Problem 30 - Precalculus Mathematics for Calculus

Question

Perform the indicated operations. (a) \(\frac{2-\frac{3}{4}}{\frac{1}{2}-\frac{1}{3}}\) (b) \(\frac{\frac{2}{5}+\frac{1}{2}}{\frac{1}{10}+\frac{3}{15}}\)

Step-by-Step Solution

Answer

(a) \(\frac{15}{2}\); (b) 3.

Step 1: Solve the numerator of (a)

First, solve the expression in the numerator of part (a): \(2 - \frac{3}{4}\). Convert 2 to a fraction with a denominator of 4: \(\frac{8}{4}\). Now subtract: \(\frac{8}{4} - \frac{3}{4} = \frac{5}{4}\).

Step 2: Solve the denominator of (a)

Next, solve the expression in the denominator of part (a): \(\frac{1}{2} - \frac{1}{3}\). Find a common denominator, which is 6: \(\frac{3}{6} - \frac{2}{6} = \frac{1}{6}\).

Step 3: Divide the fractions for (a)

Divide the result of the numerator by the result of the denominator for part (a): \(\frac{\frac{5}{4}}{\frac{1}{6}}\). Dividing by a fraction is the same as multiplying by its reciprocal: \(\frac{5}{4} \times \frac{6}{1} = \frac{30}{4}\). Simplify: \(\frac{30}{4} = \frac{15}{2}\).

Step 4: Solve the numerator of (b)

Now, solve the expression in the numerator of part (b): \(\frac{2}{5} + \frac{1}{2}\). Find a common denominator, which is 10: \(\frac{4}{10} + \frac{5}{10} = \frac{9}{10}\).

Step 5: Solve the denominator of (b)

Solve the expression in the denominator of part (b): \(\frac{1}{10} + \frac{3}{15}\). Find a common denominator, which is 30: \(\frac{3}{30} + \frac{6}{30} = \frac{9}{30}\). Simplify: \(\frac{9}{30} = \frac{3}{10}\).

Step 6: Divide the fractions for (b)

Divide the result of the numerator by the result of the denominator for part (b): \(\frac{\frac{9}{10}}{\frac{3}{10}}\). Dividing by a fraction is the same as multiplying by its reciprocal: \(\frac{9}{10} \times \frac{10}{3} = \frac{90}{30}\). Simplify: \(\frac{90}{30} = 3\).

Key Concepts

Operations with Fractions

When working with fractions, we perform operations just like we do with whole numbers. These operations can include addition, subtraction, multiplication, and division. Understanding how to correctly perform each type of operation with fractions is vital for solving complex problems.

Let's dive into how these operations are handled:

• Addition and Subtraction: For both operations, it’s crucial to have a common denominator, which is a shared multiple of the denominators involved.
• Multiplication: To multiply fractions, simply multiply the numerators together and the denominators together. Then, simplify the result if possible.
• Division: Dividing fractions involves an additional step. Instead of dividing directly, multiply by the reciprocal of the second fraction. To find the reciprocal, flip the numerator and the denominator.

These principles guide you in setting up and solving fraction problems correctly.

Common Denominators

Finding a common denominator is a key step when adding or subtracting fractions. A common denominator allows you to express fractions with different denominators in a shared form, making calculations straightforward.

Here’s how you find a common denominator:

• Identify the Least Common Multiple (LCM): Look for the smallest multiple that the denominators share.
• Convert the Fractions: Adjust each fraction by multiplying the numerator and the denominator by the necessary values to have equivalent fractions with the same denominator.

For instance, when adding \(\frac{2}{5} + \frac{1}{2}\), the LCM of 5 and 2 is 10. Thus, you convert both to have a denominator of 10, yielding \(\frac{4}{10} + \frac{5}{10}\). This makes addition feasible, leading to \(\frac{9}{10}\).

Finding and using common denominators simplifies the process of working with fractions, so it's an important skill to practice.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. A fraction is simplified when the numerator and denominator are as small as possible, but still represent the same value.

• Find the Greatest Common Factor (GCF): Identify the largest number that divides both the numerator and denominator without a remainder.
• Divide both terms: With the GCF determined, divide the numerator and the denominator by this number to simplify the fraction.

When simplifying fractions, like turning \(\frac{30}{4}\) into \(\frac{15}{2}\), you find the largest factor both the numerator and denominator share. Here, it’s 2, so dividing both by 2 results in the simplified form.

By ensuring that fractions are simplified, you make them easier to understand and compare, making subsequent mathematical operations more straightforward.