Question

Write each percent as a fraction in lowest terms. $$12.4 \%$$

Short answer

Question: Convert 12.4% into a fraction in its lowest terms. Answer: \(\frac{31}{250}\)

Step-by-step solution

Convert the percentage into a fraction

To convert the percentage into a fraction, divide the percentage by 100: $$\frac{12.4}{100}$$

Multiply the fraction by 10 to remove the decimal point

To remove the decimal point from the numerator, multiply both the numerator and denominator by 10: $$\frac{12.4 \times 10}{100 \times 10} = \frac{124}{1000}$$

Simplify the fraction to lowest terms

Find the greatest common divisor (GCD) of 124 and 1000. The GCD of 124 and 1000 is 4. Now, divide both the numerator and denominator by the GCD to simplify the fraction: $$\frac{124 \div 4}{1000 \div 4} = \frac{31}{250}$$ So, \(12.4 \%\) expressed as a fraction in lowest terms is: $$\frac{31}{250}$$

Key concepts

Simplifying Fractions

When you convert a percentage to a fraction, the goal is often to express it in what's called "lowest terms." This means the fraction is simplified so the numerator (the top number) and the denominator (the bottom number) are as small as possible, yet still whole numbers.

To start simplifying a fraction, you need to ensure that both the numerator and denominator do not share a common factor apart from 1. This means both numbers should not be divisible by any of the same number other than 1, making them "simplified."

If a fraction is not in lowest terms, you can simplify it by dividing both the numerator and the denominator by their Greatest Common Divisor (GCD). By doing this, you ensure that no further simplification is possible. The process of simplifying fractions is important because it makes working with fractions easier and more intuitive in mathematical problems.

Greatest Common Divisor (GCD)

Finding the Greatest Common Divisor, often abbreviated as GCD, is a key step in simplifying fractions. The GCD of two numbers is the largest number that divides both numbers without leaving a remainder.

For example, when simplifying the fraction \(\frac{124}{1000}\), it is crucial to find the GCD. In this case, the GCD is 4. This means 4 is the largest number that divides both 124 and 1000 evenly.

Once you know the GCD, you can simplify the fraction by dividing both the numerator and the denominator by this number.

There are different methods to find the GCD, such as:

• Prime Factorization: Break down both numbers into products of prime factors, then multiply the lowest power of all common primes.
• Euclidean Algorithm: Use a series of division steps, reducing the size of the numbers, until you find the GCD.

Finding the GCD is an essential skill for many areas of math beyond simplifying fractions, and it can help in various applications like number theory or algebra.

Fractions in Lowest Terms

When a fraction is in its lowest terms, it means it’s fully simplified. This is the version of the fraction where the numerator and denominator have no common factor other than 1. It's the simplest form of the fraction that still represents the same value.

Why is it essential to express a fraction in lowest terms? There are a few reasons:

• Improved clarity: Simplified fractions are easier to understand and work with, especially in complex calculations.
• Consistency: Converting a variety of fractions to their lowest terms makes it easier to compare different fractions directly.
• Accuracy: Simplified fractions provide the most accurate representation of a value or measurement in terms of its simplest ratio.

In our example, starting with \(\frac{124}{1000}\) and simplifying it to \(\frac{31}{250}\) puts the fraction in its lowest terms. This means \(\frac{31}{250}\) is the most compact and clarified form of the original fraction without changing its value. Every time you convert a percentage to a fraction, acknowledging and striving for the lowest terms helps streamline mathematical communication and problem-solving.