Problem 1 - Prealgebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Write Ratio as a Fraction

The given ratio is 8 to 6. To express this ratio as a fraction, write it with the first term as the numerator and the second term as the denominator: \( \frac{8}{6} \).

Step 2: Determine the Greatest Common Divisor (GCD)

To reduce the fraction to its lowest terms, find the greatest common divisor of the numerator (8) and the denominator (6). The factors of 8 are 1, 2, 4, 8, and the factors of 6 are 1, 2, 3, 6. The greatest common divisor is 2.

Step 3: Simplify the Fraction

Divide both the numerator and the denominator by their greatest common divisor. So, \( \frac{8}{6} \) becomes \( \frac{8 \div 2}{6 \div 2} = \frac{4}{3} \).

Step 4: Final Check

Ensure that the fraction \( \frac{4}{3} \) is in its simplest form. Since 4 and 3 have no common factors other than 1, \( \frac{4}{3} \) is indeed the simplest form.

Key Concepts

Understanding Ratios

A ratio compares two quantities, showing how many times one value contains or is contained within the other. It is a way to represent relationships between numbers. Ratios can be written in three different ways: with a colon (8:6), with the word "to" (8 to 6), or as a fraction (\( \frac{8}{6} \)).

• The first number in the ratio is known as the antecedent.
• The second number is called the consequent.

Writing ratios as fractions lets us see the relationship clearly and allows easy manipulation through arithmetic operations. Be careful not to confuse ratios with fractions; they represent different concepts. Ratios require comparison, while fractions represent division.

Finding the Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is a crucial concept in mathematics. It is the largest positive integer that divides each of the given numbers without leaving a remainder. For example, in the ratio 8 to 6 expressed as a fraction \( \frac{8}{6} \), we find the GCD to simplify it.

• List out the factors of both numbers. For 8, the factors are 1, 2, 4, and 8. For 6, they are 1, 2, 3, and 6.
• The largest common factor is 2, so 2 is the GCD of 8 and 6.

The GCD is an essential tool for simplifying fractions, reducing them to their simplest form by dividing both the numerator and the denominator by the GCD.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator share no factors other than 1. This process makes it easier to understand and compare fractions. Using our fraction from the ratio \( \frac{8}{6} \):

• First, find the GCD of the numerator and denominator, which we know is 2.
• Divide both the numerator and the denominator by their GCD: \( \frac{8 \div 2}{6 \div 2} = \frac{4}{3} \).

This fraction, \( \frac{4}{3} \), is now simplified. It is essential to ensure that fractions are in their simplest form for accuracy and simplicity in calculations. When both the numerator and denominator cannot be further divided by any common factors, the fraction is in its lowest terms.