Problem 10 - Prealgebra

Question

Find the following sums and differences, and reduce to lowest terms. (Add or subtract as indicated.) $$\frac{2}{5}+\frac{3}{5}+\frac{4}{5}$$

Step-by-Step Solution

Answer

The sum is \( \frac{9}{5} \), which is already in lowest terms.

Step 1: Identify the Fractions

The given expression is composed of three fractions: \( \frac{2}{5}, \frac{3}{5}, \frac{4}{5} \). These fractions have the same denominator.

Step 2: Add the Numerators

Since all fractions have the same denominator, we add their numerators directly: \( 2 + 3 + 4 = 9 \).

Step 3: Keep the Common Denominator

The denominator remains the same when fractions are added with a common denominator, so the denominator is \( 5 \).

Step 4: Write the Resulting Fraction

Combine the sum of the numerators and the common denominator to write the result as a fraction: \( \frac{9}{5} \).

Step 5: Check for Simplification

Check if \( \frac{9}{5} \) can be simplified. As 9 and 5 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Common Denominator

When you're working with fractions, finding a common denominator is a crucial step for adding or subtracting them. Thankfully, it can be a lot easier than it sounds! A common denominator is essentially a shared denominator that allows two or more fractions to be compared or combined. In the given exercise, since all fractions, \( \frac{2}{5}, \frac{3}{5}, \frac{4}{5} \), already have the same denominator of 5, this process is simplified.

To handle fractions with different denominators:

• Find the least common multiple (LCM) of the denominators.
• Convert each fraction to an equivalent fraction with the LCM as the new denominator.

But in cases like this exercise, where the denominators are already identical, you can skip these steps and focus directly on the numerators!

Simplifying Fractions

Simplifying fractions is all about making them as neat as possible by reducing them to their simplest form. This happens when the numerator and the denominator have no common factors other than 1.

Let's take another look at the fraction from the exercise, \( \frac{9}{5} \). The numbers 9 and 5 don’t share any common factors except for the number 1. That means this fraction is already as simple as it gets.

If you're asked to simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both by the GCD to find the simplest form.

The ability to simplify fractions is crucial as it improves readability and is generally more aesthetically pleasing.

Numerators and Denominators

Understanding numerators and denominators is key when working with fractions. In a fraction like \( \frac{a}{b} \), \( a \) is the numerator and \( b \) is the denominator.

• The numerator tells you how many parts you have.
• The denominator tells you how many parts make up a whole.

In the exercise, the fractions \( \frac{2}{5}, \frac{3}{5}, \frac{4}{5} \) have 2, 3, and 4 as numerators, respectively, and 5 as a common denominator.

When adding the fractions, you only sum the numerators while the denominator remains unchanged since it represents the out of total parts. So, the combined numerator, 9 from \( 2+3+4 \), is placed over the common denominator to form the new fraction, \( \frac{9}{5} \). Knowing the difference between these two components helps in maneuvering through operations with fractions efficiently.