Question

Express the following fractions as ratios. Reduce to lowest terms. \(\frac{2}{10}\) ______

Short answer

1:5

Step-by-step solution

Write the Fraction as a Ratio

Fractions can be expressed as ratios using a colon. For the fraction \( \frac{2}{10} \), write it as \( 2:10 \). This is the direct expression of the given fraction in the form of a ratio.

Reduce the Ratio to Lowest Terms

To reduce the ratio \( 2:10 \) to its lowest terms, divide both numbers by their greatest common divisor (GCD). The GCD of 2 and 10 is 2.

Simplify the Ratio

Divide both terms of the ratio by the GCD. \( \frac{2}{2} = 1 \) and \( \frac{10}{2} = 5 \), simplifying the ratio to \( 1:5 \).

Key concepts

Reducing Fractions

Reducing fractions is an essential skill in mathematics that helps simplify expressions and make calculations easier. Reducing a fraction means simplifying it to its smallest possible equivalent by ensuring the numerator and denominator share no common factors other than 1.

For example, in the fraction \( \frac{2}{10} \), both the numerator (2) and the denominator (10) can be divided by the common factor 2. This process reduces it to \( \frac{1}{5} \).

• Identify common factors: Look for numbers that divide both the numerator and the denominator.
• Simplify: Divide both the numerator and denominator by their greatest common factor.

Reducing fractions helps in making complex math problems simpler and enhances the accuracy of calculations.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. Knowing how to find the GCD is crucial for reducing fractions effectively.

Let's take the numbers 2 and 10 from our example. The divisors of 2 are 1 and 2, while the divisors of 10 are 1, 2, 5, and 10. The common divisors are 1 and 2, with the largest being 2. Thus, the GCD of 2 and 10 is 2.

• List the divisors: Write down all divisors for each number.
• Identify common divisors: Determine which numbers are divisors of both the numerator and denominator.
• Select the largest: Choose the highest number from the common divisors list.

Understanding the GCD simplifies the process of reducing fractions and simplifying ratios.

Simplifying Ratios

Simplifying ratios is very much like reducing fractions. Ratios demonstrate the relationship between two quantities. They can be difficult to interpret when not simplified, but once simplified, they provide a clearer understanding of the comparison.

The fraction \( \frac{2}{10} \) is expressed as the ratio \( 2:10 \). Simplifying it involves dividing each part of the ratio by the GCD, which is 2 in this case, resulting in a simplified ratio of \( 1:5 \).

• Express as a ratio: Write the fraction using a colon, such as 2:10.
• Calculate the GCD: Find the greatest common divisor of the two numbers.
• Divide: Use the GCD to divide both parts of the ratio, obtaining a simpler form.

Simplifying ratios not only clarifies proportions but also helps in practical applications like scaling recipes, maps, or models.