Question

Combine and simplify. Don't use your calculator for these numerical problems. The practice you get working with common fractions will help you when doing algebraic fractions. $$\frac{3}{5}+\frac{2}{5}$$

Short answer

\(1\)

Step-by-step solution

Identify Common Denominator

The fractions share the same denominator, which is 5. This means they can be combined directly.

Add the Numerators

Since the denominators are the same, add the numerators: 3 + 2.

Combine the Fractions

After adding the numerators, place the result over the common denominator. The new fraction is \(\frac{3+2}{5}\).

Simplify the Fraction

Simplify the numerator by performing the addition: \(\frac{5}{5}\).

Reduce the Fraction if Necessary

The fraction \(\frac{5}{5}\) simplifies to 1, because any number divided by itself equals 1.

Key concepts

Common Denominators

Understanding common denominators is essential when working with fractions. When two or more fractions have the same denominator, they are called common denominators. This is crucial for fraction addition because it allows us to add fractions directly without making any adjustments to the fractions. It's like stacking items of the same size; you can easily see how many you have in total.

Imagine a pie divided into equal parts. If two people take slices from this pie, the slices will naturally have a common denominator since they come from the same pie. In mathematical terms, if our pieces come in fifths (like the pieces represented by \(\frac{3}{5}\) and \(\frac{2}{5}\) in our exercise), then we don't need to perform any conversions to add them. We can simply focus on the number of pieces each person has, represented by the numerators, and sum them up.

Remember, having common denominators in fractions is analogous to having a common ground or shared unit when combining quantities. Only when the denominators match can we directly add the numerators to find the sum.

Adding Numerators

Adding numerators is a straightforward process, but one that is at the heart of fraction addition when dealing with common denominators. You simply take the numbers at the top of the fractions (the numerators) and combine them. This process is akin to adding apples to apples; the units are the same, so they can be combined easily.

Let's examine the numerators from our example, 3 and 2. These represent parts of a whole that are already in the same size, so you can add them together without adjusting their size. If you have three slices of pizza and then receive two more, you'll end up with five slices. Mathematically, we show this as \(3 + 2 = 5\). The resulting number replaces the original numerators over the shared denominator to represent the new fraction, \(\frac{5}{5}\).

It's this simple action—adding one number to another—that allows us to combine fractions. The addition of numerators ultimately tells us how many parts of the whole we have when we merge two fractions with common denominators.

Fraction Addition

Adding fractions becomes less intimidating once we break down the process. Fraction addition involves combining fractions to form a new fraction representing the sum of the parts. When the denominators are the same, it's a quick operation; however, when they're different, finding a common denominator is our first step.

Our textbook example showcases one of the simplest forms of fraction addition: \(\frac{3}{5} + \frac{2}{5}\). Since both fractions are out of 5, there's no need to change them. We just add the numerators to find the sum, which is 5, and write it over the common denominator, resulting in \(\frac{5}{5}\). This demonstrates a whole, as every part of the pie is accounted for.

However, if fractions have different denominators, we'd need to find equivalent fractions that have a common denominator before we can add the numerators. This doesn't apply to our example, but it's an important step in more complex scenarios. Remember, the goal of fraction addition is to combine the parts into a whole using a shared denominator, which ultimately simplifies the fractions into a single, easily understandable number.