Problem 4 - Prealgebra

Question

Find each of the following products. (Multiply.) $$\frac{3}{5} \cdot \frac{4}{7}$$

Step-by-Step Solution

Answer

The product is \( \frac{12}{35} \).

Step 1: Multiply the Numerators

To find the product of two fractions, we start by multiplying the numerators of each fraction. In this case, multiply 3 and 4:\[ 3 \times 4 = 12 \]

Step 2: Multiply the Denominators

Next, multiply the denominators of each fraction. Multiply 5 and 7:\[ 5 \times 7 = 35 \]

Step 3: Write the Result as a Fraction

Now combine the results from Steps 1 and 2 to form a new fraction. The numerator from Step 1 is 12 and the denominator from Step 2 is 35, so:\[ \frac{12}{35} \]

Step 4: Simplify the Fraction (if necessary)

Check if the fraction \( \frac{12}{35} \) can be simplified further. Look for any common factors between the numerator and the denominator. In this case, there are no common factors apart from 1, so \( \frac{12}{35} \) is already in its simplest form.

Key Concepts

Understanding the Numerator

In fraction multiplication, the numerator plays a crucial role. It is the number above the fraction bar and signifies how many parts of the whole are being considered. In the multiplication of two fractions, you determine the numerator of the resulting fraction by multiplying the numerators of the given fractions.
For example, in the multiplication of \( \frac{3}{5} \times \frac{4}{7} \), the numerators are 3 and 4. To find the numerator of the product, simply multiply these two numbers:

• The numerator of the first fraction: 3
• The numerator of the second fraction: 4
• Multiplied together: \( 3 \times 4 = 12 \)

It's important to always start with the numerators when multiplying fractions. This ensures that you correctly calculate how many parts of the whole are combined in the product.

Deciphering the Denominator

The denominator is equally crucial in fractions, as it defines the total number of equal parts the whole is divided into. When you are multiplying fractions, the denominator helps to determine what whole the new fraction is part of. To get the denominator of the result in fraction multiplication, simply multiply the denominators of the fractions you are working with.
Consider our example: \( \frac{3}{5} \times \frac{4}{7} \).

• The denominator of the first fraction: 5
• The denominator of the second fraction: 7
• Multiplied together: \( 5 \times 7 = 35 \)

Using the denominator, you can understand how the segments form the whole in the product fraction. This product of the denominators (35 in our case) illustrates the total number of pieces the whole is divided into.

Simplifying Fractions

Simplifying fractions is an essential step in making the fraction easier to understand and work with. It involves reducing the fraction to its simplest form by ensuring that the numerator and the denominator don't have any common factors other than 1. This is done by checking if both numbers can be divided by the same non-zero integer.
In the example we considered, the fraction \( \frac{12}{35} \) needs to be simplified. To do this:

• List the factors of the numerator (12): 1, 2, 3, 4, 6, 12
• List the factors of the denominator (35): 1, 5, 7, 35
• Find the greatest common factor: In this case, it's 1

Since there are no greater common factors, the fraction \( \frac{12}{35} \) is already simplified. Understanding this step helps in making complex fractions appear simpler and more intuitive.