Problem 3 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$64 \quad to\quad 12$$

Step-by-Step Solution

Answer

The simplified ratio as a fraction is \( \frac{16}{3} \).

Step 1: Write the Ratio as a Fraction

The first step in solving this problem is to convert the given ratio into a fraction. A ratio of 64 to 12 can be written as a fraction \( \frac{64}{12} \).

Step 2: Find the Greatest Common Divisor (GCD)

To reduce the fraction to its lowest terms, we need to find the greatest common divisor of the numerator (64) and the denominator (12). Using the Euclidean algorithm, we find the GCD of 64 and 12. The GCD is 4.

Step 3: Simplify the Fraction Using the GCD

Divide both the numerator and the denominator by their greatest common divisor to simplify the fraction. Specifically, divide 64 and 12 by 4: \( \frac{64}{4} = 16 \) and \( \frac{12}{4} = 3 \).

Step 4: Write the Simplified Fraction

Now that we have simplified the ratio, we write it as the fraction \( \frac{16}{3} \). This fraction is in its lowest terms since the numerator and denominator have no common factors other than 1.

Key Concepts

Understanding Fractions

Fractions are a way of expressing numbers that are not whole. A fraction consists of two parts: the numerator, which is the top number, and the denominator, which is the bottom number. The numerator represents how many parts of a whole we are considering, while the denominator shows into how many equal parts the whole is divided.

For example, the fraction \( \frac{64}{12} \) means you have 64 parts out of 12 equal parts of something. In this case, the whole is divided into 12 parts, and you have 64 such parts.

• Numerator: The number above the line in a fraction indicates how many parts you are dealing with.
• Denominator: The number below the line tells into how many parts the whole is divided.

Fractions are a universal way to represent numbers, measurements, and even ratios.

Finding the Greatest Common Divisor (GCD)

One essential step in simplifying fractions is finding the Greatest Common Divisor (GCD). The GCD is the largest number that can divide both the numerator and the denominator without any remainder. Finding the GCD helps in reducing fractions to their simplest form.

You can find the GCD using various methods, one of which is the Euclidean Algorithm. This algorithm involves a series of division steps until you reach a remainder of zero. Let's see how it works for our example:

• Divide 64 by 12 to get a quotient and a remainder. In this case, the remainder is 4.
• Now, take 12 and divide it by 4, which gives a remainder of 0.

The last non-zero remainder, before reaching zero, is the GCD. So, the GCD of 64 and 12 is 4. By understanding how to find the GCD, you can confidently simplify fractions.

Simplifying Fractions

Simplifying fractions means reducing them to their smallest possible form, where the numerator and denominator have no common divisors other than 1. To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).

Consider the fraction \( \frac{64}{12} \). After finding the GCD, which is 4, you divide the numerator and the denominator by this number:

• Divide 64 by 4, resulting in 16.
• Divide 12 by 4, resulting in 3.

Now your simplified fraction is \( \frac{16}{3} \). Simplifying makes fractions easier to work with and understand. It's like cleaning up extra clutter to see the core value. Remember, once a fraction is simplified, it will be in its most reduced form without any common factors other than 1.