UpStudy Free Solution:

To simplify the expression \(13 \frac { 1} { 6} + 13 \frac { 1} { 3} + 13 \frac { 3} { 4} \), follow these steps:

1. Convert each mixed number to an improper fraction:

\[13 \frac { 1} { 6} = \frac { 79} { 6} , \quad 13 \frac { 1} { 3} = \frac { 40} { 3} , \quad 13 \frac { 3} { 4} = \frac { 55} { 4} \]

2. Find a common denominator for the fractions. The least common denominator (LCD) of 6, 3, and 4 is 12.

3. Convert each fraction to have the common denominator:

\[\frac { 79} { 6} = \frac { 79 \times 2} { 6 \times 2} = \frac { 158} { 12} \]

\[\frac { 40} { 3} = \frac { 40 \times 4} { 3 \times 4} = \frac { 160} { 12} \]

\[\frac { 55} { 4} = \frac { 55 \times 3} { 4 \times 3} = \frac { 165} { 12} \]

4. Add the fractions:

\[\frac { 158} { 12} + \frac { 160} { 12} + \frac { 165} { 12} = \frac { 158 + 160 + 165} { 12} = \frac { 483} { 12} \]

5. Simplify the fraction:

\[\frac { 483} { 12} = 40 \frac { 3} { 12} = 40 \frac { 1} { 4} \]

So, \(13 \frac { 1} { 6} + 13 \frac { 1} { 3} + 13 \frac { 3} { 4} = 40 \frac { 1} { 4} \).

Supplemental Knowledge

Understanding how to work with mixed numbers and improper fractions is essential for various arithmetic operations. Here are some additional insights:

1. Mixed Numbers and Improper Fractions:

- A mixed number consists of an integer and a proper fraction (e.g., \(13 \frac { 1} { 6} \)).

- An improper fraction has a numerator larger than or equal to its denominator (e.g., \(\frac { 79} { 6} \)).

2. Conversion Between Mixed Numbers and Improper Fractions:

- To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place this sum over the original denominator.

\[13 \frac { 1} { 6} = \frac { ( 13 \times 6) + 1} { 6} = \frac { 79} { 6} \]

3. Finding a Common Denominator:

- When adding fractions with different denominators, find the least common denominator (LCD).

- The LCD is the smallest number that each of the denominators divides into without leaving a remainder.

4. Converting Fractions to a Common Denominator:

- Adjust each fraction so they all have the same denominator by multiplying both the numerator and denominator by appropriate factors.

\[\frac { 79} { 6} = \frac { 79 \times 2} { 6 \times 2} = \frac { 158} { 12} \]

5. Adding Fractions:

- Once fractions have a common denominator, add their numerators while keeping the common denominator.

\[\frac { 158} { 12} + \frac { 160} { 12} + \frac { 165} { 12} = \frac { 483} { 12} \]

6. Simplifying Fractions:

- Simplify the resulting fraction by dividing both the numerator and denominator by their greatest common divisor (GCD).

7. Example for Context:

- Consider adding \(5\frac { 2} { 3} \) and \(4\frac { 3} { 4} \):

Convert to improper fractions:

\(5\frac { 2} { 3} = \frac { 17} { 3} \)

\(4\frac { 3} { 4} = \frac { 19} { 4} \)

Find LCD (12), convert, add, simplify.

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