Problem 44 - Prealgebra

Question

Simplify each of the following complex fractions. [Examples 5–7] $$\frac{5 \frac{3}{8}+\frac{5}{8}}{4 \frac{1}{4}+1 \frac{3}{4}}$$

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Convert Mixed Numbers to Improper Fractions

To simplify the expression, first convert all mixed numbers into improper fractions. For \(5\frac{3}{8}\), multiply 5 by 8 and add 3, giving \(\frac{43}{8}\). For \(4\frac{1}{4}\), multiply 4 by 4 and add 1, resulting in \(\frac{17}{4}\). For \(1\frac{3}{4}\), multiply 1 by 4 and add 3, giving \(\frac{7}{4}\). So the expression becomes \(\frac{\frac{43}{8} + \frac{5}{8}}{\frac{17}{4} + \frac{7}{4}}\).

Step 2: Simplify Numerator and Denominator

Now, simplify both the numerator and the denominator separately by adding the fractions. In the numerator, \(\frac{43}{8} + \frac{5}{8} = \frac{48}{8} = 6\). In the denominator, \(\frac{17}{4} + \frac{7}{4} = \frac{24}{4} = 6\). So the expression now is \(\frac{6}{6}\).

Step 3: Simplify the Complex Fraction

Divide the simplified numerator by the simplified denominator: \(\frac{6}{6} = 1\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of whole numbers and fractions, for example, 2\(\frac{1}{3}\). They provide a more intuitive way to express quantities greater than one but less than two numbers. In certain math problems, like the one we have, it’s essential to convert mixed numbers into improper fractions.

• To convert a mixed number into an improper fraction, multiply the whole number by the denominator of the fractional part.
• Then, add the numerator to the result.
• This total becomes the numerator of the improper fraction, while the denominator remains unchanged.

For example, to convert 5\(\frac{3}{8}\), multiply 5 by 8 (which is 40), and add 3, resulting in 43. Thus, 5\(\frac{3}{8}\) becomes \(\frac{43}{8}\).
Understanding how to convert mixed numbers is crucial for simplifying complex fractions and other calculations.

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator, such as \(\frac{9}{4}\) or \(\frac{5}{5}\). These are essential when converting mixed numbers, allowing us to handle arithmetic operations more easily.

• They can readily express numbers greater than one using a single fraction.
• Improper fractions often arise during division and multiplication operations.
• Having a grasp on converting and simplifying improper fractions is vital for many algebraic expressions.

An improper fraction makes it straightforward to perform addition, subtraction, multiplication, and division without rearranging whole and fractional parts separately. In our exercise, converting mixed numbers to improper fractions was a prerequisite to simplify the numerator and denominator before reaching the conclusion.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. It's a crucial step in problem-solving because it often reveals a simpler path to the solution.

• When simplifying fractions, check if the numerator and denominator share any factors.
• To simplify, divide both by their greatest common divisor (GCD), resulting in the simplified form.
• Simplification is the core of properly understanding the relationships within a complex expression.

In the given problem, we simplified by first converting all terms to improper fractions. Then, we performed addition operations within the numerator and denominator. Eventually, the expression was reduced to \(\frac{6}{6}\), which simplifies to 1. Knowing how to efficiently simplify is not just about getting the right answer but also grasping the underlying simplicity of complex mathematical relationships.