Problem 6 - Algebra 1: Concepts and Skills

Question

Write the ratio in simplest form. $$\frac{12}{10}$$

Step-by-Step Solution

Answer

The simplest form of \( \frac{12}{10} \) is \( \frac{6}{5} \).

Step 1: Identify the fraction

The fraction provided is \( \frac{12}{10} \).

Step 2: Find the gcd of the numerator and the denominator

The greatest common divisor of 12 and 10 is 2.

Step 3: Simplify the fraction

To simplify the fraction, divide the numerator and the denominator by the gcd. Doing this, you get \( \frac{12 \div 2}{10 \div 2} = \frac{6}{5} \).

Key Concepts

Greatest Common Divisor

In mathematics, the greatest common divisor (GCD) helps us simplify fractions by providing the largest number that can evenly divide both the numerator and the denominator. Finding the GCD is essential in reducing fractions to their simplest form. Let's see how this works with the example \( \frac{12}{10} \).

First, list the divisors of each number:

• For 12: 1, 2, 3, 4, 6, 12
• For 10: 1, 2, 5, 10

By comparing these lists, the largest common divisor is 2. Thus, the GCD of 12 and 10 is 2.

Using the GCD, we can simplify the fraction by dividing both the numerator and the denominator by 2, as shown in the solution.

Ratio

A ratio is a relationship between two numbers showing how many times the first number contains the second. In other words, it compares different quantities. We often express ratios as fractions. In the example given, we have the fraction \( \frac{12}{10} \), which represents the ratio between 12 and 10.

When simplifying a ratio, finding the GCD allows us to express this relationship in the simplest way possible. This makes it easier to understand and use in practical applications. Simplifying \( \frac{12}{10} \) by dividing both parts by the GCD, 2, results in the simpler ratio \( \frac{6}{5} \).
This final ratio still communicates the same relationship but is in its simplest form.

Numerator and Denominator

The terms numerator and denominator are fundamental in understanding fractions and ratios. The numerator is the top number in a fraction showing how many parts we have, and the denominator is the bottom number indicating into how many parts the whole is divided.

For the fraction \( \frac{12}{10} \), 12 is the numerator, and 10 is the denominator. Simplifying involves dividing both numbers by the GCD. In this case, dividing 12 and 10 by 2 results in the new fraction \( \frac{6}{5} \), where 6 is the new numerator and 5 the new denominator.

Understanding the role of the numerator and denominator is key in working with fractions, ensuring accurate calculations and interpretations of ratios.