Problem 224 - Intermediate Algebra

Question

In the following exercises, simplify. $$ 8\left(\frac{15}{16}-\frac{5}{6}\right) $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{5}{6} \).

Step 1 - Identify the fractions inside the parentheses

The given expression is: \[ 8\left(\frac{15}{16}-\frac{5}{6}\right) \] First, identify the fractions \( \frac{15}{16} \) and \( \frac{5}{6} \).

Step 2 - Find a common denominator

To subtract the fractions, find a common denominator for \( 16 \) and \( 6 \). The least common multiple (LCM) of \( 16 \) and \( 6 \) is \( 48 \).

Step 3 - Convert fractions to have the common denominator

Convert \( \frac{15}{16} \) and \( \frac{5}{6} \) to fractions with a denominator of \( 48 \). \[ \frac{15}{16} = \frac{15 \times 3}{16 \times 3} = \frac{45}{48} \] \[ \frac{5}{6} = \frac{5 \times 8}{6 \times 8} = \frac{40}{48} \]

Step 4 - Subtract the fractions

Now that both fractions have the same denominator, subtract them: \[ \frac{45}{48} - \frac{40}{48} = \frac{5}{48} \]

Step 5 - Multiply by the number outside the parentheses

Multiply the result by \( 8 \): \[ 8 \left( \frac{5}{48} \right) = \frac{8 \times 5}{48} = \frac{40}{48} \]

Step 6 - Simplify the resulting fraction

Simplify \( \frac{40}{48} \) by dividing the numerator and the denominator by their greatest common divisor (GCD), which is \( 8 \): \[ \frac{40 \div 8}{48 \div 8} = \frac{5}{6} \]

Key Concepts

fractions

Fractions represent a part of a whole. They consist of a numerator, the top number, and a denominator, the bottom number. When working with fractions, you often need to perform operations like addition, subtraction, multiplication, or division.

For example, in the fraction \( \frac{15}{16} \), 15 is the numerator and 16 is the denominator. This means you have 15 parts out of a total of 16 parts. Understanding how numerators and denominators interact is crucial for simplifying and performing operations with fractions.

common denominator

When adding or subtracting fractions, the denominators must be the same. This is called finding a common denominator.

To find a common denominator, look for the least common multiple (LCM) of the denominators. In our exercise, the denominators were 16 and 6. The LCM of these numbers is 48.

Here's how to find the common denominator for \( \frac{15}{16} \) and \( \frac{5}{6} \):

• Multiply \( \frac{15}{16} \) by \( \frac{3}{3} \) to get \( \frac{45}{48} \).
• Multiply \( \frac{5}{6} \) by \( \frac{8}{8} \) to get \( \frac{40}{48} \).

Now, both fractions have a denominator of 48, making them easy to subtract.

greatest common divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and denominator of a fraction without leaving a remainder. Finding the GCD helps in simplifying fractions.

In our exercise, after subtracting the fractions and multiplying, we got \( \frac{40}{48} \). To simplify this fraction, find the GCD of 40 and 48, which is 8.

To simplify \( \frac{40}{48} \), divide both the numerator and denominator by the GCD:

• \( 40 \div 8 = 5 \)
• \( 48 \div 8 = 6 \)

Thus, \( \frac{40}{48} \) simplifies to \( \frac{5}{6} \). This step is crucial for achieving the simplest form of a fraction.