Problem 3 - Prealgebra

Question

Write each percent as a fraction with denominator \(100 .\) $$60 \%$$

Step-by-Step Solution

Answer

60% as a fraction with a denominator of 100 is \( \frac{60}{100} \), simplified to \( \frac{3}{5} \).

Step 1: Understand the Percent

The problem asks us to express the given percent, which is 60%, as a fraction with a denominator of 100. A percent is a way to express a number as a part of 100.

Step 2: Write the Percent as a Fraction

To convert the percent to a fraction, we use the definition of percent: 60% means 60 per 100. Thus, we can write the percent as the fraction \( \frac{60}{100} \).

Step 3: Simplify the Fraction

The fraction \( \frac{60}{100} \) can be simplified by finding the greatest common divisor (GCD) of 60 and 100, which is 20. Divide the numerator and the denominator by 20. \( \frac{60 \div 20}{100 \div 20} = \frac{3}{5} \).

Key Concepts

Understanding Percentages

Percentages are a mathematical way of expressing a number as a part of 100. This term is derived from Latin where 'per centum' means 'by the hundred'.
For example, if you have 60% of a value, it essentially means you have 60 out of a possible 100 parts of that whole value.
It's a useful method for comparison, allowing various values to be expressed on a common scale.

• Think of percentages as slices of a pie, where the pie is always divided into 100 equal pieces.
• If you have 25% of something, it means you have 25 slices of a 100-slice pie.

To solve problems involving percentages, we often convert them to fractions or decimals. This allows for easier calculations, such as adding, subtracting or multiplying quantities.

What Are Fractions?

Fractions represent a numerical value that is not whole but rather a part of a whole. They are composed of a numerator and a denominator. The numerator is the top number, indicating how many parts are being considered, while the denominator is the bottom number, showing into how many equal parts the whole is divided.
Fraction's core function is to express divisions that don't result in whole numbers. For instance, \(\frac{1}{2}\) denotes half of a whole item.
Here are quick pointers about fractions:

• Numerator: the part of the fraction that tells us how many parts we have.
• Denominator: shows into how many parts something is divided.
• Fractions simplify calculations, especially in measurement and algebra.

Understanding fractions is key to converting percentages and performing many mathematical operations seamlessly.

Simplifying Fractions

Simplifying fractions involves reducing a fraction to its smallest form, where the numerator and denominator have no common divisors, except for 1. This process generally makes calculations easier and results more understandable.
Here's how you can simplify fractions step-by-step:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• The resulting fraction is the simplified version.

For example, to simplify \(\frac{60}{100}\), find the GCD of 60 and 100, which is 20.
By dividing both 60 and 100 by 20, the fraction reduces to \(\frac{3}{5}\). This reduced form is easier to understand and use.
Simplifying fractions is particularly useful in making data and mathematical calculations more digestible.