Math Lesson 3.1.4 - Simplifying Fractions

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Welcome to our Math lesson on Simplifying Fractions, this is the fourth lesson of our suite of math lessons covering the topic of The Meaning of Fractions. Equivalent Fractions. Simplifying Fractions, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Simplifying Fractions

Simplifying fractions means writing them in the lowest terms possible. In other words, a fraction is fully simplified when it is not possible to divide its numerator and denominator by the same number, i.e. when numerator and denominator are relatively prime.

For example, the fraction 12/25 is simplified to the lowest terms as, although we can divide individually 12 and 25 by other numbers besides 1 and themselves (i.e. none of them is a prime number), we are not able to divide both of them simultaneously by the same number. This means these two numbers are relatively prime, as explained in math tutorial 1.6.

Example 3

Which of the following fractions can be simplified and which not?

Solution 3

We must check whether numerator and denominator of the above fractions have any common factor or not. We have:

Thus, the fraction 15/27 can be simplified by 3 to obtain 5/9.

As for 18/54, we have

Thus, we can simplify 18 and 54 by 2 × 3 × 3 = 18, to obtain 1/3.

For 14/15, we cannot identify any common factor because we have

For 21/35, we have

Thus, when dividing both the numerator and denominator of 21/35 by 7, we obtain the simplest form of this fraction, i.e. 3/5.

As for 78/93, we have

Thus, simplifying by 3 both numerator and denominator of 78/93, we obtain the simplest form of this fraction, i.e. 26/31.

Finally, for 55/88, we obtain

We can find any missing component of a given fraction when another equivalent fraction is known. Look at the following example:

Example 4

What is the value of "m" that produces equivalent fraction in the cases below?

Solution 4

1. We simplify the known fraction to the simplest terms. We have
   15/21 = (15 ÷ 3)/(21 ÷ 3) = 5/7
   Thus, we have
   5/7 = a/35
   Hence, since the second denominator is 5 times greater than the first one, so must be the numerator of the second fraction in respect to the first denominator too. We have
   5/7 = (a ÷ 5)/(35 ÷ 5)
   Thus, a = 25 because 25 ÷ 5 = 5.
2. We use the same approach as in the previous example. Thus, first we simplify the known fraction to the lowest terms, i.e.
   20/44 = (20 ÷ 4)/(44 ÷ 4) = 5/11
   Hence, we have
   5/11 = 15/a
   Therefore, since 5 × 3 = 15, then 11 × 3 = a. This means a = 33. We can apply twice or more the multiplication and division of numerator and denominator by the same number to check whether a fraction is equivalent to another or not instead of expressing both fractions to the lowest terms. Thus, if we have the fractions 27/45 and 18/30, we can write
   27/45 = (27 ÷ 9)/(45 ÷ 9) = 3/5 = (3 × 6)/(5 × 6) = 18/30
   In this way, we proved that the two original fractions are equivalent.

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7. Continuing learning fractions - read our next math tutorial: Proper and Improper Fractions. Mixed Numbers

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