Problem 11 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$\frac{7}{3} \text { to } \frac{6}{3}$$

Step-by-Step Solution

Answer

The simplified ratio is \( \frac{7}{6} \).

Step 1: Understand the Problem

We are given two fractions: \( \frac{7}{3} \) to \( \frac{6}{3} \), and we need to express the ratio of these two fractions as a fraction in its simplest form.

Step 2: Set Up the Ratio

To express the ratio \( \frac{7}{3} \) to \( \frac{6}{3} \) as a single fraction, we write it as a division of the two fractions: \( \frac{7}{3} \div \frac{6}{3} \).

Step 3: Simplify the Division

Dividing by a fraction is equivalent to multiplying by its reciprocal. This means we have: \( \frac{7}{3} \times \frac{3}{6} \).

Step 4: Perform the Multiplication

Multiply the numerators and the denominators: \( \frac{7 \times 3}{3 \times 6} = \frac{21}{18} \).

Step 5: Simplify the Fraction

To simplify \( \frac{21}{18} \), find the greatest common divisor (GCD), which is 3. Divide both the numerator and denominator by 3: \( \frac{21 \div 3}{18 \div 3} = \frac{7}{6} \).

Step 6: Final Step: Verify Simplification

Verify that \( \frac{7}{6} \) is in its simplest form. The GCD of 7 and 6 is 1, confirming that this fraction is already simplified.

Key Concepts

Simplifying Fractions

When we deal with fractions, simplifying means finding an equivalent fraction where the numerator and the denominator have no common factors other than 1. This process involves breaking down both the top and bottom numbers to their smallest possible form.

Simplifying can be achieved by identifying a common number that both parts can be divided by evenly. This common number is called the Greatest Common Divisor (GCD). Simplifying fractions not only makes calculations smoother but also helps in better understanding the comparison between two quantities.

• Check the common factors of the numerator and denominator.
• Divide both numbers by their GCD.
• Ensure the resulting fraction is in the simplest form, with no common factors left aside from 1.

It is important to remember that simplifying a fraction doesn’t change the value, it merely reduces it to its most concise form so we can make easy comparisons or perform other mathematical operations efficiently.

Greatest Common Divisor

The Greatest Common Divisor, or GCD, is a key concept when working with fractions. It is the largest number that divides two or more integers without leaving a remainder. When simplifying a fraction, knowing the GCD helps us determine by which number we can evenly reduce the numerator and the denominator.

For example, to simplify the fraction \( \frac{21}{18} \), we first list the factors of 21 and 18:

• Factors of 21: 1, 3, 7, 21
• Factors of 18: 1, 2, 3, 6, 9, 18

The largest common factor in both lists is 3, hence the GCD is 3.

Dividing both the numerator and the denominator by the GCD provides the simplest form of the fraction, \( \frac{21}{18} \) becomes \( \frac{7}{6} \). Using the GCD is an efficient method to ensure our fraction is expressed in its most digestible form, maintaining accuracy in every step of mathematical computations.

Multiplying Fractions

Multiplying fractions is straightforward and follows a simple rule. When given two fractions, multiplying them means multiplying their numerators to get the new numerator, and multiplying their denominators to get the new denominator.

In the context of simplifying ratios, you might encounter a scenario where you need to multiply a fraction by the reciprocal of another fraction, particularly when dividing them. Remember, dividing by a fraction is equivalent to multiplying by its reciprocal.

• Write the reciprocal: Switch the numerator and the denominator of the divisor fraction.
• Multiply the fractions: Multiply numerators across the fractions and denominators across the fractions.
• Simplify the resulting product if needed.

As an example, the division \( \frac{7}{3} \div \frac{6}{3} \) is equivalent to \( \frac{7}{3} \times \frac{3}{6} \). The result \( \frac{21}{18} \) is then simplified by identifying the GCD and dividing. Multiplying fractions, while simple, forms the foundation for numerous mathematical operations and problem-solving strategies, particularly in the realm of ratios.