Harnessing Properties: How to Add Three Fractions or Mixed Numbers with Ease

Adding three fractions or mixed numbers is easier if you regroup before you compute: pair the two that share a denominator, or pair the ones whose fractional parts make a whole. Below: how the commutative and associative properties make that legal, two worked examples, and practice questions.

However, by utilizing the properties of addition, this process can be simplified and made more intuitive. In this guide, we’ll explore how to add three fractions or mixed numbers by leveraging these properties.

Step-by-step Guide to Add Three Fractions or Mixed Numbers with Ease:

1. Understanding Addition Properties: 

There are two main properties of addition that can be helpful:

, Commutative Property: The order in which numbers are added doesn’t affect the sum. For example, \(a + b = b + a\).

, Associative Property: The way numbers are grouped in addition doesn’t affect the sum. For example, \(a + (b + c) = (a + b) + c\).

2. Converting Mixed Numbers to Improper Fractions:

If you’re dealing with mixed numbers, convert them to improper fractions. This makes the addition process more straightforward.

3. Finding the Least Common Denominator (LCD): 

Determine the smallest number into which all the denominators can divide. This will ensure that the fractions are of comparable sizes.

4. Adjusting Each Fraction to the LCD:

Modify each fraction so that they all have the LCD as their denominator.

5. Performing the Addition: 

With the same denominator in place, simply add up all the numerators to get the final answer.

6. Converting Back to Mixed Numbers (if necessary): 

If the result is an improper fraction, and you need a mixed number, convert it back.

Example 1: 

Add \(\frac{1}{4}\), \(\frac{2}{8}\), and \(1 \frac{1}{2}\).

Solution: 

First, convert the mixed number: \(1 \frac{1}{2}\) becomes \(\frac{3}{2}\).

The LCD for 4, 8, and 2 is 8. Adjusting the fractions:

, \(\frac{1}{4}\) becomes \(\frac{2}{8}\).

, \(\frac{2}{8}\) remains the same.

, \(\frac{3}{2}\) becomes \(\frac{12}{8}\).

Adding them up, the result is \(\frac{16}{8}\), which is equal to 2.

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Example 2: 

Add \(\frac{1}{3}\), \(\frac{2}{6}\), and \(2 \frac{1}{2}\).

Solution: 

First, convert the mixed number: \(2 \frac{1}{2}\) becomes \(\frac{5}{2}\).

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The LCD for 3, 6, and 2 is 6. Adjusting the fractions:

, \(\frac{1}{3}\) becomes \(\frac{2}{6}\).

, \(\frac{2}{6}\) remains the same.

, \(\frac{5}{2}\) becomes \(\frac{15}{6}\).

Adding them up, the result is \(\frac{19}{6}\), which can be expressed as \(3 \frac{1}{6}\).

Practice Questions: 

1. Add \(\frac{1}{5}\), \(\frac{2}{10}\), and \(1 \frac{3}{4}\).

2. Add \(\frac{3}{7}\), \(\frac{2}{14}\), and \(2 \frac{1}{2}\).

3. Add \(\frac{4}{9}\), \(\frac{2}{3}\), and \(3 \frac{1}{3}\).

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Answers:

1. \(2 \frac{9}{20}\)

2. \(4 \frac{1}{14}\)

3. \(6 \frac{5}{9}\)

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