x-2
new denominator -- x+3
new numerator ---- x+1
(x-2)/x + 3/20 = (x+1)/(x+3)
times 20x(x+3) , the LCD
20(x+3)(x+1) + 3x(x+3) = 20x(x+1)
expanding and simplifying gave me
x^2 + 3x - 40 = 0
(x-5)(x+8) = 0
x = 5 or x=-8
if x=5, the original fraction was 3/5
if x=-8 the original fraction was -10/-8 or 5/4
check for 3/5 , new fraction would be 6/8 or 3/4
3/5 + 3/20 = 12/20 + 3/20 = 15/20 = 3/4
but for 5/4, new fraction would be 8/7
5/4 + 3/20 = 28/20 = 7/5 ≠ 8/7
BUT, if we take the unsimplified fraction -10/-8 , new fraction would be -7/-5 = 7/5
So the original fraction would be 3/5 for sure, but
also the unsimplified fraction -10/-8
To solve this problem, we can start by setting up equations for the given information.
Let's assume that the original fraction is represented as x/y, where x is the numerator and y is the denominator.
According to the problem, the numerator of the fraction is 2 less than its denominator. So we can write the equation:
x = y - 2 ---- (Equation 1)
When both the numerator and denominator are increased by 3, the fraction is increased by 3/20. We can write the equation for this situation as:
(x + 3) / (y + 3) = x/y + 3/20 ---- (Equation 2)
We have two equations (Equation 1 and Equation 2) and two unknowns (x and y) to solve for. Let's simplify Equation 2 and substitute the value of x from Equation 1.
Expanding Equation 2, we get:
x/(y+3) + 3/(y+3) = x/y + 3/20
Now, multiplying both sides of the equation by (y+3) and simplifying, we have:
x + 3 = x(y + 3)/y + 3(y + 3)/20
Substituting the value of x from Equation 1 (x = y - 2), we get:
y - 2 + 3 = (y - 2)(y + 3)/y + 3(y + 3)/20
Expanding and simplifying this equation, we have:
y - 2 + 3 = (y^2 + 3y - 2y - 6)/y + (3y + 9)/20
Combining like terms, we get:
y + 1 = [y^2 + y - 6 + (3y + 9)]/y
Simplifying further:
y + 1 = (y^2 + 4y + 3)/y
Cross-multiplying to eliminate the fraction:
y(y + 1) = y^2 + 4y + 3
Expanding this equation:
y^2 + y = y^2 + 4y + 3
Canceling out the y^2 terms, we get:
y = 4y + 3
Now, bring all terms with y on one side:
y - 4y = 3
Simplifying, we have:
-3y = 3
Finally, solving for y by dividing both sides by -3, we get:
y = -1
Substituting this value back into Equation 1 (x = y - 2), we find the value of x:
x = -1 - 2 = -3
So the original fraction is -3/-1, which simplifies to 3/1 or simply 3.