Problem 34 - Prealgebra

Question

Find the following sums. (Add.) \(6+5 \frac{3}{5}+8 \frac{2}{5}\)

Step-by-Step Solution

Answer

The sum is 20.

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. Thus, \(5 \frac{3}{5}\) becomes \(\frac{5 \times 5 + 3}{5} = \frac{28}{5}\), and \(8 \frac{2}{5}\) becomes \(\frac{8 \times 5 + 2}{5} = \frac{42}{5}\).

Step 2: Rewrite the Sum as Addition of Fractions

Rewrite the given sum by replacing the mixed numbers with the improper fractions found in Step 1. The expression becomes: \[6 + \frac{28}{5} + \frac{42}{5}.\]

Step 3: Convert Whole Number to Fraction with Common Denominator

Convert the whole number 6 to a fraction with the common denominator of 5. This gives us \(\frac{6 \times 5}{5} = \frac{30}{5}\).

Step 4: Add the Fractions

Now that all terms are fractions with the same denominator, add them together: \[\frac{30}{5} + \frac{28}{5} + \frac{42}{5} = \frac{30 + 28 + 42}{5} = \frac{100}{5}.\]

Step 5: Simplify the Resulting Fraction

Finally, simplify \(\frac{100}{5}\) to obtain the sum. \[\frac{100}{5} = 20.\]

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. For example, in the mixed number \(5 \frac{3}{5}\), the whole number is 5, and the fraction is \(\frac{3}{5}\).
These numbers can often be more intuitive in real-life scenarios, like measuring ingredients in a recipe or telling time, as they give an immediate sense of quantity.
However, for arithmetic operations such as addition or subtraction, it's usually simpler to work with improper fractions. Understanding how to convert between these forms is crucial.

• Multiply the whole number by the denominator of the fraction.
• Add the result to the numerator of the fraction.
• Place this sum over the original denominator to form an improper fraction.

By practicing with mixed numbers and improper fractions, you'll find it easier to tackle various math problems.

Improper Fractions

Improper fractions have a numerator larger than or equal to the denominator, like \(\frac{28}{5}\). They represent values greater than or equal to one whole unit.
While improper fractions might appear unusual at first, they are very handy for arithmetic calculations. This is especially true in addition, subtraction, and multiplication of fractions.

• Improper fractions are straightforward to use because you can perform operations directly with them.
• They eliminate confusion and make calculations consistent.

Converting from a mixed number to an improper fraction is a skill that enhances mathematical fluency. Once you become comfortable with these conversions, you'll handle fractions effortlessly.

Common Denominator

To add or subtract fractions, each must have the same denominator, known as a common denominator. This common base allows you to easily combine fractions, ensuring that the parts you are adding or subtracting are of the same size.
For example, in the exercise, converting the whole number 6 to a fraction \(\frac{30}{5}\) ensures all terms have a denominator of 5.

• When fractions share the same denominator, focus only on the numerators for addition or subtraction.
• Finding a common denominator simplifies the process and minimizes errors.

Remember that adopting a single denominator for all fractions before performing any calculations is a critical step in dealing with fractions.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This step helps you present the result more clearly.
In our exercise, after adding all terms, we got \(\frac{100}{5}\). By dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 5, we simplified it to 20.

• Simplification makes fractions easier to understand and compare.
• It's a final step in ensuring that your answer is accurate and clear.

Mastering simplification means you can always express fractions in their cleanest form, which is a vital skill in both math and real-world applications.