Question

Write each ratio in simplest form. \(\frac{2}{10}\)

Short answer

The ratio \(\frac{2}{10}\) simplifies to \(\frac{1}{5}\).

Step-by-step solution

Identify the Greatest Common Factor (GCF)

To simplify the ratio \(\frac{2}{10}\), we begin by finding the greatest common factor of both the numerator (2) and the denominator (10). The factors of 2 are 1 and 2. The factors of 10 are 1, 2, 5, and 10. The greatest common factor is 2.

Divide by the GCF

Next, divide both the numerator and the denominator of \(\frac{2}{10}\) by the greatest common factor, which is 2. Dividing the numerator gives \(\frac{2}{2} = 1\) and dividing the denominator gives \(\frac{10}{2} = 5\).

Write the Simplified Ratio

After dividing both terms by their GCF, the simplified ratio is \(\frac{1}{5}\). This is the simplest form because 1 and 5 have no common factors other than 1.

Key concepts

Greatest Common Factor

Determining the greatest common factor (GCF) is an essential step in simplifying ratios. It involves finding the highest number that evenly divides both the numerator and the denominator in a fraction. For example, when simplifying the ratio \(\frac{2}{10}\), we first look at the factors for each number:

• The factors of 2 are 1 and 2.
• The factors of 10 are 1, 2, 5, and 10.

The largest factor that both numbers share is the greatest common factor, which in this case is 2. The GCF helps to determine how much you can "reduce" the numbers while keeping their relationship the same in simplified form.

Numerator and Denominator

In any fraction, the numerator is the number on top, and the denominator is the number on the bottom. They represent the two parts of a ratio. For our exercise \(\frac{2}{10}\), 2 is the numerator, and 10 is the denominator. Understanding these terms helps in knowing which numbers you're working with while simplifying a ratio.

When simplifying, you divide both the numerator and the denominator by their greatest common factor. This process shrinks the size of both numbers while maintaining the proportion between them. This is essential to maintain the same value of the ratio, but in a simpler way. By doing so, we transform \(\frac{2}{10}\) into \(\frac{1}{5}\) by dividing each by 2.

Simplest Form

The simplest form of a ratio or fraction is achieved when the numerator and the denominator have no common factors other than 1. This is also known as reducing a fraction. In the example of simplifying \(\frac{2}{10}\), we reached the simplest form when we found \(\frac{1}{5}\) because 1 and 5 do not share any common factors other than 1.

This process ensures that the ratio is presented in the smallest possible numbers, which can make it easier to understand and work with. Simplifying ratios helps highlight the core relationship between the numbers, free of any extraneous factors.