Question

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

Short answer

\(1:9\)

Step-by-step solution

Understand the Concept

A ratio compares two quantities by division. Expressing a fraction as a ratio involves writing it in the form "a:b" where the fraction is expressed as the ratio of the numerator to the denominator.

Write the Fraction as a Ratio

The fraction \(\frac{1}{9}\) can be expressed as the ratio \(1:9\), where 1 is the numerator and 9 is the denominator.

Reduce the Ratio to Lowest Terms

Since the greatest common divisor (GCD) of 1 and 9 is 1, the ratio \(1:9\) is already in its lowest terms. No further simplification is required.

Key concepts

Ratios

A ratio is a way to express the relationship between two numbers by showing how many times one value contains or is contained within the other. It is typically written using the colon symbol, like this: "a:b". This form highlights the comparison between two different quantities.
For example, when we have fraction \( \frac{1}{9} \), it can be reshaped into a ratio \(1:9\) to show the relationship directly.
Ratios are crucial in identifying proportions in various fields, such as recipes, map reading, and even business analytics. By converting fractions into ratios, we can easily interpret and communicate the size comparison between components. It's a practical application of fractions that provides a clearer visual understanding of their relative sizes.

Numerator and Denominator

In the world of fractions, the numerator and the denominator have specific roles. The numerator is the top part of the fraction and represents a number of parts being considered out of a whole.
The denominator, found on the bottom, tells us into how many equal parts the whole is divided.
For instance, in \( \frac{1}{9} \), the numerator is 1, indicating one count of something. The denominator is 9, suggesting the whole is divided into nine equal parts.
The relationship between the numerator and the denominator is fundamental since it shapes the value of the fraction. It is through this interaction we derive key concepts such as ratios, simplification, and comparison between fractions.

Greatest Common Divisor (GCD) Reduction

The greatest common divisor (GCD) is a critical tool for reducing fractions or ratios to their simplest forms. The GCD of two numbers is the largest integer that can exactly divide both numbers with no remainder.
For example, the fraction \( \frac{1}{9} \) as a ratio \(1:9\), has the numbers 1 and 9. The GCD is 1 because it is the highest number that divides them both entirely.
Reducing a ratio using the GCD involves dividing both parts of the ratio by this greatest common factor, which simplifies it to its most concise form.
Since \(1:9\) cannot be reduced further without altering its proportions, it remains in its simplest form. Understanding how to find and use the GCD ensures we efficiently simplify ratios and fractions, making calculations and comparisons more manageable.

Simplification of Fractions

Simplification is the process of reducing fractions to their simplest form while retaining their overall value. This is important for ease of use, especially in complex mathematical operations.
A fraction is simplified by finding the GCD of the numerator and the denominator and dividing both by this number.
Taking our original fraction \( \frac{1}{9} \) as an example, since the GCD is 1, it means there is no further simplification possible. The fraction is already in its simplest form, meaning the numerator and denominator share no other common divisors.
Simplifying fractions helps in better understanding, comparing, and performing arithmetic operations. It emphasizes why understanding core concepts like GCD and fraction composition is vital for mastering mathematical expressions.