Problem 46 - Prealgebra

Question

The problems below review some basic concepts of addition of fractions and mixed numbers. Add each of the following and reduce all answers to lowest terms. $$\frac{1}{5}+\frac{3}{10}$$

Step-by-Step Solution

Answer

\( \frac{1}{5} + \frac{3}{10} = \frac{1}{2} \)

Step 1: Find a Common Denominator

To add the fractions \( \frac{1}{5} \) and \( \frac{3}{10} \), we need a common denominator. The denominators are 5 and 10. The least common multiple (LCM) of 5 and 10 is 10, so we will use 10 as the common denominator.

Step 2: Convert to Equivalent Fractions

We need to convert \( \frac{1}{5} \) so that it has the same denominator as \( \frac{3}{10} \). Multiply the numerator and denominator of \( \frac{1}{5} \) by 2 (since \( 5 \times 2 = 10 \)) to get: \( \frac{1}{5} = \frac{2}{10} \). Now both fractions are: \( \frac{2}{10} \) and \( \frac{3}{10} \).

Step 3: Add the Fractions

With a common denominator, we add the numerators: \( \frac{2}{10} + \frac{3}{10} = \frac{5}{10} \).

Step 4: Simplify the Fraction

The fraction \( \frac{5}{10} \) can be simplified by dividing the numerator and the denominator by 5 (the greatest common divisor): \( \frac{5}{10} = \frac{1}{2} \). This is the fraction in its simplest form.

Key Concepts

Common Denominator

When adding fractions, the first task is to find a common denominator. This involves identifying a common number that both denominators can divide evenly into. Understanding the concept of a common denominator is crucial because it enables us to combine fractions smoothly. Consider the fractions \( \frac{1}{5} \) and \( \frac{3}{10} \); here, 5 and 10 are the denominators. To add these fractions, they must share the same denominator. It's much like moving from different measurement units to a consistent one, ensuring that the fractions are alike.To find a common denominator, one effective strategy is to determine the Least Common Multiple (LCM). The LCM of two numbers is the smallest number that both can divide into without leaving a remainder. In this case, both 5 and 10 can divide into 10. Thus, 10 is the least common denominator, enabling us to proceed with fraction addition.

Least Common Multiple

The Least Common Multiple, or LCM, is an important concept in mathematics. It refers to the smallest multiple that two or more numbers share. Finding the LCM is particularly useful when working with fractions that have unlike denominators. For example, to add the fractions \( \frac{1}{5} \) and \( \frac{3}{10} \), we must first find their LCM.**How to Find the LCM:**

• List the multiples of each number. For 5: 5, 10, 15, and so on. For 10: 10, 20, 30, and so on.
• Identify the smallest multiple that appears in both lists. In this example, 10 is the first common multiple, making it the LCM.

With the LCM found, each fraction can be easily converted to have this common denominator. Here, \( \frac{1}{5} \) becomes \( \frac{2}{10} \), aligning perfectly with \( \frac{3}{10} \) for a straightforward addition.

Simplifying Fractions

Once fractions are added, simplifying them ensures that the result is as straightforward and tidy as possible. Simplifying involves reducing the fraction to its simplest form, where the numerator and denominator have no common divisors other than 1. In our example, adding \( \frac{2}{10} \) and \( \frac{3}{10} \) gives us \( \frac{5}{10} \).**Steps to Simplify:**

• Find the greatest common divisor (GCD) of the numerator and the denominator. For \( \frac{5}{10} \), the GCD is 5.
• Divide both the numerator and the denominator by the GCD. Thus, \( \frac{5}{10} \) becomes \( \frac{1}{2} \).

The fraction \( \frac{1}{2} \) cannot be reduced further, as there are no common factors left. Simplifying fractions makes them cleaner and easier to work with in future calculations.