Problem 32 - Basic College Mathematics

Question

Multiply. $$ \frac{3}{7} \cdot \frac{4}{5} $$

Step-by-Step Solution

Answer

\( \frac{12}{35} \)

Step 1: Multiply the Numerators

First, multiply the numerators (the top numbers) of the fractions. For the given problem, multiply 3 and 4. \( 3 \times 4 = 12 \)

Step 2: Multiply the Denominators

Next, multiply the denominators (the bottom numbers) of the fractions. For the given problem, multiply 7 and 5. \( 7 \times 5 = 35 \)

Step 3: Form the New Fraction

Now, place the product of the numerators over the product of the denominators to form the new fraction. \( \frac{12}{35} \)

Step 4: Simplify the Fraction (if possible)

Check if the fraction can be simplified. Since 12 and 35 have no common factors other than 1, the fraction \( \frac{12}{35} \) is already in its simplest form.

Key Concepts

Numerators and Denominators

To understand fraction multiplication, we first need to know what numerators and denominators are. In any fraction, the numerator is the number above the fraction line, and the denominator is the number below. These two parts serve different functions.

For example, in the fraction \( \frac{3}{7} \):

• 3 is the numerator
• 7 is the denominator

The numerator shows how many parts we have, while the denominator indicates into how many equal parts the whole is divided. It's crucial to differentiate between them, as they play specific roles when multiplying fractions.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it.

For example, the fraction \( \frac{12}{35} \) is already in its simplest form because there are no common factors between 12 and 35 other than 1. Here are simple steps to simplify any fraction:

• Find the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.
• Write down the simplified fraction.

Simplified fractions are easier to understand and work with, especially in more complicated math problems.

Fraction Multiplication

Multiplying fractions might seem complex at first, but it's straightforward once you grasp the steps. Here's a simple guide:

• Multiply the numerators (top numbers) together.
• Multiply the denominators (bottom numbers) together.
• Form a new fraction from these products.
• Simplify the resulting fraction if possible.

Let's use our exercise as an example:
Given fractions \( \frac{3}{7} \) and \( \frac{4}{5} \),

• Multiply the numerators: \(3 \times 4 = 12\)
• Multiply the denominators: \(7 \times 5 = 35\)
• Form the new fraction: \( \frac{12}{35} \)
• Check if it can be simplified. In this case, \( \frac{12}{35} \) is already simplified.

Practice these steps, and soon fraction multiplication will become second nature!