Problem 43 - Prealgebra

Question

Find each sum or difference. Write in simplest form. $$ \frac{2}{9}+\frac{5}{9} $$

Step-by-Step Solution

Answer

The sum is \( \frac{7}{9} \) in simplest form.

Step 1: Check for a Common Denominator

The fractions \( \frac{2}{9} \) and \( \frac{5}{9} \) already have a common denominator, which is 9. This means we can directly proceed to adding the numerators.

Step 2: Add the Numerators

Since the denominators are the same, we add the numerators: \( 2 + 5 = 7 \).

Step 3: Write the Resulting Fraction

The result of the sum is \( \frac{7}{9} \).

Step 4: Simplify the Fraction, if Possible

Check if the fraction \( \frac{7}{9} \) can be simplified further. Since 7 is a prime number and does not divide 9, \( \frac{7}{9} \) is already in its simplest form.

Key Concepts

Common Denominator

When working with fractions, a common denominator is a shared number in the denominator position of two or more fractions. This is an essential part of addition and subtraction of fractions because it ensures uniformity. In our example, \( \frac{2}{9} + \frac{5}{9} \), both fractions already share the common denominator of 9.

Having the same denominator is crucial because it allows us to add the numerators directly. When fractions do not initially have a common denominator, you must manipulate them to create one. This often involves finding the least common multiple (LCM) of the denominators and converting each fraction.

• For simple calculations, if two fractions already have the same denominator, you can skip straight to adding their numerators. This makes the process much quicker.
• In more complex problems, you may need to convert fractions to have this shared baseline denominator — the "common ground" for accurate addition or subtraction.
• Always start by checking whether fractions are already aligned with a common denominator to ease the process.

Simplifying Fractions

A simplified fraction is one that is expressed in the simplest possible form. This means there are no common factors (other than one) between the numerator and the denominator, allowing the fraction to be reduced to the lowest terms.

In our example, \( \frac{7}{9} \), we check whether both numbers share any factor other than 1. The number 7 is a prime number, meaning it only has factors of 1 and itself. 9 cannot be divided evenly by 7, hence \( \frac{7}{9} \) is already in its simplest form.

• To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).
• For prime numerators, like in \( \frac{7}{9} \), simplification often leads directly to identifying the fraction as already simplified.
• Always double-check simplification possibilities to ensure you're working with the simplest form possible.

Numerators and Denominators

Understanding numerators and denominators is key to mastering fraction operations. A numerator is the top part of a fraction representing how many parts we consider. On the other hand, the denominator, the bottom part, indicates into how many equal parts the whole is divided.

In our example of \( \frac{2}{9} + \frac{5}{9} \), we have parts of 9 being added. The numerator 2 signifies two parts, while 5 signifies five parts, all from a whole made up of 9 parts. When the denominator is the same, we only focus on adding the numerators, treating the parts they represent as equivalent.

• The denominator sets the "pace" or scale of the fraction, making it crucial for precise arithmetic operations.
• Numerators communicate the quantity dealt with, so understanding their role in calculations helps ensure accuracy.
• The relationship between numerators and denominators must be carefully maintained for correct fraction operations.