After studying this article on Fractions with common denominators, students will understand the concept of fractions with common denominators and the methods to add and subtract them. They will learn to identify examples, perform calculations, and apply this knowledge in various mathematical contexts.
Fractions with Common Denominators
A fraction represents a part of a whole and is expressed as \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator. The denominator should not be equal to zero. Fractions with common denominators are fractions that have the same denominator.
A common denominator is a number that is a common multiple of the denominators of two or more fractions. Having a common denominator allows us to perform arithmetic operations like addition and subtraction on those fractions.
To know more about the basics of fractions, please visit the page on Fractions class 6. This page contains comprehensive information tailored to the understanding of the general basics of fractions.
Examples of Fractions with Common Denominators
Let us now look at some examples of fractions with like denominators.
1. \(\frac{3}{5}\) and \(\frac{4}{5}\)
2. \(\frac{7}{10}\), \(\frac{2}{10}\), and \(\frac{5}{10}\)
These examples share the same denominators 5 and 10, respectively. Some other examples of fractions with common denominator are
1. \(\frac{3}{5}\) and \(\frac{2}{5}\): Since both fractions have 5 as the denominator, they have common denominators.
2. \(\frac{7}{9}\), \(\frac{5}{9}\), and \(\frac{1}{9}\): All these fractions have the same denominator of 9.
3. \(\frac{4}{12}\) and \(\frac{8}{12}\): Both these fractions share the common denominator of 12.
4. \(\frac{5}{20}\), \(\frac{15}{20}\), and \(\frac{10}{20}\): These three fractions have the same denominator of 20.
5. \(\frac{6}{4}\) and \(\frac{2}{4}\): These fractions have the common denominator of 4.
When fractions have common denominators, it makes arithmetic operations like addition and subtraction more straightforward since the denominators remain the same, and only the numerators are combined or subtracted.
Finding Common Denominators
To find common denominators for fractions, follow these steps:
1. List the Multiples: List the multiples of the denominators.
2. Identify the Least Common Multiple (LCM): Find the least common multiple of the denominators.
3. Convert the Fractions: Multiply the numerators and denominators of the fractions by the appropriate factor so that the denominator of each fraction is the LCM.
Example
Consider the fractions \(\frac{2}{3}\) and \(\frac{3}{4}\).
- Multiples of 3: 3, 6, 9, 12, 15, …
Multiples of 4: 4, 8, 12, 16, … - LCM of 3 and 4 is 12.
- Convert the fractions:
\(\frac{2}{3} \times \frac{4}{4} = \frac{8}{12}\)
\(\frac{3}{4} \times \frac{3}{3} = \frac{9}{12}\)
Adding and Subtracting Fractions with Common Denominators
When adding or subtracting fractions with common denominators, the denominator remains the same, and only the numerators are added or subtracted.
Adding Fractions with Common Denominators
Example 1
\[
\frac{3}{5} + \frac{4}{5} = \frac{3+4}{5} = \frac{7}{5}
\]
Example 2
\[
\frac{7}{10} + \frac{2}{10} = \frac{7+2}{10} = \frac{9}{10}
\]
Subtracting Fractions with Common Denominators
Example 1
\[
\frac{4}{5} – \frac{3}{5} = \frac{4-3}{5} = \frac{1}{5}
\]
Example 2
\[
\frac{7}{10} – \frac{5}{10} = \frac{7-5}{10} = \frac{2}{10} = \frac{1}{5}
\]
Having a common denominator makes the process of adding and subtracting fractions straightforward. You simply perform the arithmetic operation on the numerators and keep the common denominator.
Questions
1. What is a common denominator? Can you provide an example?
2. How would you add \(\frac{3}{8} + \frac{5}{8}\)? What would be the result?
Answers
1. A common denominator is when two or more fractions have the same value in the denominator. Example: \(\frac{5}{6}\) and \(\frac{3}{6}\).
2. To add \(\frac{3}{8} + \frac{5}{8}\), you keep the denominator the same and add the numerators: \(\frac{3+5}{8} = \frac{8}{8} = 1\).
Practice Problems
Given below are 10 questions related to ‘fractions with common denominators,’ including answers and hints. Practice these questions for better conceptual clarity.
MCQs
Question 1. What is the sum of \(\frac{2}{7}\) and \(\frac{3}{7}\)?
a) \(\frac{1}{7}\)
b) \(\frac{5}{7}\)
c) \(\frac{7}{7}\)
d) \(\frac{3}{7}\)
Answer
Answer: b) \(\frac{5}{7}\)
Hint: When adding fractions with common denominators, simply add the numerators.
Question 2. Which of the following fractions does not have a common denominator with (\frac{4}{5}\)?
a) \(\frac{2}{5}\)
b) \(\frac{3}{5}\)
c) \(\frac{5}{4}\)
d) \(\frac{1}{5}\)
Answer
Answer: c) \(\frac{5}{4}\)
Hint: Look for fractions with a denominator of 5.
Word Problems
Question 3. Ravi is an artist and wants to mix two different shades of blue paint to create the perfect shade for his new painting. He has \(\frac{3}{8}\) of a liter of sky blue paint and \(\frac{5}{8}\) of a liter of ocean blue paint. What is the total amount of blue paint Ravi has when he mixes both shades together?
Answer
Solution:
Since both fractions have a common denominator of 8, they can be added directly:
\[
\frac{3}{8} + \frac{5}{8} = \frac{3 + 5}{8} = \frac{8}{8} = 1
\]
Ravi has 1 liter of blue paint when he mixes both shades together.
Question 4. Seema and her friends are sharing cookies at a party. Seema ate \(\frac{4}{6}\) of a cookie, and her friend Anu ate \(\frac{5}{6}\) of a cookie. How much more did Anu eat than Seema?
Answer
Answer: 10 students
Solution:
**Solution:**
First, notice that the fractions have a common denominator, so we can subtract them directly:
\[
\frac{5}{6} – \frac{4}{6} = \frac{5 – 4}{6} = \frac{1}{6}
\]
Anu ate \(\frac{1}{6}\) of a cookie more than Seema.
Question 5. In a community garden, Rashmi is responsible for planting roses, and Kunal is responsible for planting sunflowers. Rashmi planted \(\frac{5}{12}\) of the total garden area with roses, and Kunal planted \(\frac{7}{12}\) of the total garden area with sunflowers. What fraction of the garden area was left unplanted?
Answer
Answer: \(\frac{1}{2}\)
Solution:
First, we’ll find the total fraction of the garden area that was planted by adding the fractions representing the parts planted by Rashmi and Kunal, since they have a common denominator:
\[
\frac{5}{12} + \frac{7}{12} = \frac{5 + 7}{12} = \frac{12}{12} = 1
\]
This means that the entire garden area was planted.
Next, we’ll find the fraction of the garden area left unplanted. Since the entire garden area is considered to be 1, and the entire area was planted, the unplanted fraction is:
\[
1 – 1 = 0
\]
Or, we can represent this as a fraction with a common denominator of 12:
\[
\frac{12}{12} – \frac{12}{12} = \frac{0}{12}
\]
So, the fraction of the garden area left unplanted is \(\frac{0}{12}\), which means no part of the garden was left unplanted.
True/False Type
Question 6. Adding \(\frac{5}{8}\) and \(\frac{3}{8}\) results in \(\frac{8}{8}\).
Answer
Answer: True
Hint: Add the numerators while keeping the denominator the same.
Question 7. \(\frac{3}{6}\) and \(\frac{1}{3}\) have a common denominator.
Answer
Answer: False
Hint: Check whether the denominators are equal or can be simplified to the same value.
Fill in the Blanks Type
Question 8. The sum of \(\frac{2}{3}\) and \(\frac{1}{3}\) is (_______).
Answer
Answer: \(\frac{3}{3}\)
Hint: Just add the numerators.
Question 9. When subtracting fractions with common denominators, the denominator (_______).
Answer
Answer: remains the same
Hint: Focus on the operation with the numerators.
Matrix Match Type
Question 10. Match the following fractions with their equivalent sums or differences.
| Column A | Column B |
|---|---|
| P: \(\frac{1}{9} + \frac{2}{9}\) | i: \(\frac{3}{9}\) |
| Q: \(\frac{4}{5} – \frac{2}{5}\) | ii: \(\frac{2}{5}\) |
| R: \(\frac{3}{7} + \frac{1}{7}\) | iii: \(\frac{4}{7}\) |
Answer
Answer: P-i, Q-iii, R-ii
Hint: Perform the addition or subtraction in each option in Column A and match with Column B.
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