Problem 47 - Prealgebra

Question

Find the LCD for each of the following; then use the methods developed in this section to add or subtract as indicated. $$\frac{3}{4}+\frac{1}{8}+\frac{5}{6}$$

Step-by-Step Solution

Answer

The sum of the fractions is \(\frac{41}{24}\).

Step 1: Identify the Denominators

List the denominators of all the fractions: 4, 8, and 6.

Step 2: Find the Least Common Denominator (LCD)

Identify the least common denominator by finding the least common multiple of the denominators 4, 8, and 6. The LCM of 4, 8, and 6 is 24.

Step 3: Convert Fractions to Have the Same Denominator

Convert each fraction to an equivalent fraction with the LCD of 24. \[\frac{3}{4} = \frac{3 \times 6}{4 \times 6} = \frac{18}{24}, \frac{1}{8} = \frac{1 \times 3}{8 \times 3} = \frac{3}{24}, \frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24}.\]

Step 4: Add the Fractions

Since the fractions now have a common denominator, add their numerators while keeping the denominator the same. \[\frac{18}{24} + \frac{3}{24} + \frac{20}{24} = \frac{18 + 3 + 20}{24} = \frac{41}{24}.\]

Step 5: Simplify if Necessary

Check if the resulting fraction \(\frac{41}{24}\) can be simplified. Since 41 is a prime number and does not divide into 24, the fraction is already in its simplest form.

Key Concepts

Least Common Multiple

The Least Common Multiple (LCM) is a key concept in arithmetic, especially when dealing with fractions. It refers to the smallest positive integer that is divisible by each of the numbers you are considering. For instance, if you have a set of numbers like 4, 8, and 6, finding their LCM means finding the smallest number that all of them divide into without leaving a remainder.

To discover the LCM of two or more numbers, like the denominators in the fractions \(\frac{3}{4}\), \(\frac{1}{8}\), and \(\frac{5}{6}\), follow these steps:

• List the multiples of each number.
• Identify the smallest multiple common to each list.

In this exercise, the multiples are the count-by numbers for 4, 8, and 6. When calculated, the least amount that can be found in all lists is 24. Thus, the LCM of 4, 8, and 6 is 24, which also becomes the Least Common Denominator (LCD) needed for adding or subtracting fractions efficiently.

Fractions

Fractions represent parts of a whole and are composed of a numerator (the top part) and a denominator (the bottom part). Understanding fractions is crucial when you need to perform arithmetic operations like addition and subtraction, especially with different denominators.

When adding or subtracting fractions, a common denominator is required. This ensures each fraction piece is comparable to all others. In the given problem:

• The fraction \(\frac{3}{4}\) indicates 3 parts out of 4.
• The fraction \(\frac{1}{8}\) indicates 1 part out of 8.
• The fraction \(\frac{5}{6}\) indicates 5 parts out of 6.

By finding a common multiple of the denominators (4, 8, and 6) and converting each fraction to an equivalent with the same denominator, these fractions can be easily added by aligning them over a standardized bottom number, as shown through the LCM process.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same portion of a whole. They are essential in arithmetic operations where you must have a common base to compare or combine them.

To create equivalent fractions with a common denominator, multiply both the numerator and the denominator by the same factor so that the value of the fraction doesn't change.

For example, transforming \(\frac{3}{4}\) into an equivalent fraction with a denominator of 24 involves multiplying both parts by 6:

• Numerator: \(3 \times 6 = 18\)
• Denominator: \(4 \times 6 = 24\)
• Equivalent Fraction: \(\frac{18}{24}\)

Repeat this process for \(\frac{1}{8}\) by multiplying by 3, and \(\frac{5}{6}\) by multiplying by 4 to transform them to \(\frac{3}{24}\) and \(\frac{20}{24}\), respectively. This step ensures all fractions can be expressed over a unified denominator, facilitating their addition into a single fraction.