Problem 25 - Algebra 2 and Trigonometry

Question

In \(25-30,\) perform the indicated operations and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{3}{5} \times \frac{5}{9} \div \frac{4}{3} $$

Step-by-Step Solution

Answer

The simplified result is \(\frac{1}{4}\). There are no undefined variables in this operation.

Step 1: Multiply the fractions

First, multiply the first two fractions: \( \frac{3}{5} \times \frac{5}{9} \). Multiply the numerators together and the denominators together:\[\frac{3 \times 5}{5 \times 9} = \frac{15}{45}.\]Simplify by dividing both the numerator and the denominator by their greatest common divisor, which is 15:\[\frac{15}{45} = \frac{1}{3}.\]

Step 2: Convert division to multiplication

When dividing fractions, divide by a fraction by multiplying by its reciprocal. So, perform the operation:\[\frac{1}{3} \div \frac{4}{3}\]This is equivalent to:\[\frac{1}{3} \times \frac{3}{4}.\]

Step 3: Perform the multiplication

Now multiply the fractions from Step 2:\[\frac{1}{3} \times \frac{3}{4}.\]Multiply the numerators and the denominators:\[\frac{1 \times 3}{3 \times 4} = \frac{3}{12}.\]Simplify \( \frac{3}{12} \) by dividing both numerator and denominator by their greatest common divisor, which is 3:\[\frac{3}{12} = \frac{1}{4}.\]

Step 4: Determine undefined values

The original fractions are undefined if the denominators are zero. In the multiplications, the original fraction has denominators of 5, 9, and 3. The operation \( \frac{4}{3} \) introduces a potential for division by zero if it were the reciprocal zero. However, since 3 is not zero, and 9 and 5 are constants, no variable is introduced, the problem does not have undefined variable values in this specific multiplication and division.

Key Concepts

Multiplication of Fractions

Multiplying fractions is straightforward once you understand the rule of multiplying across the tops (numerators) and the bottoms (denominators) of the fractions. It's similar to regular multiplication, but since fractions represent parts of a whole, we pair numbers that indicate parts to form a new fraction.
Here's a quick guide:

• Multiply the numerators together to get a new numerator.
• Multiply the denominators together to get a new denominator.

In our example, we multiplied \( \frac{3}{5} \times \frac{5}{9} \), which resulted in \( \frac{15}{45} \). To simplify, we divided both by their greatest common divisor, 15, resulting in \( \frac{1}{3} \).
Remember, simplifying right after multiplying can make further steps easier and more efficient.

Division of Fractions

Dividing fractions might seem confusing at first, but it's quite simple once you know the trick. The key is to multiply by the reciprocal, or the flipped version, of the fraction you are dividing by.
Here's how to do it:

• Change the division sign to a multiplication sign between the fractions.
• Flip the second fraction, turning it upside down to get its reciprocal.

In the exercise, \( \frac{1}{3} \div \frac{4}{3} \) became \( \frac{1}{3} \times \frac{3}{4} \) by flipping \( \frac{4}{3} \) to \( \frac{3}{4} \).
This method converts a division problem into an easier-to-manage multiplication problem, leading to a quick solution.

Simplifying Fractions

Simplifying fractions is all about finding an equivalent fraction that uses the smallest numbers possible for both the numerator and the denominator. This process doesn't change the value of the fraction — just its appearance.
Here's a step-by-step guide:

• Find the greatest common divisor (GCD) of both the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

In our simplification, the fraction \( \frac{15}{45} \) was reduced to \( \frac{1}{3} \) using the GCD of 15, and \( \frac{3}{12} \) was simplified to \( \frac{1}{4} \) with the GCD of 3.
Reducing fractions to their simplest form is crucial because it makes them easier to understand and use in further calculations. It's also a beneficial skill to enhance your mathematical accuracy and efficiency.