Problem 37 - Prealgebra

Question

Change each percent to a fraction in lowest terms. $$60 \%$$

Step-by-Step Solution

Answer

60% as a fraction in lowest terms is \( \frac{3}{5} \).

Step 1: Understand what percent means

A percent is a part per hundred. So, when we have 60%, it means 60 per 100, or \( \frac{60}{100} \).

Step 2: Write the percent as a fraction

Convert 60% to a fraction by writing it as \( \frac{60}{100} \).

Step 3: Simplify the fraction

To simplify \( \frac{60}{100} \), find the greatest common divisor (GCD) of 60 and 100. The GCD is 20. Divide both the numerator and the denominator by 20: \( \frac{60 \div 20}{100 \div 20} = \frac{3}{5} \).

Step 4: Verify the simplification

Check that \( \frac{3}{5} \) is in its simplest form by ensuring that 3 and 5 have no common factors other than 1. This confirms \( \frac{3}{5} \) is the fraction in lowest terms.

Key Concepts

Simplifying Fractions

Simplifying fractions means making a fraction as simple as possible. We often refer to this as putting a fraction in its 'lowest terms.' The key here is to reduce the numbers in the fraction while keeping the same overall value. To do this, we find a number that both the top number (numerator) and bottom number (denominator) of the fraction can be divided by. This number helps us "shrink" both parts of the fraction by the same amount.
Many people think simplifying is hard, but it's simple when you practice. Look at the example we have: the fraction \( \frac{60}{100} \). To simplify, we must divide the top and bottom by their greatest common factor. This makes calculating more straightforward and turning any percent into a simple, easy-to-understand fraction.

Greatest Common Divisor

The greatest common divisor, or GCD, of two numbers is the highest number that evenly divides both numbers. Finding the GCD is essential for simplifying fractions. The method usually involves listing out factors or using Euclid's algorithm, which is faster.
In the context of our example, we want to simplify \( \frac{60}{100} \). We find that 20 is the GCD of 60 and 100, because:

• 20 is a factor of 60 (\(60 \div 20 = 3\)).
• 20 is also a factor of 100 (\(100 \div 20 = 5\)).

Thus, using 20 to simplify the fraction ensures we are reducing both parts by the largest possible number. Finding the GCD is like finding a helpful "key" that unlocks the simplest version of the fraction.

Fraction in Lowest Terms

A fraction is in its lowest terms when it cannot be simplified any further. This means there are no numbers other than 1 that can divide both its numerator and denominator. Simplifying fractions to their lowest terms makes them easier to work with. It's often crucial in solving mathematical problems efficiently.
By dividing both the numerator and denominator of \( \frac{60}{100} \) by the GCD, which is 20, we get \( \frac{3}{5} \). This fraction, \( \frac{3}{5} \), has no numbers other than 1 that can divide both 3 and 5 evenly:

• The only factors of 3 are 1 and 3.
• The only factors of 5 are 1 and 5.

Therefore, \( \frac{3}{5} \) is truly in its simplest form. Working with fractions in their lowest terms not only simplifies calculations but also enhances understanding of mathematical concepts.