Question

Express the following fractions as ratios. Reduce to lowest terms. \(\frac{1}{9}\) ______

Short answer

The fraction \( \frac{1}{9} \) is expressed as the ratio 1:9, already in lowest terms.

Step-by-step solution

Understand the Fraction

A fraction represents a division of the numerator by the denominator. Here, the fraction is \( \frac{1}{9} \), which means 1 divided by 9.

Convert Fraction to Ratio

A fraction can be expressed as a ratio by setting the numerator as the first term and the denominator as the second term. For \( \frac{1}{9} \), the ratio is 1:9.

Simplify the Ratio

Check if the ratio can be simplified further by finding the greatest common divisor (GCD) of the two numbers. Here, both 1 and 9 have a GCD of 1, so the ratio is already in its lowest terms.

Key concepts

Understanding Ratios

Ratios are a way to compare two quantities by using division. For example, when you have a fraction like \( \frac{1}{9} \), it can also be expressed as a ratio, which would be written as 1:9. This tells us how many times one part relates to another. It’s a simple comparison.

• The numerator (top number) becomes the first term in a ratio.
• The denominator (bottom number) becomes the second term in a ratio.

This means that for the fraction \( \frac{1}{9} \), the relationship between the numerator and denominator is expressed directly in the ratio as 1:9.
It's important to remember that ratios and fractions both represent parts of a whole in mathematical discussions.

Numerator and Denominator in Fractions

The terms numerator and denominator play a crucial role in understanding fractions and ratios.

• **Numerator**: This is the top part of the fraction. It indicates how many parts you have.
• **Denominator**: This is the bottom part. It tells you the total number of equal parts into which the whole is divided.

For the fraction \( \frac{1}{9} \), *1* is the numerator, and *9* is the denominator. This essentially means one part out of nine equal parts. The position of these numbers is key in understanding how the fraction behaves in mathematical calculations. Their specific roles help you convert fractions to ratios correctly.

Simplifying to Lowest Terms

To simplify a ratio or a fraction to its lowest terms means to reduce both the numerator and the denominator to their smallest values while keeping the same ratio or fraction. This involves finding the greatest common divisor (GCD) of the numbers in question.

• **Find the GCD**: This is the largest number that can divide both the numerator and the denominator without leaving a remainder.
• **Divide Both Terms by the GCD**: Once found, both the numerator and denominator are divided by their GCD.

In our example of the ratio 1:9, the greatest common divisor is 1. Dividing both 1 and 9 by 1 doesn’t change the numbers, so the ratio remains 1:9. Hence, it is already in its lowest terms. Learning to simplify effectively ensures clarity and accuracy in mathematical results.