Problem 35 - Beginning and Intermediate Algebra

Question

Add or subcract as indicared. Write the answer in lower terms. See Erample 5 $$ \frac{4}{5}+\frac{1}{5} $$

Step-by-Step Solution

Answer

The sum is 1.

Step 1: Identify the denominators

Both fractions have the same denominator, which is 5.

Step 2: Add the numerators

Add the numerators of the fractions since the denominators are the same. This means you add 4 and 1.

Step 3: Simplify the fraction (if necessary)

After adding the numerators, we get \( \frac{5}{5} \). Simplify this fraction, if possible.

Step 4: Simplify \( \frac{5}{5} \) to a whole number

\( \frac{5}{5} = 1 \). Since the numerator and the denominator are the same, they divide evenly to 1.

Key Concepts

Like Denominators

When adding or subtracting fractions, it's crucial to have like denominators. This means both fractions need to share the same bottom number. Having the same denominator allows you to easily add or subtract the numerators without any complicated adjustments. This works directly because

• the denominator indicates how many parts the whole is divided into,
• and sharing the same denominator means both fractions are split into equally sized parts.

In our exercise, both fractions, \( \frac{4}{5} \) and \( \frac{1}{5} \), already have like denominators of 5. This allows us to seamlessly add the numerators together, knowing all parts are of the same size. Finding and using like denominators simplifies the process of fraction addition or subtraction. In situations where fractions don't initially have like denominators, adjustments are needed to make them the same, but our example skips this due to its simplicity.

Simplification

Simplification is the process of reducing a fraction to its simplest form. This means making the numerator and denominator as small as possible while keeping the fraction equivalent to the original. Simplification is most helpful in making fractions easier to understand and work with.

• A simplified fraction allows for more straightforward comparison with other numbers and fractions.
• It reduces errors when performing further mathematical operations.

In our step-by-step solution, the addition of the numerators 4 and 1 gives us \( \frac{5}{5} \). This fraction can be simplified because both the numerator and the denominator are the same. Simplifying \( \frac{5}{5} \) means identifying that it equals a whole number. Simplification is always a good final step in fraction problems, ensuring answers are presented in the most understandable format.

Whole Number Conversion

Converting a fraction to a whole number can seem tricky at first, but it occurs when the numerator and the denominator are equal. This equality indicates the fraction is equivalent to 1 whole unit. In our example, after adding the fractions, we reached \( \frac{5}{5} \). Since 5 divided by 5 equals 1, this fraction converts to the whole number 1.Getting to this final form requires.

• Identifying when the numerator and denominator match,
• then confirming that dividing them yields a complete number with no remainder,
• which implies each part perfectly fills the whole once.

Many students find this conversion step satisfying as it provides a clear, concise answer to the problem. Remember, any time a fraction’s numerator matches its denominator, you can convert it to the whole number 1 for a neat and clean result.