Problem 44 - Prealgebra

Question

Find the LCD for each of the following; then use the methods developed in this section to add or subtract as indicated. $$\frac{23}{70}+\frac{29}{84}$$

Step-by-Step Solution

Answer

The result of the addition is \(\frac{283}{420}\), which is in its simplest form.

Step 1: Find the Prime Factorization

To find the Least Common Denominator (LCD) of \(70\) and \(84\), start by finding the prime factorization of each number. \(70\) can be factored into prime factors as \(2 \times 5 \times 7\), while \(84\) factors into \(2^2 \times 3 \times 7\).

Step 2: Determine the LCD

The LCD is found by taking the highest power of each prime factor that appears in the factorizations. For the numbers \(70\) and \(84\), the prime factors are \(2, 3, 5,\) and \(7\). Therefore, the LCD is \(2^2 \times 3^1 \times 5^1 \times 7^1 = 420\).

Step 3: Rewrite Fractions with the LCD

Convert each fraction to have the LCD as its denominator. For \(\frac{23}{70}\), multiply by \(\frac{6}{6}\) to get \(\frac{138}{420}\). For \(\frac{29}{84}\), multiply by \(\frac{5}{5}\) to get \(\frac{145}{420}\).

Step 4: Add the Fractions

Now that both fractions have the same denominator, we can add them: \(\frac{138}{420} + \frac{145}{420} = \frac{283}{420}\).

Step 5: Simplify (if necessary)

Check if \(\frac{283}{420}\) can be simplified. Since 283 is a prime number and does not divide 420, this fraction is already in its simplest form.

Key Concepts

Prime Factorization

Understanding prime factorization is a fundamental skill in mathematics, especially when dealing with fractions. Prime factorization involves breaking down a number into its basic building blocks—prime numbers. A prime number is a number greater than one that has no positive divisors other than one and itself. For example:

• The number 70 can be expressed as the product of its prime factors, which are 2, 5, and 7. Thus, the prime factorization of 70 is \(2 \times 5 \times 7\).
• Similarly, the number 84 can be broken down into \(2^2 \times 3 \times 7\).

Knowing how to perform prime factorization is crucial for finding the Least Common Denominator (LCD) of fractions, as it allows you to account for all prime factors of both numbers involved.

Fractions Addition

Adding fractions requires a common denominator, which is where the Least Common Denominator (LCD) comes into play. Once the LCD is determined, it allows fractions to be rewritten with this denominator, enabling straightforward addition.

• For two fractions, like \(\frac{23}{70}\) and \(\frac{29}{84}\), the LCD is 420.
• Each fraction is converted to an equivalent fraction using the LCD: \(\frac{23}{70}\) becomes \(\frac{138}{420}\) and \(\frac{29}{84}\) becomes \(\frac{145}{420}\).
• Once converted, these fractions can be added directly: \(\frac{138}{420} + \frac{145}{420} = \frac{283}{420}\).

Using the LCD ensures the fractions are compatible for addition and simplifies subsequent operations.

Simplifying Fractions

Often in mathematics, after performing operations such as addition, it is necessary to simplify the result. Simplifying a fraction involves reducing it to its smallest possible equivalent form.

• This process checks if there are any common factors between the numerator and the denominator.
• For example, after adding the fractions \(\frac{138}{420}\) and \(\frac{145}{420}\), we get \(\frac{283}{420}\). While 283 is a prime number and does not divide 420, verifying no simplification is needed.

By ensuring the fraction is in its simplest form, we can achieve a cleaner and often easier-to-understand result.

Mathematics Curriculum

The concept of Least Common Denominator is integral to a student's understanding of fractions within the mathematics curriculum. It encompasses a variety of skills from basic arithmetic to more complex problem-solving techniques.

• Students learn prime factorization, paving the way for understanding common denominators.
• Understanding how to find and apply the LCD facilitates operations like addition and subtraction of fractions.
• The ability to simplify fractions taps into logical reasoning and number sense.

These foundational skills are essential, serving not just in fraction work but across various aspects of mathematics education, helping students build a robust mathematical toolkit.