Question

Write each percent as a fraction in lowest terms. $$20 \%$$

Short answer

Question: Convert 20% to a fraction in its lowest terms. Answer: $$ \frac{1}{5} $$

Step-by-step solution

Convert the percentage to a fraction

To convert 20% to a fraction, simply place the percentage over 100, like this: $$ \frac{20}{100} $$

Simplify the fraction

Now, we'll simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 20 and 100 is 20. Let's divide both numerator and denominator by the GCD to simplify the fraction: $$ \frac{20 \div 20}{100 \div 20} = \frac{1}{5} $$ So, the fraction in its lowest terms is: $$ 20 \% = \frac{1}{5} $$

Key concepts

Simplifying Fractions

Simplifying fractions is an essential part of working with fractions. It means reducing a fraction to its simplest form. This is when the numerator and denominator are as small as possible but still hold the same value.
For example, if you have a fraction like \( \frac{20}{100} \), you can simplify it to \( \frac{1}{5} \).
The simplified fraction represents the same value as the original, but in the neatest way possible.

• The purpose of simplifying is to make calculations easier.
• A simplified fraction is more understandable when comparing or computing with other numbers.

To simplify a fraction, you divide the top and bottom by their largest possible common divisor. Let's dive into finding this important number next.

Greatest Common Divisor

Finding the greatest common divisor (GCD), also known as the greatest common factor (GCF), is the key step in simplifying fractions. The GCD of two numbers is the largest number that divides both without leaving a remainder.
For example, to simplify \( \frac{20}{100} \), you need to find the GCD of 20 and 100, which is 20.
Once known, this number allows you to reduce the fraction properly. You'll divide both the numerator and the denominator by the GCD.

• To find the GCD, list out the factors of each number and identify the largest one they share.
• It's possible to use Euclid’s algorithm—or if the numbers are small, simply identify the common factors by inspection.

Getting the GCD right is crucial for ensuring the fraction is truly in its simplest form.

Fractions in Lowest Terms

"Fractions in lowest terms" means that the fraction cannot be reduced any further. Once you have used the greatest common divisor to simplify a fraction, you have found its lowest terms.
Using our earlier example, the fraction \( \frac{20}{100} \) in lowest terms is \( \frac{1}{5} \).
This form is often preferred because it's the most concise representation of the value.

• It helps a great deal with mental math and further calculations because smaller numbers are simpler to work with.
• Fractions in lowest terms are also easier to compare with other fractions.
• In real life, lowest terms are helpful when dividing up resources or interpreting data efficiently.

The fraction \( \frac{1}{5} \) is more manageable than \( \frac{20}{100} \) and conveys the same exact value while being clearer.