Problem 19 - Calculate With Confidence

Question

Change each of the following percentages to a ratio, and reduce to lowest terms. 45% __________

Step-by-Step Solution

Answer

45% as a ratio is 9:20.

Step 1: Convert Percentage to Fraction

To convert a percentage to a fraction, divide the percentage by 100. So, 45% becomes \( \frac{45}{100} \).

Step 2: Simplify the Fraction

Reduce the fraction \( \frac{45}{100} \) to its lowest terms by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 45 and 100 is 5.

Step 3: Divide by the GCD

Divide both the numerator and the denominator by the GCD. So, \( \frac{45}{100} \) simplifies to \( \frac{45 \div 5}{100 \div 5} = \frac{9}{20} \).

Step 4: Convert the Simplified Fraction to a Ratio

Express the simplified fraction as a ratio. \( \frac{9}{20} \) can be written as the ratio 9:20.

Key Concepts

Ratios

Ratios are a way to compare two quantities, showing how many times one value contains or is contained within the other. They give insight into the relative sizes of two numbers. For example, if you have a ratio of 9:20, it means for every 9 parts of one item, there are 20 parts of another.
Using ratios helps in solving problems involving proportions. It's important to remember that ratios can be simplified just like fractions to make them easier to work with. When simplifying a ratio, you divide both parts of the ratio by the greatest common divisor, just as you would with a fraction. This often gives a cleaner and more understandable form.

Fractions

Fractions emphasize the part of a whole concept. A fraction consists of two numbers, the numerator and the denominator, separated by a horizontal line. The numerator is the top number, indicating how many parts are being considered. The denominator is the bottom number, representing the total number of equal parts.
For instance, the fraction \( \frac{9}{20} \) tells us that we are dealing with 9 parts out of a total of 20 equal sections.
Understanding fractions is essential for converting percentages into a different form, doing division, and tackling part-to-whole relationships. Always be sure that you can simplify fractions when possible to ease computation and interpretation.

Simplifying Fractions

Simplifying fractions is an important step in working with ratios and percentages. A fraction is simplified when both the numerator and the denominator have no common factors other than 1. This is achieved by dividing both by their greatest common divisor (GCD).
**Steps to Simplify a Fraction:**

• Identify the greatest common divisor of the numerator and the denominator.
• Divide both the numerator and denominator by this number.
• The resulting fraction is your simplified fraction.

A simplified fraction is not only easier to understand but also makes further calculations more manageable. Continuing with our example, \( \frac{45}{100} \) simplifies to \( \frac{9}{20} \) when both parts are divided by 5.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is the largest number that divides two or more integers without leaving a remainder. Understanding the GCD is key in simplifying fractions because it helps find the common factor that allows reductions.
**Finding the GCD:**

• List all factors of each number.
• Identify the common factors from the lists.
• The highest number in this list is the GCD.

Using this method, the GCD of 45 and 100 is 5, allowing us to confirm the simplification of \( \frac{45}{100} \) to \( \frac{9}{20} \). Knowing how to determine the GCD is practical in reducing not only fractions, but also simplifying ratios to make complex problems easier to handle.