Problem 223 - Intermediate Algebra

Question

In the following exercises, simplify. $$ 12\left(\frac{9}{20}-\frac{4}{15}\right) $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{11}{5}\).

Step 1: Simplify the expression inside the parentheses

First, find a common denominator for the fractions \(\frac{9}{20} \) and \(\frac{4}{15}\). The least common multiple of 20 and 15 is 60. Rewrite both fractions with the denominator 60. \(\frac{9}{20} = \frac{9 \times 3}{20 \times 3} = \frac{27}{60}\) and \(\frac{4}{15} = \frac{4 \times 4}{15 \times 4} = \frac{16}{60}\).

Step 2: Subtract the fractions

Perform the subtraction: \(\frac{27}{60} - \frac{16}{60} = \frac{27 - 16}{60} = \frac{11}{60}\).

Step 3: Multiply by the constant

Now multiply the simplified fraction by 12: \(12 \times \frac{11}{60}\). Simplify by performing the multiplication: \(12 \times \frac{11}{60} = \frac{12 \times 11}{60} = \frac{132}{60}\).

Step 4: Simplify the final fraction

Simplify \(\frac{132}{60}\) by dividing the numerator and the denominator by their greatest common divisor (GCD). GCD of 132 and 60 is 12. So, \(\frac{132 \text{ ÷ } 12}{60 \text{ ÷ } 12} = \frac{11}{5}\). Thus, the answer is \(\frac{11}{5}\).

Key Concepts

Common Denominators

Finding a common denominator is key when subtracting fractions. A common denominator is a shared multiple of the denominators of two or more fractions. This makes it possible to combine the fractions into a single one.

In our exercise, we needed a common denominator for \(\frac{9}{20} \) and \(\frac{4}{15} \). The least common multiple (LCM) of 20 and 15 is 60. Converting each fraction to a denominator of 60 ensures they can be subtracted directly:

\[ \frac{9}{20} = \frac{9 \times 3}{20 \times 3} = \frac{27}{60} \]
\[ \frac{4}{15} = \frac{4 \times 4}{15 \times 4} = \frac{16}{60} \]
We can now subtract \(\frac{27}{60} \) and \(\frac{16}{60} \) easily.

Fraction Subtraction

Subtracting fractions with the same denominator is straightforward. You subtract the numerators and keep the same denominator.

In our example, we subtract \(\frac{27}{60} \) and \(\frac{16}{60} \):

\[ \frac{27}{60} - \frac{16}{60} = \frac{27 - 16}{60} = \frac{11}{60} \]
Notice that only the numerators are subtracted. The shared denominator (60) remains the same. It is this consistency that makes using a common denominator so important in fraction subtraction.

Constant Multiplication

After subtracting the fractions, we need to multiply by a constant. This operation requires multiplying the numerator by the constant, keeping the denominator the same.

In our case, we have \(\frac{11}{60} \) which we multiply by 12:

\[ 12 \times \frac{11}{60} = \frac{12 \times 11}{60} = \frac{132}{60} \]
It's important to perform this multiplication carefully to avoid mistakes. The next step involves simplifying the resultant fraction.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is crucial in simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder.

To simplify \(\frac{132}{60} \), we find the GCD of 132 and 60. This value is 12. Dividing both the numerator and denominator by their GCD simplifies the fraction:

\[ \frac{132 \text{ ÷ } 12}{60 \text{ ÷ } 12} = \frac{11}{5} \]
This yields the final simplified answer. Finding the GCD helps us reduce fractions to their simplest form, making them easier to understand and work with.