Question

Express the following ratios as fractions. Reduce to lowest terms. 8: 6 ______

Short answer

The simplified fraction is $\frac{4}{3}$.

Step-by-step solution

Identify the Ratio

The given ratio is 8:6, which we want to express as a fraction.

Write as a Fraction

Write the given ratio as a fraction: $\frac{8}{6}$.

Simplify the Fraction

To simplify $\frac{8}{6}$, find the greatest common divisor (GCD) of 8 and 6, which is 2.

Divide Numerator and Denominator by the GCD

Divide both the numerator and the denominator by their GCD (2): $\frac{8÷2}{6÷2}=\frac{4}{3}$.

Key concepts

Understanding Ratios

Ratios are a way to express the relationship between two numbers. They show how many times one number contains another. When you see a ratio like "8:6," you are looking at a comparison of two quantities. You can think of it as saying, "For every 8 of something, there are 6 of something else."

Ratios are very useful in everyday situations, such as cooking, where you might need to maintain the ratio of ingredients for a recipe. They are also common in mathematics, where ratios can show the proportion between different quantities.

To convert a ratio to a fraction, you simply write the first number as the numerator and the second number as the denominator. So, the ratio 8:6 becomes the fraction $\frac{8}{6}$. This step represents the first step in simplifying or working with ratios in fraction form.

Simplifying Fractions

Simplifying fractions is all about making the fraction as simple as possible. A simplified fraction is one where the numerator and the denominator have no common divisors other than 1. When we simplify, it helps us understand the fraction better and makes calculations easier.

To simplify $\frac{8}{6}$, we divide both the numerator (8) and the denominator (6) by their greatest common divisor (GCD), turning it into $\frac{4}{3}$. This fraction is simpler, yet still represents the same ratio as $\frac{8}{6}$.

Here's how you can simplify any fraction:

• Find the greatest common divisor of both the numerator and denominator.
• Divide both parts of the fraction by this number.
• The fraction you end up with is the simplest form.

Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is the largest number that divides two numbers without leaving a remainder. It is essential when simplifying fractions, as it helps reduce a fraction to its simplest form.

When you take the fraction $\frac{8}{6}$ and want to simplify it, you first need to find the GCD of 8 and 6. In this case, the GCD is 2, which is the largest number that can divide both 8 and 6 fully.

To find the GCD:

• List the divisors of each number.
• Identify the largest common number in both lists.

Once you have the GCD, you use it to divide both the numerator and the denominator of the fraction, simplifying it to $\frac{4}{3}$.

Understanding how to find and use the GCD is crucial for mastering the art of simplifying fractions, making mathematical problems easier to solve and understand.