Problem 14 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$\frac{5}{3} \text { to } \frac{1}{3}$$

Step-by-Step Solution

Answer

The ratio is 5 in lowest terms.

Step 1: Understand the Meaning of the Ratio

The ratio \( \frac{5}{3} \text{ to } \frac{1}{3} \) compares the fraction \( \frac{5}{3} \) to \( \frac{1}{3} \). To write this ratio as a fraction, express it as \( \frac{\frac{5}{3}}{\frac{1}{3}} \).

Step 2: Simplify the Complex Fraction

To simplify \( \frac{\frac{5}{3}}{\frac{1}{3}} \), recognize that dividing by a fraction is the same as multiplying by its reciprocal. Thus, \( \frac{\frac{5}{3}}{\frac{1}{3}} = \frac{5}{3} \times \frac{3}{1} \).

Step 3: Perform the Multiplication

Multiply \( \frac{5}{3} \) by \( \frac{3}{1} \): \[ \frac{5}{3} \times \frac{3}{1} = \frac{5 \times 3}{3 \times 1} = \frac{15}{3} \].

Step 4: Simplify the Resulting Fraction

Now, simplify \( \frac{15}{3} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 3. \[ \frac{15}{3} = 5 \].

Key Concepts

Understanding Fractions

Fractions are a way to represent a part of a whole. They consist of a numerator (top number) and a denominator (bottom number). In the fraction \( \frac{5}{3} \), 5 is the numerator, and 3 is the denominator. This means you have 5 parts of something that was divided into 3 equal parts.

In simple terms, fractions are about dividing things into equal sections and knowing how many parts you have. Understanding fractions is crucial as they are the foundation for many calculations related to dividing and ratios.

• Numerator: The number of parts you have.
• Denominator: The number of parts the whole is divided into.

Always remember, fractions like \( \frac{5}{3} \) are used to express ratios, proportions, and to perform arithmetic operations like addition, subtraction, multiplication, and division of fractions.

Simplifying Fractions

Simplifying fractions means finding an equivalent fraction in its simplest form, where the numerator and the denominator are as small as possible. This is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Consider the fraction \( \frac{15}{3} \). The greatest common divisor of 15 and 3 is 3. By dividing both the top and bottom numbers by 3, we transform \( \frac{15}{3} \) into its simplest form, which is 5.

The process of simplifying fractions is vital because it makes calculations easier and expressions cleaner. It is especially useful in solving mathematical problems like simplifying complex expressions or working with ratios.

• Locate GCD: Find the highest number that divides both numerator and denominator.
• Divide: Simplify by dividing numerator and denominator by GCD.

With practice, simplifying fractions will become second nature and will help you tackle more complex mathematical tasks.

Understanding Complex Fractions

Complex fractions are fractions where the numerator, the denominator, or both contain fractions themselves. They can seem intimidating at first, but with a step-by-step approach, they become manageable.

For instance, the complex fraction \( \frac{\frac{5}{3}}{\frac{1}{3}} \) can be simplified by recognizing that dividing by a fraction is the same as multiplying by its reciprocal. Thus, \( \frac{\frac{5}{3}}{\frac{1}{3}} \) becomes \( \frac{5}{3} \times \frac{3}{1} \).

Understanding how to simplify complex fractions is critical in algebra and higher-level math exercises where such fractions frequently appear. By mastering this concept, you improve your overall mathematical understanding and problem-solving skills.

• Recognize: Identify complex fractions in expressions.
• Reciprocal: Use the reciprocal method to simplify.
• Multiply: Transform the division of fractions to multiplication.

With practice, the simplification of complex fractions becomes a valuable tool in your math toolkit.