Question

Write each percent as a fraction in lowest terms. $$30 \%$$

Short answer

Question: Convert 30% into a fraction in its lowest terms. Answer: $$\frac{3}{10}$$

Step-by-step solution

Convert the percent to a fraction

We are given a percent: $$30 \%$$. In order to convert it into a fraction, we just need to write this percent over 100. So, the fraction is: $$\frac{30}{100}$$

Simplify the fraction to its lowest terms

Now, we need to simplify the fraction to its lowest terms. To do this, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 30 and 100 is 10. Now, divide both the numerator and the denominator by the GCD: $$\frac{30}{100} = \frac{30\div10}{100\div10} = \frac{3}{10}$$ So the answer is: $$30 \% = \frac{3}{10}$$

Key concepts

Simplifying Fractions

When converting percentages to fractions, it is helpful to simplify the fraction to its most basic form. Simplifying, or reducing a fraction means to make the fraction as simple as possible.

Simplification of fractions involves dividing the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor. For instance, if you start with the fraction \(\frac{30}{100}\), this fraction can be simplified by recognizing that both 30 and 100 are divisible by 10. When you divide both by 10, you get the simplified fraction \(\frac{3}{10}\).

Simplifying not only makes the fraction easier to understand but also makes it easier when you need to perform further arithmetic operations, such as addition or multiplication of fractions. It can be particularly useful for visualizing fractions in a real-world context, making math more intuitive and relatable.

Greatest Common Divisor

The Greatest Common Divisor (GCD), also known as the greatest common factor, is the largest number that divides evenly into two or more numbers without leaving a remainder. It's a key concept in simplifying fractions.

To find the GCD of two numbers, you can list out the factors of each number and find the largest factor that appears in both lists. For instance, the factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30 and the factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, 100. The largest number that appears in both lists is 10, which makes it the GCD of 30 and 100.

Moreover, there are various algorithms such as the Euclidean algorithm that can be used for finding the GCD in a more systematic and efficient manner, especially with larger numbers.

Numerical Fractions

Numerical fractions represent a part of a whole number and are composed of a numerator and a denominator.

Most commonly, fractions are used to express numbers that are not whole, such as the result of dividing one integer by another. In the context of the problem \(30\%\) becoming \(\frac{30}{100}\), we're representing a part of 100, which can then be simplified. All percentages can be expressed as a fraction over 100 since the term 'percent' literally means 'per hundred'.

Fractions can appear in different forms – proper fractions, where the numerator is less than the denominator (\(\frac{3}{10}\)), improper fractions, where the numerator is greater than or equal to the denominator, and mixed numbers, which combine a whole number with a fraction. Understanding how to convert between these forms and operate with them is fundamental in mathematics.