How to add fractions with different denominators?

Adding fractions with different denominators

In this article, we will learn how to add fractions with different denominators. Adding fractions with the same denominator is straightforward.

When the denominators are different, a common denominator must be found before the fractions can be added. Once the common denominator is found, the fractions can be expressed in terms of this denominator, and the numerators can be added together to get the final result.

Important Definitions

• Fraction: A representation of a part of a whole, expressed as a ratio of two numbers, the numerator and the denominator. It is denoted as $\frac{a}{b}$, where $a$ is the numerator, and $b$ is the denominator.
• Denominator: The number below the line in a fraction, indicating the total number of equal parts in the whole.
• Numerator: The number above the line in a fraction, indicating how many parts of the whole are taken.
• Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers.

Steps For Adding Fractions with Unlike Denominators

• Identify the Denominators: Determine the denominators of the fractions you are adding.
• Find the Least Common Multiple (LCM): Calculate the LCM of the denominators to find the common denominator. To find the LCM of the denominators, you can utilize the prime factorization method.
• Convert to Common Denominator: Multiply the numerators and denominators by the necessary factor to make the denominators the same.
• Add the Numerators: Once the denominators are the same, add the newly calculated numerators, keeping the common denominator.
• Simplify if Necessary: Reduce the resulting fraction to its simplest form if possible.

Let us understand these steps with the help of solved examples given below

How to add fractions with different denominators

Let’s take the example of adding the fractions \(\frac{3}{4}\) and \(\frac{5}{6}\), and follow the given steps:

1. Identify the Denominators: The denominators are 4 and 6.
2. Find the Least Common Multiple (LCM): The LCM of 4 and 6 is indeed 12.
3. Convert to Common Denominator:
   • To convert \(\frac{3}{4}\) to a denominator of 12, we multiply both numerator and denominator by 3, resulting in \(\frac{9}{12}\).
   • To convert \(\frac{5}{6}\) to a denominator of 12, we multiply both numerator and denominator by 2, resulting in \(\frac{10}{12}\).
4. Add the Numerators:
   \[
   \frac{9}{12} + \frac{10}{12} = \frac{19}{12}
   \]
5. Simplify if Necessary: The resulting fraction \(\frac{19}{12}\) is already in its simplest form.
   So, the sum of \(\frac{3}{4}\) and \(\frac{5}{6}\) is \(\frac{19}{12}\).

How does identifying the LCM facilitate the addition of fractions with different denominators?

• Identifying the LCM (Least Common Multiple) of the denominators helps in finding the common denominator for the fractions being added. This is essential because fractions can only be added when their denominators are the same. The LCM provides the smallest common denominator, maintaining the simplest form of the fractions. By converting the fractions to have this common denominator, the addition process is simplified, ensuring that the integrity of the fractions is maintained.

Adding Fractions with Unlike Denominators Examples

Here are some examples that demonstrate the procedure for adding fractions with unlike denominators. This series of examples is intended to provide practice and understanding for students in classes 11 and 12 studying physics in an Indian school.

Example 1: Adding \(\frac{2}{5}\) and \(\frac{3}{4}\)

1. Identify the Denominators: 5 and 4.
2. Find the Least Common Multiple (LCM): 20.
3. Convert to Common Denominator:
\[
\frac{2}{5} \times \frac{4}{4} = \frac{8}{20}, \quad \frac{3}{4} \times \frac{5}{5} = \frac{15}{20}
\]
4. Add the Numerators: \(\frac{8}{20} + \frac{15}{20} = \frac{23}{20}\).
5. Simplify if Necessary: The fraction is in its simplest form.

Example 2: Adding \(\frac{7}{8}\) and \(\frac{5}{6}\

1. Identify the Denominators: 8 and 6.
2. Find the Least Common Multiple (LCM): 24.
3. Convert to Common Denominator:
\[
\frac{7}{8} \times \frac{3}{3} = \frac{21}{24}, \quad \frac{5}{6} \times \frac{4}{4} = \frac{20}{24}
\]
4. Add the Numerators: \(\frac{21}{24} + \frac{20}{24} = \frac{41}{24}\).
5. Simplify if Necessary: The fraction is in its simplest form.

Example 3: Adding \(\frac{3}{9}\) and \(\frac{5}{6}\)

1. Identify the Denominators: 9 and 6.
2. Find the Least Common Multiple (LCM): 18.
3. Convert to Common Denominator:
\[
\frac{3}{9} \times \frac{2}{2} = \frac{6}{18}, \quad \frac{5}{6} \times \frac{3}{3} = \frac{15}{18}
\]
4. Add the Numerators: \(\frac{6}{18} + \frac{15}{18} = \frac{21}{18}\).
5. Simplify if Necessary: \(\frac{21}{18} = \frac{7}{6}\), after dividing both numerator and denominator by 3.

These examples illustrate the method of adding fractions with unlike denominators, and how to find the common denominator using the LCM, and then simplify if necessary.

Questions for Adding Fractions with Unlike Denominators

Given below are some practice questions and answers related to adding fractions with unlike denominators

Question 1:
Add \(\frac{1}{3}\) and \(\frac{2}{5}\).

Answer

1. Denominators: 3 and 5.
2. LCM: 15.
3. Common Denominator: \(\frac{5}{15}\) and \(\frac{6}{15}\).
4. Sum: \(\frac{11}{15}\).

Question 2:
What is the sum of \(\frac{4}{6}\) and \(\frac{3}{8}\)?

Answer

1. Denominators: 6 and 8.
2. LCM: 24.
3. Common Denominator: \(\frac{16}{24}\) and \(\frac{9}{24}\).
4. Sum: \(\frac{25}{24}\).

Question 3:
If you have two fractions with prime denominators, such as 3 and 7, what will be the LCM? How does this help in adding fractions like \(\frac{5}{3}\) and \(\frac{2}{7}\)?

Answer

The LCM of prime numbers like 3 and 7 is simply the product of the two numbers, so the LCM is 21. This allows us to add the fractions as follows:
1. Common Denominator: \(\frac{5}{3} \times \frac{7}{7} = \frac{35}{21}\), \(\frac{2}{7} \times \frac{3}{3} = \frac{6}{21}\).
2. Sum: \(\frac{41}{21}\).

Question 4:
Add \(\frac{3}{4}\) and \(\frac{5}{6}\) and simplify the result if possible.

Answer

1. Denominators: 4 and 6.
2. LCM: 12.
3. Common Denominator: \(\frac{9}{12}\) and \(\frac{10}{12}\).
4. Sum: \(\frac{19}{12}\), already in its simplest form.

Question 5:
Find the sum of \(\frac{7}{10}\) and \(\frac{3}{8}\), and express the result in the simplest form.

Answer

1. Denominators: 10 and 8.
2. LCM: 40.
3. Common Denominator: \(\frac{28}{40}\) and \(\frac{15}{40}\).
4. Sum: \(\frac{43}{40}\), already in its simplest form.

Multiple Choice Questions

Question 1:
What is the sum of \(\frac{1}{2}\) and \(\frac{1}{3}\)?

A) \(\frac{2}{5}\)

B) \(\frac{5}{6}\)

C) \(\frac{1}{2}\)

D) \(\frac{3}{4}\)

Answer

B) \(\frac{5}{6}\)

Question 2:
If you are adding fractions with denominators that are both prime numbers, what is the common denominator?

A) Sum of the two denominators

B) Difference between the two denominators

C) Product of the two denominators

D) Half the product of the two denominators

Answer

C) Product of the two denominators

Question 3:
What is the result of \(\frac{3}{4} + \frac{2}{5}\)?

A) \(\frac{15}{20}\)

B) \(\frac{7}{9}\)

C) \(\frac{23}{20}\)

D) \(\frac{11}{10}\)

Answer

C) \(\frac{23}{20}\)

Question 4:
When adding \(\frac{4}{7}\) and \(\frac{3}{5}\), what is the Least Common Multiple (LCM) of the denominators?

A) 12

B) 15

C) 24

D) 35

Answer

D) 35

Question 5:
What is the sum of \(\frac{2}{8}\) and \(\frac{1}{4}\) in its simplest form?

A) \(\frac{1}{2}\)

B) \(\frac{3}{4}\)

C) \(\frac{5}{8}\)

D) \(\frac{4}{8}\)

Answer

A) \(\frac{1}{2}\)

---

For further study on handling fractions, you can explore multiplying fractions with different denominators and learn about the denominator and divisor. These resources provide comprehensive information on related concepts.