Problem 22 - Beginning and Intermediate Algebra

Question

Multiply or divide as indicated. Write the answer in lowest terms. $$\frac{7}{8} \cdot \frac{3}{21}$$

Step-by-Step Solution

Answer

The simplified answer is \( \frac{1}{8} \).

Step 1: Multiply the Numerators

Multiply the numerators of the fractions. For the given fractions \( \frac{7}{8} \) and \( \frac{3}{21} \), the numerators are 7 and 3, respectively. The product is:\[ 7 \times 3 = 21 \]

Step 2: Multiply the Denominators

Multiply the denominators of the fractions. For the given fractions \( \frac{7}{8} \) and \( \frac{3}{21} \), the denominators are 8 and 21, respectively. The product is:\[ 8 \times 21 = 168 \]

Step 3: Write the Fraction

Combine the results of the numerator and denominator multiplication into a single fraction:\[ \frac{21}{168} \]

Step 4: Reduce to Lowest Terms

To simplify \( \frac{21}{168} \), find the greatest common divisor (GCD) of 21 and 168. Both numbers are divisible by 21. Dividing both by 21 gives:\[ \frac{21 \div 21}{168 \div 21} = \frac{1}{8} \]

Key Concepts

Numerators and Denominators

Fractions are composed of two main parts: numerators and denominators. The numerator is the top number in a fraction, while the denominator is at the bottom. These components are crucial when performing operations such as addition, subtraction, multiplication, and division with fractions.

In the process of fraction multiplication, both the numerators and denominators are independently multiplied. For instance, in the given example with the fractions \(\frac{7}{8}\) and \(\frac{3}{21}\), the numerators 7 and 3 are multiplied to produce 21. Similarly, the denominators 8 and 21 are multiplied to form 168.

Understanding how numerators and denominators interact is key to mastering fraction operations. It's important to keep them separate during multiplication to ensure that you're structuring the new fraction correctly after multiplication.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form where the numerator and denominator have no common divisors other than 1. This can make fractions easier to read, compare, and compute with.

To simplify, after finding the product of the numerators and denominators, we need to look at \(\frac{21}{168}\) from the example. Both 21 and 168 can be divided by the same number to find their simplest form. This requires us to dive into the concept of the greatest common divisor (GCD), which will be discussed in the next section.

It's always a good practice to check if a fraction can be simplified, especially after multiplication, as it often results in more manageable numbers.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD is crucial when simplifying fractions, as it helps reduce the fraction to its simplest possible terms.

For \(\frac{21}{168}\), we need to determine the GCD of 21 and 168. Through detailed checking or calculation (factoring or using the Euclidean algorithm, for example), it is found that both can be divided by 21. Dividing both by 21 simplifies the fraction to \(\frac{1}{8}\).

• Identify factors of both numbers.
• Find the largest number common to both lists of factors.
• Divide the numerator and denominator by this GCD.

Understanding and applying the GCD is a vital skill for managing and simplifying fractions effectively.