The ratio of the numerator to the denominator of a certain fraction is one to four. if three is added to the numerator and subtracted from the denominator, the new fraction reduces to one-third. what is the original fraction?

Let x = the numerator part of the original fraction. Since the ratio of the numerator to denominator is one to four, this means 4x is the denominator.

is our original fraction.

Now we work with adding and subtracing to get to something that looks like one-third.

We write the second half as x / 3x because the fraction can simplify to 1/3. The rest of the problem is solved by cross multiplying (essentially, clearing fractions and simplifying) and solving for x.

(3x)(x + 3) = (4x -3)(x)

3x² + 9x = 4x² - 3x

Since it's a quadratic, let's set one side equal to zero and use the Zero Product Property (ZPP).

9x = x² - 3x

0 = x² - 12x

x is a common factor in x²-12x, so we can use the ZPP and factor

0 = x (x - 12)

So x = 0 or x =12.

Having zero as an answer won't work as we'd be dividing by zero. Let's check x = 12.

12/48 is the starting fraction, which simplifies to 1/4

12+3 / 48 -3 = 15 / 45 = 1/3.

Thus, 12/48 was the starting fraction.

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