Problem 113 - Algebra 1: Concepts and Skills

Question

Subtract. Write the answer as a fraction in simplest form. (Skills Review p. 768) $$ 8 \%-\frac{1}{20} $$

Step-by-Step Solution

Answer

The result of the operation \(8 \% -\frac{1}{20}\) is \(\frac{3}{100}\).

Step 1: Convert Percentage to Fraction

First, convert 8% to a fraction. A percentage is a number out of 100, so 8% can be written as \(\frac{8}{100}\) or it can be further reduced to \(\frac{2}{25}\) by dividing both numerator and denominator by 4.

Step 2: Perform Subtraction

Now, subtract \(\frac{1}{20}\) from \(\frac{2}{25}\). They both need to have the same denominator in order to subtract. The least common multiple of 20 and 25 is 100. \(\frac{2}{25} = \frac{8}{100}\) and \(\frac{1}{20} = \frac{5}{100}\). Therefore the subtraction becomes \(\frac{8}{100} - \frac{5}{100}\) which equals \(\frac{3}{100}\).

Step 3: Simplify Fraction

The fraction \(\frac{3}{100}\) is already in its simplest form, because 3 and 100 do not share any factors aside from 1.

Key Concepts

Percentage to Fraction Conversion

Converting a percentage to a fraction is a valuable skill in mathematics. When you see a percentage, remember it is a number out of 100. For example, 8% means 8 out of 100. To convert this into a fraction, write it as \( \frac{8}{100} \).

• Start with the percentage number, like 8%.
• Place this number over 100, giving you \( \frac{8}{100} \).

Now, this fraction can often be simplified. Simplification involves dividing the numerator and the denominator by a common factor. Here, 4 is a factor of both 8 and 100, so divide both by 4 to get \( \frac{2}{25} \). Remember to always look for the greatest common factor to ensure your fraction is in the simplest form.

Fraction Subtraction

Subtracting fractions may seem tricky at first, but becoming comfortable with it is crucial.Firstly, ensure both fractions have the same denominator. For our example of fractions \( \frac{2}{25} \) and \( \frac{1}{20} \), the denominators 25 and 20 need a common denominator.

• Find the Least Common Multiple (LCM). The LCM of 25 and 20 is 100.
• Convert \( \frac{2}{25} \) to \( \frac{8}{100} \) and \( \frac{1}{20} \) to \( \frac{5}{100} \).

Now, with both fractions over the same denominator, subtract the numerators: \( 8 - 5 = 3 \). This gives the fraction \( \frac{3}{100} \).Remember, the key steps are finding the common denominator and then confident subtraction.

Simplest Form

Writing a fraction in its simplest form means ensuring the numerator and the denominator have no common divisors other than 1. This makes it more understandable and standardized.For \( \frac{3}{100} \), examine the numbers 3 and 100.

• The prime factors of 3 are just 3.
• The prime factors of 100 are 2 and 5.

As 3 and 100 share no common prime factors, \( \frac{3}{100} \) is already in its simplest form. To simplify other fractions, find the greatest common divisor and divide both the numerator and the denominator by this number.Practicing this method helps in ensuring fractions are as reduced as possible.