Problem 58 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\frac{5}{6} \cdot \frac{14}{15}$$

Step-by-Step Solution

Answer

The product is \( \frac{7}{9} \).

Step 1: Multiply the Numerators

To multiply the fractions \( \frac{5}{6} \) and \( \frac{14}{15} \), start by multiplying the numerators: \( 5 \times 14 = 70 \).

Step 2: Multiply the Denominators

Next, multiply the denominators: \( 6 \times 15 = 90 \).

Step 3: Form the New Fraction

Combine the results from Step 1 and Step 2 to form a new fraction: \( \frac{70}{90} \).

Step 4: Simplify the Fraction

Find the greatest common divisor (GCD) of 70 and 90, which is 10. Divide both the numerator and the denominator by their GCD to simplify: \( \frac{70 \div 10}{90 \div 10} = \frac{7}{9} \).

Key Concepts

Numerator and Denominator

When we talk about fractions, two terms often come up: numerator and denominator. The numerator is the top number of a fraction. It tells us how many parts we have. On the other hand, the denominator is the bottom number. It indicates the total number of equal parts the whole is divided into.

For example, in the fraction \( \frac{5}{6} \), the numerator is 5, and the denominator is 6. So, we are talking about 5 parts out of a total of 6.

• The numerator (top number) shows how many parts are being considered.
• The denominator (bottom number) shows how many total parts there are.

Understanding these two components is crucial as they help us in operations like addition, subtraction, and especially multiplication of fractions.

Simplifying Fractions

Simplifying fractions means reducing them to their smallest form without changing their value. A fraction is said to be simplified when the numerator and denominator have no common factors other than 1.

Continuing with our earlier example, after multiplying the numerators and denominators, we get a new fraction: \( \frac{70}{90} \). To simplify, we need to find the largest number that divides both 70 and 90 evenly. This number is known as the greatest common divisor.

• The simplified form is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD).
• This process retains the original value of the fraction.

Simplified fractions are easier to understand and work with, especially in further calculations.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD), also known as the greatest common factor, is the largest number that can evenly divide two or more numbers.

To simplify a fraction, finding the GCD of the numerator and the denominator is essential. For example, in the fraction \( \frac{70}{90} \), you need to determine the GCD of 70 and 90. Here, the greatest number that divides both 70 and 90 without leaving a remainder is 10.

• The GCD helps simplify fractions by reducing the fraction to its smallest terms.
• You can use various methods to find the GCD, such as listing the factors or using the Euclidean algorithm.

Once the GCD is determined, both the numerator and the denominator can be divided by it to simplify the fraction. For \( \frac{70}{90} \), dividing both by 10 yields \( \frac{7}{9} \), which is the simplest form of the fraction.