Question

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

Short answer

The ratio equivalent to 45% is 9:20.

Step-by-step solution

Convert Percentage to Fraction

To change a percentage to a ratio, we first convert the percentage to a fraction. A percentage is a portion out of 100, so we write 45% as the fraction \( \frac{45}{100} \).

Simplify the Fraction

Next, we reduce the fraction \( \frac{45}{100} \) to its lowest terms. We find the greatest common divisor (GCD) of 45 and 100, which is 5. We then divide both the numerator and the denominator by 5. \( \frac{45 \div 5}{100 \div 5} = \frac{9}{20} \).

Express as a Ratio

Finally, we express the simplified fraction \( \frac{9}{20} \) as a ratio, which is 9:20.

Key concepts

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This process makes fractions easier to work with and interpret. The concept of simplifying involves taking a fraction and dividing both the numerator (the top number) and the denominator (the bottom number) by the same number until no further simplification is possible.

To simplify a fraction:

• Find a number that divides both the numerator and the denominator exactly.
• Divide both the numerator and denominator by this number.
• Repeat the process until you cannot find any number except 1 that divides both parts of the fraction.

In the example of converting 45% to a fraction, we have \(\frac{45}{100}\). Simplifying it, we divide both by their greatest common divisor (GCD), which is 5, giving us the simplest form \(\frac{9}{20}\). By simplifying fractions, we can achieve their most reduced form, which is quick and easy to read.

Ratios

A ratio is a comparison between two quantities showing the relative size of one quantity to the other. Ratios are often used to compare values and can be written in several ways, like '9:20', '9 to 20', or as a fraction \(\frac{9}{20}\).

Converting a percentage to a ratio involves converting the percentage first into a fraction and then simplifying that fraction to its smallest terms. In our example, 45% becomes the ratio '9:20' after the process of simplification. Ratios are handy for representing proportions and comparisons in an intuitive way. This approach helps in easily understanding the size relationship between two numbers.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can exactly divide both the numerator and the denominator of a fraction. Finding the GCD is a crucial step in simplifying fractions since it helps determine the smallest terms of the fractions.

Here's how you can find the GCD:

• List the factors of both the numerator and the denominator.
• Identify the largest factor that appears in both lists.
• Divide both the numerator and the denominator by this number.

In the example given, the GCD of 45 and 100 is 5. We use this GCD to simplify the fraction \(\frac{45}{100}\) to \(\frac{9}{20}\). Understanding the GCD is fundamental when you wish to simplify fractions to their least terms efficiently.