Adding Fractions with Like and Unlike Denominators

Chapter 29: Adding Fractions with Like Denominators

Concept Explanation:

Adding fractions with like denominators is straightforward, as you simply add the numerators while keeping the denominator the same.

For instance:

$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$

Common Mistakes:

Neglecting the Denominator: Always remember to keep the denominator unchanged.
Improper Addition: Ensure you are accurately adding the numerators.

Helpful Tips:

Visual Models: Drawing fraction circles can help visualize how fractions are combined.
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Use Examples**: Practicing with simple fractions first can build confidence.

Hard Questions:

Q1: Calculate $\frac{2}{5} + \frac{3}{5}$ .

Step-by-Step Solution:

Since the denominators are the same, add the numerators:

$\frac{2 + 3}{5} = \frac{5}{5} = 1$

Answer:

The result is $1$ .

Q2: Calculate $\frac{4}{7} + \frac{2}{7}$ .

Step-by-Step Solution:

Since the denominators are the same, add the numerators:

$\frac{4 + 2}{7} = \frac{6}{7}$

Answer:

The result is $\frac{6}{7}$ .

Q3: Calculate $\frac{5}{12} + \frac{1}{12}$ .

Step-by-Step Solution:

Since the denominators are the same, add the numerators:

$\frac{5 + 1}{12} = \frac{6}{12} = \frac{1}{2}$

Answer:

The result is $\frac{1}{2}$ .

Chapter 30: Adding Fractions with Unlike Denominators

Concept Explanation:

When adding fractions with unlike denominators, the first step is to find a common denominator. This involves calculating the least common multiple (LCM) of the denominators.

Find the LCM: The least common multiple of the denominators provides the common denominator.
Rewrite Each Fraction: Adjust each fraction to have the common denominator.
Add the Numerators: Once the denominators are the same, add the numerators.

Common Mistakes:

Failing to Find the LCM: Not identifying the least common denominator leads to incorrect sums.
Incorrectly Converting Fractions: Ensure that each fraction is properly adjusted to the common denominator.

Helpful Tips:

Practice with Small Numbers: Start with simpler fractions to build confidence before tackling more complex problems.
Visual Representation: Drawing bar models can help illustrate how fractions are combined.

Hard Questions:

Q1: Add $\frac{1}{4} + \frac{1}{6}$ .

Step-by-Step Solution:

The LCM of 4 and 6 is 12.
Rewrite the fractions:

$\frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}$

Add:

$\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$

Answer:

The result is $\frac{5}{12}$ .

Q2: Add $\frac{2}{3} + \frac{1}{6}$ .

Step-by-Step Solution:

The LCM of 3 and 6 is 6.
Rewrite the fractions:

$\frac{2}{3} = \frac{4}{6}$

Add:

$\frac{4}{6} + \frac{1}{6} = \frac{5}{6}$

Answer:

The result is $\frac{5}{6}$ .

Q3: Add $\frac{3}{8} + \frac{1}{4}$ .

Step-by-Step Solution:

The LCM of 8 and 4 is 8.
Rewrite the second fraction:

$\frac{1}{4} = \frac{2}{8}$

Add:

$\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$

Answer:

The result is $\frac{5}{8}$ .

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