Problem 55 - Algebra 1: Concepts and Skills

Question

Write the percent as a fraction or as a mixed number in simplest form. (Skills Review p. 768 ) $$ 45 \% $$

Step-by-Step Solution

Answer

The fraction in its simplest form is \( \frac{9}{20} \).

Step 1 - Writing Percentage as Fraction

First, write down 45% as a fraction of 100. So, we get \( \frac{45}{100} \).

Step 2 - Simplifying the Fraction

To simplify this fraction, find the greatest common divisor (GCD) of 45 and 100, which is 5. Then divide both the numerator and the denominator by this common divisor. This gives us \( \frac{45 ÷ 5}{100 ÷ 5} = \frac{9}{20} \).

Key Concepts

Simplifying Fractions

Fractions are simplified to make them easier to understand and work with. Simplifying means reducing a fraction to its simplest form where the numerator and denominator have no common factors other than 1. To simplify a fraction, you identify the greatest common factor (GCF) of both the numerator and the denominator.
For example, let's take the fraction \( \frac{45}{100} \), which comes from converting a percentage such as 45% into a fraction. To simplify it, we would first need to know the biggest number that divides evenly into both 45 and 100.
Once we find this number, we divide both the numerator (45) and the denominator (100) by this number. In this situation, the fraction simplifies to \( \frac{9}{20} \), because the greatest common divisor we use is 5. When you finish, you should double-check that the new fraction (\( \frac{9}{20} \)) cannot be reduced further.

Greatest Common Divisor

The greatest common divisor (GCD) is a very useful tool when simplifying fractions. It is the largest number that can divide both the numerator and the denominator of a fraction without any remainder. Finding the GCD ensures that you are reducing the fraction to its lowest form.
To find the GCD of two numbers, you can list all the factors of each number and see which is the largest one they have in common. Alternatively, you can use the more methodical Euclidean algorithm, which involves a process of division and finding remainders until you reach a remainder of zero.
For the fraction \( \frac{45}{100} \), as we've seen, the GCD is 5 because both 45 and 100 share this factor, and it is the largest one they both share. Once the GCD is found, dividing both terms of the fraction by the GCD results in the simplified fraction \( \frac{9}{20} \). Identifying the GCD is a crucial step in ensuring our fractions are expressed in the simplest form possible.

Mixed Numbers

Mixed numbers are used when dealing with fractions that represent values greater than one whole unit. They express numbers as a combination of a whole number and a proper fraction. Although the context here is simply converting a percentage into a basic fraction form, understanding mixed numbers is important in more complex conversions and arithmetic operations.
For instance, if you were converting a percentage like 125%, you would have \( \frac{125}{100} \) which is equivalent to \( 1 \frac{1}{4} \). The whole number, 1, is how many times 100 completely fits into 125, and the fraction \( \frac{1}{4} \) is what remains.
Although 45% does not need to be expressed as a mixed number, comprehending this concept allows you to tackle similar problems in mathematics where numbers exceed whole values. Having different forms, like mixed numbers, broadens your ability to work effectively with fractions across different contexts.