Problem 108 - Algebra 1: Concepts and Skills

Question

Subtract. Write the answer as a fraction in simplest form. (Skills Review p. 768) $$ \frac{7}{8}-80 \% $$

Step-by-Step Solution

Answer

The result of subtracting 80% from 7/8 in simplest form is \( \frac{3}{40} \).

Step 1: Convert Percentage to Fraction

First, convert 80% into a fraction. The conversion can be done by dividing the percentage by 100. So, 80% becomes 80/100 or \( \frac{4}{5} \).

Step 2: Subtraction of Fractions

The next step is subtracting \( \frac{4}{5} \) from \( \frac{7}{8} \). Since the denominators are different for the fractions involved, it's necessary to find the least common denominator (LCD). The LCD of 5 and 8 is 40. The fractions then become \( \frac{35}{40} \) and \( \frac{32}{40} \). Subtracting these gives \( \frac{3}{40} \).

Step 3: Simplification

Finally, simplify the fraction to its lowest terms. Since the numerator and the denominator don't have any common factors (other than 1), \( \frac{3}{40} \) is already in its simplest form.

Key Concepts

Convert Percentage to Fraction

When working with percentages and fractions together, it's crucial to convert percentages to fractions to perform operations like addition and subtraction. This process involves understanding that a percentage represents a part of a whole, namely per one hundred. For instance, 80% means 80 out of every 100, or more simply, 80 divided by 100. To convert a percentage to a fraction, we divide the percentage value by 100 and reduce the resulting fraction if possible.

Least Common Denominator (LCD)

When it comes to operations with fractions, having a common denominator is essential for addition or subtraction. The least common denominator (LCD) is the smallest number that is a common multiple of the denominators of two or more fractions. To find the LCD, you can list the multiples of each denominator and find the smallest multiple that appears in all lists, or you can use the prime factorization method to find the LCD. In our case, the multiples of 8 are 8, 16, 24, 32, 40, ..., and the multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, .... As we see, 40 is the first number that appears in both lists, making it the LCD.

• Find multiples of each denominator.
• Identify the smallest common multiple.
• Convert fractions to equivalent fractions with the LCD as the new denominator for easy subtraction.

Simplify Fractions

Simplifying fractions is an essential step to present your answer in its most basic form. The primary goal is finding the greatest common factor (GCF) of the numerator and the denominator, then dividing both by the GCF. If the numerator and the denominator have no common factors (other than 1), the fraction is already in its simplest form. Simplifying a fraction not only makes it easier to understand but also helps in comparing fractions. For the subtraction problem given, the resulting fraction \( \frac{3}{40} \) is already simplified, as 3 and 40 do not share any common factors.

The steps to simplify a fraction include:

• Find the GCF of the numerator and denominator.
• Divide both numerator and denominator by the GCF.
• If there is no GCF apart from 1, the fraction is already in its simplest form.