Multiplying Fractions with Different Denominators

Multiplying fractions with different denominators is the focus of this article. The method includes understanding fractions, finding the least common multiple (LCM), and step-by-step multiplication, providing a solid foundation for advanced mathematical concepts.

If you are looking to calculate the LCM of specific numbers, you may utilize the LCM Calculator. This tool can effortlessly determine the LCM of two or more numbers, providing a quick and accurate result. Similarly, if you need to find the Greatest Common Divisor (GCD) of numbers, you can make use of the GCD Calculator.

Multiplying Fractions with Different Denominators

Multiplying fractions with different denominators involves finding the LCM, converting to equivalent fractions, multiplying, and simplifying the result.

Let us first start by defining fractions and what are fractions with, unlike denominators and fractions with common denominators.

Definition of Fractions

A fraction is a mathematical expression that represents the division of one integer by another integer. A fraction is written in the form of \(\frac{a}{b}\), where:

• \(a\) is the numerator (the top part of the fraction),
• \(b\) is the denominator (the bottom part of the fraction), and
• \(b \neq 0\).

So we now know what are fractions. Now, let us proceed to establish the definitions of fractions with dissimilar denominators and fractions with identical denominators.

1. Unlike Denominators: Unlike denominators refer to fractions where the denominators are not the same.
   Example:- Consider the fractions \(\frac{2}{3}\) and \(\frac{5}{4}\). Here, the denominators are 3 and 4, respectively. Since these numbers are not the same, we say that these fractions have unlike denominators. (learn more in the article Fractions with different denominators)
2. Common Denominator: Common denominators refer to fractions having the same number in the denominator, allowing for straightforward mathematical operations.
   Example:- Consider the fractions \(\frac{3}{8}\) and \(\frac{5}{8}\) have a common denominator of 8, making them easily comparable or combinable.

Steps for Multiplying Fractions with Different Denominators

Given below are the steps for multiplying fractions with different (unlike) denominators

1. Find the Least Common Multiple (LCM): Determine the smallest number that is a multiple of both denominators.
2. Convert to Equivalent Fractions: Change the fractions to equivalent fractions with the common denominator (LCM).
3. Multiply the Fractions: Multiply the numerators together and the denominators together.
4. Simplify the Result: Reduce the resulting fraction to its simplest form by dividing both the numerator and denominator by their greatest common divisor (GCD).

Let us now look at how to follow these steps for multiplying fractions with unlike denominators

Step 1: Find the Least Common Multiple (LCM)

The first step is to find the least common multiple of the denominators. The LCM is the smallest number that is a multiple of both denominators.

For example, if you have two fractions \(\frac{1}{3}\) and \(\frac{2}{5}\), the LCM of 3 and 5 is 15.

Step 2: Convert to Equivalent Fractions

Next, you need to convert the fractions to equivalent fractions with the common denominator (LCM).

For \(\frac{1}{3}\), multiply both the numerator and denominator by 5 to get \(\frac{5}{15}\).
For \(\frac{2}{5}\), multiply both the numerator and denominator by 3 to get \(\frac{6}{15}\).

Step 3: Multiply the Fractions

Now, multiply the numerators together and the denominators together:

\(\frac{5}{15} \times \frac{6}{15} = \frac{5 \times 6}{15 \times 15} = \frac{30}{225}\)

Step 4: Simplify the Result

Finally, simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD):

\(\frac{30}{225} = \frac{2}{15}\)

Solved Examples for multiplying fractions with unlike denominators

Given below are two more examples illustrating the process of multiplying fractions with different denominators:

Example 1

Multiply \(\frac{2}{3}\) and \(\frac{4}{5}\):

1. Find the LCM: LCM of 3 and 5 is 15.
2. Convert to Equivalent Fractions: \(\frac{2}{3} = \frac{10}{15}\) and \(\frac{4}{5} = \frac{12}{15}\).
3. Multiply the Fractions: \(\frac{10}{15} \times \frac{12}{15} = \frac{120}{225}\).
4. Simplify the Result: \(\frac{120}{225} = \frac{8}{15}\).

Example 2

Multiply \(\frac{3}{4}\) and \(\frac{7}{6}\):

1. Find the LCM: LCM of 4 and 6 is 12.
2. Convert to Equivalent Fractions: \(\frac{3}{4} = \frac{9}{12}\) and \(\frac{7}{6} = \frac{14}{12}\).
3. Multiply the Fractions: \(\frac{9}{12} \times \frac{14}{12} = \frac{126}{144}\).
4. Simplify the Result: \(\frac{126}{144} = \frac{7}{8}\).

These examples demonstrate the systematic approach to multiplying fractions with different denominators, following the steps of finding the LCM, converting to equivalent fractions, performing the multiplication, and simplifying the result.

Questions

1. Multiply the fractions \(\frac{2}{3}\) and \(\frac{3}{5}\). What is the result?
2. How do you find the LCM of two numbers? Explain with an example.
3. Why is it necessary to find a common denominator when multiplying fractions with different denominators?

Answer

1. \(\frac{2}{3} \times \frac{3}{5} = \frac{6}{15} = \frac{2}{5}\)
2. The LCM of two numbers is found by dividing the product of the numbers by their GCD. For example, for 8 and 12, \(\text{LCM}(8, 12) = \frac{8 \times 12}{\text{GCD}(8, 12)} = 24\).
3. Finding a common denominator ensures that the fractions represent the same “whole” and hence can be multiplied together accurately.