Problem 76 - Beginning and Intermediate Algebra

Question

Perform the following operations. Write answers in lowest terms. $$ 4 \frac{3}{7}+\frac{31}{7} $$

Step-by-Step Solution

Answer

The answer is \(\frac{62}{7}\).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \( 4 \frac{3}{7} \) to an improper fraction. Multiply the whole number \(4\) by the denominator \(7\), which gives \(4 \times 7 = 28\). Then add the numerator \(3\), resulting in \(28 + 3 = 31\). Therefore, \(4 \frac{3}{7} = \frac{31}{7}\).

Step 2: Add Fractions with the Same Denominator

Add \(\frac{31}{7}\) and \(\frac{31}{7}\). Since they have the same denominator, simply add the numerators: \(31 + 31 = 62\). Thus, \(\frac{31}{7} + \frac{31}{7} = \frac{62}{7}\).

Step 3: Check for Lowest Terms

The fraction \(\frac{62}{7}\) is already in its simplest form because 62 and 7 have no common factors other than 1.

Key Concepts

Understanding Improper Fractions

An improper fraction occurs when the numerator (top number) is greater than or equal to the denominator (bottom number). This means the fraction represents a number greater than one. In the problem, 4 \( \frac{3}{7} \) needed to be transformed into an improper fraction before performing any operations.
To convert a mixed number, like 4 \( \frac{3}{7} \), into an improper fraction, follow these steps:

• Multiply the whole number part (4) by the denominator (7): \( 4 \times 7 = 28 \).
• Add the result to the numerator (3): \( 28 + 3 = 31 \).
• The improper fraction is \( \frac{31}{7} \).

This step is critical since improper fractions make it straightforward to perform addition, subtraction, and other operations on fractions with different or similar denominators.

Mixing with Mixed Numbers

Mixed numbers combine whole numbers and fractions. They offer a more intuitive understanding of quantities greater than one. However, for mathematical operations, converting them into improper fractions is beneficial.

When dealing with mixed numbers, it's crucial to remember:

• They are easier to understand at a glance, showing both the whole number and the fraction part.
• For operations like addition or subtraction, converting to improper fractions simplifies the process.
• Conversion is reversible; once calculations are complete, you can change improper fractions back to mixed numbers for a more intuitive result.

Let's consider our example, where 4 \( \frac{3}{7} \) was converted to \( \frac{31}{7} \) for ease of addition, illustrating the practicality of switching forms in calculations.

Simplifying to the Simplest Form

Simplifying a fraction to its simplest form means reducing it so that no smaller whole number except 1 divides both the numerator and the denominator. This is an essential final step to ensure answers are precise and easily interpretable.

For the problem, the solution was \( \frac{62}{7} \). To check:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• If the GCD is 1, the fraction is already simplified.

In our case, 62 and 7 have no common factors other than 1, confirming \( \frac{62}{7} \) is indeed in its simplest form. Simplifying fractions conveys the number clearly and accurately, avoiding unnecessary complexity in a mathematical expression.