in a fraction twince the numerator is 2 more than the denominator . if 3 is added to the numerator and to the denominator , the new fraction is 2/3 . find the original fraction.
n / (2n -2)
(n + 3)/(2n+1) = 2/3
3(n+3) = 2 (2n+1)
3 n + 9 = 4n + 2
n = 7
so
n/(2n-2) = 7/12
Let the denominator be x.
Numerator will= x+2
If 3 is added,
Numerator will be x+2+3 = x +5
Denominator will be x+3
Now,
X+5/ x+3 = 2/3
By cross multiply,
X+5 × 3 =3x + 15,
X+3×2=2x + 6,
Now,
3x+15=2x+6
3x-2x=6-15
X= -9
Numerator=-9 and denominator=-9×2= -18
To solve this problem, let's start by setting up the equation for the given information.
Let's assume the original fraction is a/b, where 'a' is the numerator and 'b' is the denominator.
According to the first piece of information, "in a fraction, the numerator is 2 more than the denominator." Mathematically, this can be expressed as:
a = b + 2
According to the second piece of information, "if 3 is added to the numerator and the denominator, the new fraction is 2/3." Mathematically, this can be expressed as:
(a + 3) / (b + 3) = 2/3
Now, let's solve these equations step by step.
First, substitute the value of 'a' from the first equation into the second equation:
(b + 2 + 3) / (b + 3) = 2/3
Simplifying, we get:
(b + 5) / (b + 3) = 2/3
Next, cross-multiply to eliminate the fractions:
3(b + 5) = 2(b + 3)
Expanding the equation, we get:
3b + 15 = 2b + 6
Now, solve for 'b' by subtracting 2b from both sides and subtracting 6 from both sides:
3b - 2b = 6 - 15
b = -9
Now that we have the value of 'b', we can substitute it back into the first equation to find 'a':
a = (-9) + 2
a = -7
Therefore, the original fraction is -7/-9, which can be simplified to 7/9.