Question

Express the following fractions as ratios. Reduce to lowest terms. \(\frac{5}{10}\) ______

Short answer

The fraction \( \frac{5}{10} \) expressed as a ratio is 1:2.

Step-by-step solution

Identify the Fraction

The given fraction is \( \frac{5}{10} \). To express it as a ratio, we will first simplify this fraction to its lowest terms.

Simplify the Fraction

To simplify \( \frac{5}{10} \), find the greatest common divisor (GCD) of 5 and 10, which is 5. Divide both the numerator and the denominator by 5. \[ \frac{5 \div 5}{10 \div 5} = \frac{1}{2} \] Now the fraction is simplified to \( \frac{1}{2} \).

Express as Ratio

A ratio compares two quantities and is often written using a colon. The fraction \( \frac{1}{2} \) can be expressed as the ratio 1:2. Thus, \( \frac{5}{10} \) simplifies to the ratio 1:2.

Key concepts

Understanding Ratios

Ratios are a way to compare two quantities by showing the relationship between them. They can be expressed in different forms, such as "3 to 4," "3:4," or as a fraction \( \frac{3}{4} \). Understanding how to express fractions as ratios allows you to quickly see how one value relates to another. For example, in a classroom with 8 boys and 12 girls, the ratio of boys to girls is 8:12. By reducing this ratio, we can express it in the simplest form, which would be 2:3 when you divide both numbers by 4 (the greatest common divisor of 8 and 12).

It's important to remember that the order of the terms in a ratio matters. In the above example, the ratios of boys to girls (8:12) and girls to boys (12:8) are different, highlighting that the first number corresponds to boys and the second to girls.

Simplifying Fractions

Simplifying fractions is crucial when working with ratios because it allows for clarity and simplicity in representation. A fraction is simplified by finding its simplest or lowest terms. This means reducing it to a point where the numerator and the denominator are as small as possible while still retaining the same value. For instance, the fraction \( \frac{5}{10} \) can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 5. Thus, \( \frac{5}{10} \) simplifies to \( \frac{1}{2} \).

• Find the greatest common divisor (GCD) that both numbers share.
• Divide both the numerator and the denominator by this number.
• The result is the fraction in its simplest form.

Simplifying fractions into their simplest form makes it easier to compare and interpret ratios in everyday scenarios or mathematical problems.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) of two numbers is the largest positive integer that divides each of the numbers without leaving a remainder. It is also known as the greatest common factor (GCF). The GCD is particularly useful when simplifying fractions or ratios, as it helps to reduce them to their simplest form. To find the GCD, you can use several methods, one of the easiest being the listing method.

Let's discuss how you can determine the GCD:

• List out all the factors of each number.
• Identify the largest factor that both numbers share.

For example, consider the numbers 5 and 10. The factors of 5 are {1, 5}, and the factors of 10 are {1, 2, 5, 10}. The largest common factor between them is 5, which is the GCD.

Understanding the GCD is beneficial, especially when simplifying fractions like \( \frac{5}{10} \) to \( \frac{1}{2} \), helping to express the equivalent ratio of 1:2.