# The numerator of a fraction is 5 less than its denominator. If 1 is added to the numerator and to the denominator, the new fraction is `2/3` . Find the fraction. Please help as I don't understand.

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hardydl | Certified Educator

Start by writing any missing number as a variable.

If “The numerator of the fraction is five less than the denominator,” then you could write the denominator as x and the numerator as x-5.

That means that the missing fraction is (x-5)/x.

Now you must “add one to the numerator and denominator.”

So the new fraction is (x-5+1)/9x+1) or (x-4)/(x+1).

According to the question, this equals 2/3.

Therefore, we can set up an equation: (x-4)/(x+1) = 2/3.

To solve, cross multiply: 3(x-4) = 2(x+1)

Solve for x:

3x-12 = 2x+2

x=14

Remember, this is only x. You must still plug it into the fraction: (14-5)/14.

Simplify: (9/14).

gsenviro | Certified Educator

Suppose the denominator is x,

then the numerator is x-5, since numerator is 5 less than denominator.

And the required fraction is (x-5)/x

If we use the given scenario, when 1 is added to the numerator and denominator , the fraction becomes 2/3

Adding 1 to both numerator and denominator:

(x-5+1)/(x+1) = 2/3

or 3 (x-4) = 2(x+1)

or 3x-12 = 2x+2

or 3x-2x = 2+12     that is x =14 and x-5 = 9

Thus the required fraction (x-5)/x = 9/14.

check: adding 1 to both numerator and denominator, we get (9+1)/(14+1) = 1015 = 2/3, which is given information.