Problem 52 - Prealgebra

Question

Find the LCD for each of the following; then use the methods developed in this section to add or subtract as indicated. $$\frac{1}{8}+\frac{1}{4}+\frac{1}{5}+\frac{1}{10}$$

Step-by-Step Solution

Answer

The sum is \( \frac{27}{40} \).

Step 1: Identify the Denominators

List out the denominators of each fraction to identify which values we need the Least Common Denominator (LCD) for. The fractions are \( \frac{1}{8} \), \( \frac{1}{4} \), \( \frac{1}{5} \), and \( \frac{1}{10} \), so the denominators are 8, 4, 5, and 10.

Step 2: Find the Least Common Denominator (LCD)

To find the LCD, identify the least common multiple of the denominators. The prime factorizations are: - 8 is \( 2^3 \) - 4 is \( 2^2 \) - 5 is \( 5 \) - 10 is \( 2 \times 5 \).The LCD will be the highest power of each prime number present: \( 2^3 = 8 \) and \( 5 \). Thus, LCD is \( 8 \times 5 = 40 \).

Step 3: Convert Fractions to Equivalent Fractions with LCD

Convert each fraction to have a denominator of 40.- \( \frac{1}{8} = \frac{1 \times 5}{8 \times 5} = \frac{5}{40} \)- \( \frac{1}{4} = \frac{1 \times 10}{4 \times 10} = \frac{10}{40} \)- \( \frac{1}{5} = \frac{1 \times 8}{5 \times 8} = \frac{8}{40} \)- \( \frac{1}{10} = \frac{1 \times 4}{10 \times 4} = \frac{4}{40} \).

Step 4: Add the Fractions

Add together the fractions that now have a common denominator:\[ \frac{5}{40} + \frac{10}{40} + \frac{8}{40} + \frac{4}{40} = \frac{5+10+8+4}{40} = \frac{27}{40}. \]

Step 5: Simplify the Result (if necessary)

Since \( \frac{27}{40} \) is already in its simplest form (as 27 and 40 have no common factors other than 1), no further simplification is necessary.

Key Concepts

Understanding Fractions

Fractions are a way of representing parts of a whole. They consist of a numerator, which is the top number, and a denominator, which is the bottom number. For instance, in the fraction \( \frac{1}{8} \), 1 is the numerator and 8 is the denominator. The numerator tells us how many parts we have, while the denominator tells us how many equal parts the whole is divided into.
Fractions can represent divisions, parts of a set, or ratios. When dealing with fractions, it's crucial to understand how they relate to each other through their numerators and denominators. This understanding helps us perform operations such as addition, subtraction, multiplication, and division involving fractions effectively.

Addition of Fractions

To add fractions, all fractions involved need the same denominator, known as a common denominator. Without this, sums can't be accurately calculated, as the parts represented by each fraction aren't equal.
When fractions share the same denominator, they describe parts of a whole that are equally divided, allowing numerators to be simply added together. In cases where denominators differ, like in \( \frac{1}{8} + \frac{1}{4} + \frac{1}{5} + \frac{1}{10} \), a common denominator is essential. This is where the Least Common Denominator (LCD) comes into play, providing the smallest shared multiple of all denominators.
The LCD ensures all fractions are expressed in terms of a common base, simplifying the addition process into a straightforward calculation of numerators over this shared denominator.

Prime Factorization and Finding the Least Common Denominator (LCD)

Prime factorization involves breaking down numbers into their basic building blocks, known as prime numbers, which are numbers greater than 1 only divisible by 1 and themselves. Using prime factorization makes finding the LCD easier.
Consider the denominators 8, 4, 5, and 10. Their prime factorizations are:

• 8: \(2^3\)
• 4: \(2^2\)
• 5: \(5\)
• 10: \(2 \times 5\)

To find the LCD, identify the highest power of each prime number: \(2^3\) for 2 and 5 for itself. Multiplying these gives the LCD: \(8 \times 5 = 40\). Now all fractions can be expressed with this common denominator, readying them for addition.

Simplifying Fractions

Once fractions have been added, the resulting fraction may require simplification. Simplifying, or reducing, a fraction means adjusting it to its smallest possible form, where the numerator and denominator have only 1 as a common factor. This is done by dividing both the top and bottom by their greatest common divisor (GCD).
For example, if we ended up with \( \frac{27}{40} \), we must determine if it can be simplified. Since 27 factors as \(3^3\) and 40 factors as \(2^3 \times 5\), the only common factor is 1. Therefore, \( \frac{27}{40} \) is already in its simplest form, needing no further simplification. Simplifying fractions not only makes results more presentable but also easier to interpret and use in subsequent calculations.