Problem 2 - Elementary Algebra

Question

Add or subtract as indicated, and express your answers in lowest terms. (Objective 1) $$\frac{3}{11}+\frac{5}{11}$$

Step-by-Step Solution

Answer

\( \frac{8}{11} \)

Step 1: Identify the operation

Look at the problem given which is \( \frac{3}{11} + \frac{5}{11} \). The operation indicated is addition.

Step 2: Confirm common denominators

Both fractions \( \frac{3}{11} \) and \( \frac{5}{11} \) have the same denominator of 11, so they are ready to be added directly.

Step 3: Add the numerators

Since the denominators are the same, add the numerators of the fractions together. \( 3 + 5 = 8 \). The sum is \( \frac{8}{11} \).

Step 4: Simplify the fraction

Check if the fraction \( \frac{8}{11} \) can be simplified by finding the greatest common divisor (GCD) of 8 and 11. Since 11 is a prime number and does not divide 8, \( \frac{8}{11} \) is already in its lowest terms.

Key Concepts

Common Denominators

In fraction addition or subtraction, having common denominators is like finding a common language. When the denominators are the same, you can easily perform the operation with the numerators. A denominator is the bottom part of a fraction and shows how many equal parts the whole is divided into.
If the denominators are different, you need to find a common denominator before adding or subtracting. This is usually done by finding the least common multiple (LCM) of the two denominators. But if the denominators are already the same, like in our exercise, you already have common denominators available. This makes it straightforward to add or subtract fractions without extra steps.

Lowest Terms

Expressing fractions in their lowest terms means simplifying them so they cannot be reduced any further. It's like tidying up to make everything neat and understandable.
To do this, divide both the numerator and the denominator by their greatest common divisor (GCD). When a fraction is already simplified, it is in its simplest form, making it easier to interpret and use in further calculations.

• For example, the fraction \(\frac{8}{11}\) from our problem is already in its lowest terms because 8 and 11 have no common divisors other than 1.

Keeping fractions in their lowest terms is crucial for clarity and consistency in math solutions.

Numerators

Numerators can be seen as the top half of a fraction, indicating how many parts of the whole are considered.
When adding fractions with common denominators, you focus on the numerators because the denominators, which describe the size of the parts, remain the same.
Let's take our example: \(\frac{3}{11} + \frac{5}{11}\).

• Add the numerators: \(3 + 5\).
• The result is \(8\), so the fraction becomes \(\frac{8}{11}\).

Just remember, only add or subtract the numerators directly when the fractions have the same denominator. This ensures that the fraction values truly represent the parts of the same whole.

Greatest Common Divisor

The greatest common divisor (GCD) is a handy tool in simplifying fractions. It’s the largest number that can completely divide both the numerator and the denominator without leaving a remainder.
To find the GCD, list out the factors of both numbers and choose the biggest one they share.

• In simple terms, if you're dealing with \(\frac{8}{11}\), you look at the factors.
• The number 8 has factors 1, 2, 4, and 8 while 11 only has 1 and 11.
• The largest common factor here is 1, which means the fraction is already in its lowest terms.

Using the GCD helps ensure that you have the simplest form of the fraction, ensuring clarity and ease in mathematical operations.