Problem 55 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$ \frac{3}{4} \cdot \frac{3}{8} $$

Step-by-Step Solution

Answer

The product is \( \frac{9}{32} \) and is already simplified.

Step 1: Multiply the Numerators

To find the product of two fractions, start by multiplying the numerators of the fractions. In this case, multiply 3 by 3:\( 3 \times 3 = 9 \)

Step 2: Multiply the Denominators

Next, multiply the denominators of the fractions. Multiply 4 by 8:\( 4 \times 8 = 32 \)

Step 3: Write the Resulting Fraction

Combine the results from the previous steps to form a new fraction. The product of the two fractions is:\( \frac{9}{32} \)

Step 4: Simplify the Fraction

Check if the fraction \( \frac{9}{32} \) can be simplified. Since 9 and 32 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Numerators

When dealing with fractions, the **numerator** is the top number. It tells you how many parts of a whole you have. In the problem \( \frac{3}{4} \cdot \frac{3}{8} \), both fractions have the number '3' as the numerator. The operation we perform on numerators in this exercise is straightforward multiplication. You multiply the numerators together to find the numerator of the resulting fraction.

Here's how it works:

• Multiply the numerators: \( 3 \times 3 = 9 \)

This result ‘9’ becomes the numerator of the fraction that results from multiplying the two fractions together. Understanding numerators is crucial because it represents the portion of the whole we are interested in.

Denominators

The **denominator** of a fraction is the bottom number. It represents the total number of equal parts the whole is divided into. For example, in \( \frac{3}{4} \cdot \frac{3}{8} \), the denominators are 4 and 8. These numbers might be different from each other, and that's perfectly okay. Just like with numerators, when multiplying fractions, we multiply the denominators to find the denominator of the product.

A look into the multiplication:

• Multiply the denominators: \( 4 \times 8 = 32 \)

This final figure, '32', is the new denominator of the resulting fraction. Knowing how denominators work helps us understand what the structure of the fraction really means.

Simplifying Fractions

**Simplifying fractions** involves making the fraction as compact as possible without changing its value. This means reducing it to the smallest numerator and denominator. After multiplying \( \frac{3}{4} \cdot \frac{3}{8} \), you end with \( \frac{9}{32} \). To simplify, check if both the numerator (9) and the denominator (32) can be divided by a common factor other than 1.

Steps for Simplifying:

• Look for common factors of 9 and 32. Here, their only common factor is 1.
• Since 1 doesn’t reduce our fraction any further, \( \frac{9}{32} \) is already in its simplest form.

Simplifying is essential for clarity and precision, ensuring that our final answer is as straightforward as it can be.

Common Factors

**Common factors** are numbers that can evenly divide two or more numbers without leaving a remainder. When reducing fractions, identifying common factors is key in simplifying them effectively. For \( \frac{9}{32} \), you check common factors to see if it can be simplified further.

Steps for Finding Common Factors:

• List factors of both numerator and denominator.
• Find the greatest shared factor.

In our case, factors of 9 are 1, 3, and 9, and factors of 32 are 1, 2, 4, 8, 16, and 32. The only shared factor is 1, which doesn't help in simplification. Recognizing common factors is a useful skill, as it’s a core part of effectively dealing with fractions in math.