Problem 38 - Prealgebra

Question

Change each percent to a fraction in lowest terms. $$40 \%$$

Step-by-Step Solution

Answer

The fraction in lowest terms is \(\frac{2}{5}\).

Step 1: Convert Percent to Fraction

To convert a percentage into a fraction, first divide the percentage by 100. For 40% this means writing it as \(\frac{40}{100}\). This fraction represents the value of 40% as a fraction of 1.

Step 2: Simplify the Fraction

Now, we simplify the fraction \(\frac{40}{100}\) by finding the greatest common divisor (GCD) of 40 and 100. The GCD is 20, so we divide both numerator and denominator by 20. Thus, \(\frac{40}{100} = \frac{40 \div 20}{100 \div 20} = \frac{2}{5}\).

Key Concepts

Simplifying Fractions

Fractions can sometimes look complicated, but simplifying them can make things much clearer! Simplifying a fraction means making the numerator (top number) and the denominator (bottom number) as small as possible, while still keeping the same value. This process helps in making calculations easier and results more understandable.

• To simplify a fraction, you first need to check if the numerator and denominator have any common factors, that is, numbers that can exactly divide both without leaving any remainder.
• Once you find these common factors, you should divide both the numerator and the denominator by the greatest of these common factors.

For example, in the fraction \( \frac{40}{100} \), both 40 and 100 can be divided by 20, which is the largest number they both share as a factor.
Thus, \( \frac{40}{100} \) simplifies to \( \frac{2}{5} \). This fraction is now in its simplest form and still represents the same original value!

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial concept when simplifying fractions. The GCD of two numbers is the largest number that divides both of them perfectly.

• Finding the GCD helps in minimizing fractions to their simplest form.
• To determine the GCD, you can list all divisors of both the numerator and the denominator and identify the largest common one.

In our example, the numbers 40 and 100 both have divisors like 1, 2, 4, 5, 10, 20, but the greatest of them is 20.
So, 20 is the GCD of 40 and 100, and we use it to simplify the fraction \( \frac{40}{100} \) to \( \frac{2}{5} \). The use of the GCD ensures that you reach the most reduced version of the fraction without losing any value of the expression.

Fraction Basics

Understanding fractions starts with knowing that they are a way to represent parts of a whole. A fraction has two parts:

• Numerator: This is the top number, indicating how many parts we have.
• Denominator: This is the bottom number, showing how many equal parts make up a whole.

When converting a percentage to a fraction, you're essentially expressing it as a fraction of 100, since a percent is a part out of a hundred. For instance, 40% is equivalent to \( \frac{40}{100} \).
This basic form can then be simplified by reducing it through a process that uses the greatest common divisor, as we previously discussed. Intermediate steps like converting percents to fractions allow us to apply mathematical processes more flexibly, aiding in clearer calculations and comparisons.