Problem 64 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\frac{76}{99} \cdot \frac{66}{38}$$

Step-by-Step Solution

Answer

The product is \( \frac{76}{57} \).

Step 1: Simplify Each Fraction

First, simplify each fraction individually. For \( \frac{76}{99} \), the greatest common divisor (GCD) of 76 and 99 is 1, so it is already simplified. For \( \frac{66}{38} \), the GCD is 2. Simplifying gives \( \frac{33}{19} \). Thus, the fractions are \( \frac{76}{99} \) and \( \frac{33}{19} \).

Step 2: Multiply the Numerators

Multiply the numerators of the two fractions: \( 76 \times 33 = 2508 \).

Step 3: Multiply the Denominators

Multiply the denominators of the two fractions: \( 99 \times 19 = 1881 \).

Step 4: Form the Product of the Fractions

Form a new fraction with the product of the numerators over the product of the denominators: \( \frac{2508}{1881} \).

Step 5: Simplify the Product Fraction

Simplify \( \frac{2508}{1881} \). The GCD of 2508 and 1881 is 3. Dividing both by 3, we get \( \frac{836}{627} \). Both 836 and 627 can be further divided by 11, giving \( \frac{76}{57} \). Finally, since 76 and 57 have no common divisor other than 1, the simplest form is \( \frac{76}{57} \).

Step 6: Reduce Further If Possible

Check if the fraction \( \frac{76}{57} \) can be reduced further by finding the GCD. Since the GCD is 1, \( \frac{76}{57} \) is the simplest form.

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a fundamental concept in simplifying fractions. It refers to the largest number that can divide two or more numbers without leaving a remainder. For example, to simplify a fraction, such as \( \frac{66}{38} \), finding the GCD is a crucial step.
To find the GCD of 66 and 38, you can list out the factors of each number:

• Factors of 66 are 1, 2, 3, 6, 11, 22, 33, and 66.
• Factors of 38 are 1, 2, 19, and 38.

The highest common factor in both lists is 2, which means the GCD is 2. This simplifies the fraction \( \frac{66}{38} \) to \( \frac{33}{19} \), by dividing both the numerator and the denominator by the GCD. By understanding how to determine the GCD, you can simplify fractions effectively.

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, which means they share no common factor other than 1.
To simplify \( \frac{2508}{1881} \), we start with finding the GCD of the two numbers. Using either the Euclidean algorithm or primary factorization, we find that the GCD is 3. Divide both the numerator and the denominator by this GCD:

• \( 2508 \div 3 = 836 \)
• \( 1881 \div 3 = 627 \)

This gives us \( \frac{836}{627} \). We can further simplify this by dividing both terms by their GCD, which is 11, yielding \( \frac{76}{57} \). Finally, verify by checking that 76 and 57 have no common divisors other than 1. This final check confirms that the fraction is now in its simplest form.

Product of Fractions

The product of fractions involves multiplying two or more fractions. To multiply fractions, simply multiply the numerators together and the denominators together. For instance, consider multiplying \( \frac{76}{99} \) and \( \frac{33}{19} \).
Step by step multiplication includes:

• Multiply the numerators: \( 76 \times 33 = 2508 \)
• Multiply the denominators: \( 99 \times 19 = 1881 \)
• Form the new fraction: \( \frac{2508}{1881} \)

Often, the resulting fraction needs to be simplified, as shown in previous sections. Using the GCD of the product of numerators and denominators helps in reducing the fraction. Understanding the multiplication and simplification of fractions is pivotal for more complex mathematical operations, enabling clearer and more accurate mathematical expressions.