the numerator of a fraction is 5 less than the denominator.if 1 is added to both the numerator and the denominator the fraction becomes 2/3. find the fraction
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To find the fraction, let's first set up an equation based on the given information.
Let the denominator of the fraction be represented by 'x'. Since the numerator is 5 less than the denominator, the numerator would be 'x - 5'.
The original fraction can be represented as (x - 5) / x.
Now, according to the second condition, if 1 is added to both the numerator and the denominator, the fraction becomes 2/3. So, we can create another equation based on this:
(x - 5 + 1) / (x + 1) = 2/3
Simplifying the equation: (x - 4) / (x + 1) = 2/3
Now, cross multiply: 3(x - 4) = 2(x + 1)
Solving this equation will give us the value of x, which represents the denominator:
3x - 12 = 2x + 2
x = 14
Now that we know the denominator is 14, we can substitute this value back into the original fraction equation to find the numerator:
Numerator = x - 5
Numerator = 14 - 5
Numerator = 9
So, the fraction is 9/14.
We can choose to write the fraction as
x/(x+5). That will satisfy the first requirement about the numerator being 5 less than the denominator.
We are also told that
(x+1)/(x+6) = 2/3
Now solve for x.
x + 1 = (2/3)x + 4
x/3 = 3
x = 9
So the fraction is 9/14.