Reducing Fractions

Let's learn how to reduce fractions.

We already learned about equivalent fractions. There are many fractions that are equivalent to the given one, but there is one special fraction.

Fraction is irreducible if numerator and denominator have no common factors.

For example, $\frac{{1}}{{3}}$ is irreducible, while $\frac{{2}}{{6}}$ is not, because 2 and 6 have common factor 2.

Two ways to find irreducible fraction:

Try to find common factor of numerator and denominator. Divide them by that factor. Repeat this step until there are no common factors. This way is similar to finding prime factorization.
This method is essentially the same as first, but you need only one step. Find Greatest Common Divisor of numerator and denominator. Divide numerator and denominator by found number.

From second way it follows another definition of irreducible fraction.

Fraction $\frac{{a}}{{b}}$ is irreducible if ${G}{C}{D}{\left({a},{b}\right)}={1}$ .

Indeed, if ${G}{C}{D}{\left({a},{b}\right)}={1}$ then ${a}$ and ${b}$ have no common factors, so $\frac{{a}}{{b}}$ is irreducible.

Example 1. Reduce fraction $\frac{{18}}{{24}}$ .

Both 18 and 24 are divisible by 2, so we divide them by 2: we get new equivalent fraction $\frac{{9}}{{12}}$ .

12 is divisible by 2, but 9 is not, so we try 3.

Both 9 and 12 are divisible by 3, so we divide them by 3: we get new equivalent fraction $\frac{{3}}{{4}}$ .

We are done because 3 and 4 have no common factors.

Answer: $\frac{{3}}{{4}}$ .

Now, let's try to use second way.

Example 2. Reduce fraction $\frac{{40}}{{60}}$ .

Find greatest common divisor: ${G}{C}{D}{\left({40},{60}\right)}={20}$ .

Divide both numerator and denominator by 20: $\frac{{\frac{{40}}{{\color{red}{{{20}}}}}}}{{\frac{{60}}{{\color{red}{{{20}}}}}}}=\frac{{2}}{{3}}$ .

Answer: $\frac{{2}}{{3}}$ .

Next example.

Example 3. Reduce fraction $\frac{{5}}{{35}}$ .

Find greatest common divisor: ${G}{C}{D}{\left({5},{35}\right)}={5}$ .

Divide both numerator and denominator by 5: $\frac{{\frac{{5}}{{\color{red}{{{5}}}}}}}{{\frac{{35}}{{\color{red}{{{5}}}}}}}=\frac{{1}}{{7}}$ .

Answer: $\frac{{1}}{{7}}$ .

Now, take pen and paper and do following exercises.

Exercise 1. Reduce fraction $\frac{{15}}{{25}}$ .

Answer: $\frac{{3}}{{5}}$ .

Next exercise.

Exercise 2. Reduce fraction $\frac{{135}}{{90}}$ .

Answer: $\frac{{3}}{{2}}$ .

Next exercise.

Exercise 3. Reduce fraction $\frac{{2}}{{9}}$ .

Answer: $\frac{{2}}{{9}}$ (already irreducible).

Next exercise.

Exercise 4. Reduce fraction $\frac{{60}}{{12}}$ .

Answer: 5. Hint: $\frac{{5}}{{1}}={5}$ .

Next exercise.

Exercise 5. Reduce fraction $-\frac{{18}}{{18}}$ .

Answer: -1.