Fractions with different denominators

In this article, we will learn about fractions with different denominators. To know more about the basics of fractions, please visit the fractions class 6 page. This page contains comprehensive information tailored to the understanding of the general basics of fractions. Also, do not forget to look at our article on fractions with common denominators.

Let us first understand how would ve define these fractions with different denominators

Fractions with different denominators refer to fractions where the bottom numbers (denominators) are not the same.

Examples of Fractions with different denominators

\(\frac{3}{4}\) and \(\frac{5}{6}\) have different denominators, 4 and 6 respectively.

Given below are some more examples of fractions with different denominators

\(\frac{1}{2}\) and \(\frac{3}{4}\)
\(\frac{3}{5}\) and \(\frac{2}{7}\)
\(\frac{4}{6}\) and \(\frac{5}{8}\)
\(\frac{7}{9}\) and \(\frac{1}{3}\)
\(\frac{2}{4}\) and \(\frac{3}{10}\)
\(\frac{5}{12}\) and \(\frac{7}{15}\)
\(\frac{3}{16}\) and \(\frac{4}{5}\)
\(\frac{6}{7}\) and \(\frac{8}{9}\)
\(\frac{9}{11}\) and \(\frac{10}{13}\)
\(\frac{2}{3}\) and \(\frac{4}{5}\)

Adding and Subtracting Fractions with Different Denominators

Step 1: Common Denominator Needed for Mathematical Operations

When adding or subtracting fractions, they must have the same denominator, referred to as a common denominator.
Having a common denominator ensures that the fractions represent the same divisions of the whole, making the addition or subtraction valid.

Step 2: Finding the Least Common Multiple (LCM)

The least common multiple (LCM) of the denominators is often used as the common denominator.
This ensures that the common denominator is the smallest possible, simplifying calculations.

Step 3: Converting to Equivalent Fractions

Fractions must be converted to equivalent fractions with the common denominator before performing the operation.
This is done by multiplying both the numerator and denominator of each fraction by the factor needed to reach the common denominator.

Step 4: Performing Operations

Once the fractions have the same denominator, they can be added or subtracted by operating on the numerators.

Step 5: Same Denominator Simplifies Operations:

With the same denominator, addition or subtraction is performed on the numerators.
The result can be further simplified if necessary.
The resulting fraction may be simplified if possible by dividing both the numerator and denominator by their greatest common divisor (GCD).

Let us look at some examples

Example 1: Adding \(\frac{3}{4}\) and \(\frac{5}{6}\)

1. Identify the Different Denominators: 4 and 6.
2. Find the LCM: The LCM of 4 and 6 is 12.
3. Convert to Equivalent Fractions:

Multiply \(\frac{3}{4}\) by \(\frac{3}{3}\) to get \(\frac{9}{12}\).
Multiply \(\frac{5}{6}\) by \(\frac{2}{2}\) to get \(\frac{10}{12}\).

4. Perform the Addition: \(\frac{9}{12} + \frac{10}{12} = \frac{19}{12}\).
5. Simplify if Needed: The result is already in simplest form.

Example 2: Subtracting \(\frac{7}{8}\) from \(\frac{5}{6}\)

1. Identify the Different Denominators: 8 and 6.
2. Find the Least Common Multiple (LCM): The LCM of 8 and 6 is 24, so this will be the common denominator.
3. Convert to Equivalent Fractions with Common Denominator:

Multiply \(\frac{7}{8}\) by \(\frac{3}{3}\) to get \(\frac{21}{24}\).
Multiply \(\frac{5}{6}\) by \(\frac{4}{4}\) to get \(\frac{20}{24}\).

4. Perform the Subtraction: \(\frac{20}{24} – \(\frac{21}{24}\) = \(\frac{-1}{24}\).
5. Simplify if Needed: The result is already in simplest form.

By converting the fractions to have the same denominator, we can perform the subtraction on the numerators while keeping the common denominator.

These examples show that finding a common denominator is not only essential for addition but also for subtraction of fractions.

Practice Problems

Attempt these practice problems to enhance your understanding of fractions with different denominators. Take time to solve each problem. Only use the hints if needed. For using hints and checking answers, click the ‘Answer’ button.

Word Problems

Question 1. Raju has \(\frac{1}{4}\) of a chocolate bar, and Sonu gives him \(\frac{1}{2}\) of a chocolate bar. How much chocolate does Raju have now?

Answer

Hint: Convert both fractions to have the same denominator (4), and then add them.
Answer: \(\frac{3}{4}\)

Question 2. A jug contains \(\frac{5}{6}\) liters of juice. If Sita pours out \(\frac{1}{3}\) litres of juice, how much juice is left in the jug?

Answer

Hint: Convert both fractions to have the same denominator (6), and then subtract them.
Answer: \(\frac{3}{6} = \frac{1}{2}\)

Question 3. Anu wants to sew a dress using \(\frac{2}{3}\) yards of blue fabric and \(\frac{4}{5}\) yards of green fabric. How much fabric will she use in total?

Answer

Hint: Convert both fractions to have the same denominator (15), and then add them.
Answer: \(\frac{22}{15}\)

Direct Addition and Subtraction

Question 4. Add \(\frac{5}{9}\) and \(\frac{2}{3}\).

Answer

Hint: Find the common denominator (9) and then add the numerators.
Answer: \(\frac{11}{9}\)

Question 5. Subtract \(\frac{7}{10}\) from \(\frac{3}{5}\).

Answer

Hint: Find the common denominator (10) and then subtract the numerators.
Answer: \(\frac{-1}{10}\)

True or False

Question 6. \(\frac{3}{4} + \frac{2}{8} = \frac{5}{8}\). True or False?

Answer

Hint: Simplify \(\frac{2}{8}\) and then add.
Answer: False

Question 7. \(\frac{5}{6} – \frac{1}{3} = \frac{1}{2}\). True or False?

Answer

Hint: Find the common denominator (6) and then subtract.
Answer: True

Multiple-Choice Questions (MCQs)

Question 8. What is \(\frac{2}{5} + \(\frac{3}{4}\)?
Options: (A) \(\frac{22}{20}\) (B) \(\frac{5}{9}\) (C) \(\frac{23}{20}\) (D) \(\frac{1}{2}\)

Answer

Hint: Find the common denominator (20) and then add.
Answer: (C) \(\frac{23}{20}\)

Question 9. What is the subtraction of \(\frac{3}{4}\) from \(\frac{5}{6}\)?
Options: (A) \(\frac{1}{12}\) (B) \(\frac{-1}{12}\) (C) \(\frac{1}{6}\) (D) \(\frac{-1}{6}\)

Answer

Hint: Find the common denominator (12) and then subtract.
Answer: (A) \(\frac{1}{12}\)

Question 10. If \(\frac{3}{4}\) is subtracted from \(\frac{3}{4}\), the result is:
Options: (A) \(\frac{6}{4}\) (B) \(\frac{3}{4}\) (C) \(\frac{0}{4}\) (D) \(\frac{1}{4}\)

Answer

Hint: Since both fractions are the same, the subtraction results in 0.
Answer: (C) \(\frac{0}{4}\), which can be simplified to 0.