Question

Express the following fractions as ratios. Reduce to lowest terms. \(\frac{1}{5}\) ______

Short answer

The fraction \( \frac{1}{5} \) as a ratio is 1:5.

Step-by-step solution

Write the fraction as a ratio

The fraction \( \frac{1}{5} \) can be written as the ratio 1:5. This is because the fraction \( \frac{1}{5} \) means 1 part of 5 equal parts, which is directly expressed as the ratio 1:5.

Simplify the Ratio

The ratio 1:5 is already in its simplest form since both numbers 1 and 5 have no common factors other than 1.

Final Answer Verification

After expressing the fraction as a ratio and simplifying, the final answer is confirmed to be accurate. The ratio 1:5 is correct and cannot be simplified further.

Key concepts

Understanding Ratios

Ratios are a fundamental concept in mathematics, often used to compare quantities directly. When we talk about a ratio, we are essentially expressing how one quantity relates to another.
For instance, in a classroom with 2 girls and 3 boys, the ratio of girls to boys is 2:3. This simply means for every 2 girls, there are 3 boys. Ratios can be written in the form of 'a to b', 'a:b', or even as a fraction, such as \( \frac{a}{b} \).

• Ratios are a way to express proportions or comparisons.
• They help us understand relationships between different quantities.
• Writing a fraction as a ratio usually involves simple conversion, like writing \( \frac{1}{5} \) as 1:5.
• Both parts of a ratio must represent the same units or categories to be meaningful.

Understanding ratios will help in various real-life applications, from recipes in cooking to mixing chemicals in a lab, ensuring consistency and proportionality.

Mastering Simplification

Simplification is the process of making something easier to understand or deal with. In mathematics, it specifically refers to reducing fractions or ratios to their simplest form, where the numbers no longer have any common factors except for 1. This makes calculations easier and results clearer.
For example, consider the fraction \( \frac{4}{8} \). To simplify, divide both the numerator (4) and the denominator (8) by their greatest common factor, which is 4. The result is \( \frac{1}{2} \), a simplified form.

• Find the greatest common divisor (GCD) of both numbers in a ratio or fraction.
• Divide both terms of the ratio or fraction by this GCD.
• Ensure the simplified ratio or fraction can't be reduced further.

Simplification makes our results cleaner and easier to work with in subsequent operations. It's a key skill for simplifying equations and solving math problems swiftly.

Fractions and Ratios Conversion

Converting between fractions and ratios is an important mathematical skill, as it allows us to view numbers in different ways depending on the context. A fraction like \( \frac{1}{5} \) signifies one part of a whole divided into five parts. When converted to a ratio, it becomes 1:5, offering a slightly different but congruent perspective.
To convert a fraction to a ratio:

• Recognize that the numerator of the fraction corresponds to the first term of the ratio.
• The fraction's denominator becomes the second term of the ratio.
• Verify if the ratio can be simplified further, though sometimes, like in our example, it's already simplified.

This conversion provides flexibility in problem-solving, allowing one to choose the form that best suits the situation at hand. Understanding both forms and how to transition between them is crucial for comprehensively tackling math problems.