Question

Express the following fractions as ratios. Reduce to lowest terms. \(\frac{2}{3}\) ______

Short answer

The ratio is 2:3.

Step-by-step solution

Understand What a Ratio Is

A ratio is a comparison between two quantities, showing the relative sizes of two or more values. Ratios can be written in various forms, such as ":", "to", or as a fraction.

Express the Fraction as a Ratio

Given the fraction \( \frac{2}{3} \), this can be expressed as a ratio using a colon. Thus, \( \frac{2}{3} \) is written as the ratio 2:3.

Reduce the Ratio to Lowest Terms

To reduce a ratio to its lowest terms, ensure that the two numbers in the ratio do not have any common factors other than 1. The numbers 2 and 3 have no common factors, so the ratio 2:3 is already in its lowest terms.

Key concepts

Understanding Fractions

Fractions are used to represent a part of a whole. They consist of two parts: a numerator and a denominator. The numerator is the top number which indicates how many parts we have, whereas the denominator, the bottom number, shows the total number of equal parts in a whole. For instance, in the fraction \( \frac{2}{3} \), 2 is the numerator and 3 is the denominator. This means we have 2 parts out of a total 3. Fractions can be an easy way to visualize division or even share parts of a set quantity. It's important to understand how fractions can be converted into other ways of representing quantities, such as percentages or ratios. Ratios are closely related, as they also compare amounts, though in a different form.

Simplifying to Lowest Terms

In mathematics, simplifying means expressing something in the simplest form. For fractions and ratios, that means reducing them to their lowest terms. To simplify a fraction, you need to find the greatest common factor (GCF) of the numerator and the denominator. The GCF is the largest number that divides both numbers without a remainder. Simplifying ensures that both numbers are as small as possible while retaining the original value of the ratio or fraction.

• Check if both numbers share a common factor.
• Divide both the numerator and the denominator by their GCF.
• If the GCF is 1, the fraction is already in its simplest form.

For the example \( \frac{2}{3} \), 2 and 3 are already simplified because they have no common factors other than 1. This means the fraction and ratio are already in their lowest terms, providing the easiest possible way to understand the relationship between the numbers.

Comparison of Quantities via Ratios

A ratio is a succinct way to compare two quantities. It tells us how much of one thing there is compared to another, providing insight into their relative sizes or amounts. This can be useful in various scenarios, such as mixing recipes or understanding scale in maps. When a fraction like \( \frac{2}{3} \) is expressed as a ratio, it is written using a colon, like 2:3. This means that for every 2 of the first quantity, there are 3 of the second, illustrating a direct comparison. Ratios provide flexibility since they can represent comparisons in multiple forms such as fractional representations or with words like "to". By using ratios, you can easily compare different quantities to see which is larger or smaller, map proportions, or identify equivalences. Understanding this concept can be a key tool in both practical applications and more theoretical math problems.