Problem 2 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$6 \quad to\quad 8$$

Step-by-Step Solution

Answer

The ratio 6 to 8 as a fraction in lowest terms is \( \frac{3}{4} \).

Step 1: Write the Ratio as a Fraction

The first step is to write the provided ratio as a fraction. The ratio given is 6 to 8, which we can write as a fraction like this: \( \frac{6}{8} \).

Step 2: Find the Greatest Common Divisor (GCD)

Now, we need to find the greatest common divisor (GCD) of the numbers involved in the fraction. The numbers are 6 and 8. The factors of 6 are 1, 2, 3, 6, and the factors of 8 are 1, 2, 4, 8. The largest common factor is 2.

Step 3: Simplify the Fraction

Divide both the numerator and the denominator of the fraction by the GCD. This means we'll divide both 6 and 8 by 2:\[ \frac{6 \div 2}{8 \div 2} = \frac{3}{4} \].

Step 4: Verify the Fraction is in Lowest Terms

Check that the fraction \( \frac{3}{4} \) is in its simplest form. The numbers 3 and 4 have no common divisors other than 1, confirming it is in lowest terms.

Key Concepts

Understanding Ratios to Fractions

A ratio is a way to show a relationship between two numbers. It tells us how much of one thing we have compared to another. For example, the ratio 6 to 8 can be represented as the fraction \( \frac{6}{8} \). This is because a ratio of 6 to 8 means that for every 6 parts of something, there are 8 parts of another thing.

Here's how you can convert a ratio to a fraction:

• Write the first number of the ratio (6) as the numerator (the top number) of the fraction.
• Write the second number of the ratio (8) as the denominator (the bottom number) of the fraction.

It's important to note that when converting ratios to fractions, it's similar to describing a part of a whole in mathematics. By understanding this conversion, you can simplify or adjust these expressions easily.

Finding the Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is crucial when you want to simplify a fraction. The GCD is the largest number that can divide both the numerator and the denominator without leaving a remainder. For the fraction \( \frac{6}{8} \), we need to find the GCD of 6 and 8.

Steps to find the GCD:

• List out the factors of each number.
• The factors of 6 are 1, 2, 3, and 6.
• The factors of 8 are 1, 2, 4, and 8.
• Identify the largest common factor, which is 2 in this case.

By finding the GCD, you can simplify the fraction to its simplest form, making it easier to work with and understand its value.

Simplifying Fractions to Lowest Terms

After finding the greatest common divisor, you can simplify the fraction to its lowest terms. This means there is no number, except 1, that can divide both the numerator and the denominator.

To simplify the fraction \( \frac{6}{8} \) using the GCD:

• Divide both the numerator (6) and the denominator (8) by the GCD (2): \( \frac{6 \div 2}{8 \div 2} = \frac{3}{4} \).
• Check that 3 and 4 have no common divisors other than 1, which confirms that \( \frac{3}{4} \) is in its lowest terms.

This process makes fractions much simpler to use in calculations and comparisons, as it reduces them to their most basic form.