Problem 31 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{1}{3}+\frac{3}{4}-\frac{3}{7} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{55}{84}\).

Step 1: Find a Common Denominator

To add or subtract fractions, they must have the same denominator. The denominators are 3, 4, and 7. Find the least common multiple (LCM) of these numbers. The LCM of 3, 4, and 7 is 84.

Step 2: Convert Fractions to Equivalent Fractions with the Common Denominator

Convert each fraction to have a denominator of 84. For \(\frac{1}{3}\): \(\frac{1}{3} = \frac{1 \times 28}{3 \times 28} = \frac{28}{84}\) For \(\frac{3}{4}\):\(\frac{3}{4} = \frac{3 \times 21}{4 \times 21} = \frac{63}{84}\) For \(\frac{3}{7}\):\(\frac{3}{7} = \frac{3 \times 12}{7 \times 12} = \frac{36}{84}\)

Step 3: Perform the Addition and Subtraction

Add and subtract the numerators of the equivalent fractions:\(\frac{28}{84} + \frac{63}{84} - \frac{36}{84} = \frac{28 + 63 - 36}{84}\)Calculate the numerator: \(28 + 63 - 36 = 55\)Thus, the expression becomes \(\frac{55}{84}\).

Step 4: Simplify the Result (if possible)

Check if \(\frac{55}{84}\) can be simplified. The greatest common divisor (GCD) of 55 and 84 is 1, meaning \(\frac{55}{84}\) is already in its simplest form.

Key Concepts

Least Common Multiple (LCM)

The Least Common Multiple (LCM) is a crucial concept when working with fractions. It's the smallest multiple that two or more numbers share. Imagine you want to add or subtract fractions; they need to have the same denominator.
For instance, if you're working with fractions that have denominators of 3, 4, and 7, you need to find a common ground. That's where the LCM comes in.
To find the LCM of these numbers, you can list the multiples of each number:

• Multiples of 3: 3, 6, 9, 12, 15, …
• Multiples of 4: 4, 8, 12, 16, 20, …
• Multiples of 7: 7, 14, 21, 28, …

Now, identify the smallest multiple they all share. In this case, the LCM of 3, 4, and 7 is 84. This is why you convert each fraction to have 84 as the denominator before adding or subtracting. Finding the LCM ensures the fractions have a common base, simplifying the computation process.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same part of a whole. This is important when finding a common denominator for fractions. You create equivalent fractions to ensure the denominators match, making addition and subtraction possible.
To convert to equivalent fractions, multiply both the numerator and the denominator by the same number.For example, with the fraction \( \frac{1}{3} \), to create a denominator of 84, multiply both parts by 28:\[\frac{1}{3} = \frac{1 \times 28}{3 \times 28} = \frac{28}{84}\]

• For \( \frac{3}{4} \), multiply numerator and denominator by 21 to get \( \frac{63}{84} \).
• For \( \frac{3}{7} \), multiply by 12 to arrive at \( \frac{36}{84} \).

This process aligns the fractions by the same denominator, enabling straightforward addition or subtraction. Remember, equivalent fractions don’t change the value of the original fraction, they make calculations easier.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) helps you simplify fractions. It’s the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD can make a fraction simpler and easier to work with.
When you check if a fraction is simplified, look for common divisors of the numerator and denominator. If the GCD is 1, the fraction is in its simplest form. In the case of \( \frac{55}{84} \), you'll find:

• Prime factors of 55: 5, 11
• Prime factors of 84: 2, 3, 7

No common prime factors exist, so the GCD is 1.
Thus, \( \frac{55}{84} \) is already simplified. Identifying the GCD ensures fractions are reduced to their simplest form for easier arithmetic operations.