Problem 3 - Elementary Algebra

Question

For exercises \(3-6\), evaluate or simplify. $$ \frac{3}{20} \cdot \frac{2}{15} $$

Step-by-Step Solution

Answer

\( \frac{1}{50} \)

Step 1 - Multiply the Numerators

Multiply the numerators of both fractions together. This means you take the top numbers of the fractions and multiply them: ewline ewline \(\frac{3}{20} \times \frac{2}{15} = \frac{3 \times 2}{20 \times 15}\)

Step 2 - Multiply the Denominators

Next, multiply the denominators of both fractions together. This means you take the bottom numbers of the fractions and multiply them: ewline ewline \(\frac{3 \times 2}{20 \times 15} = \frac{6}{300}\)

Step 3 - Simplify the Fraction

Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by that number. The GCD of 6 and 300 is 6: ewline ewline \(\frac{6}{300} \rightarrow \frac{6 \rightarrow 1}{300 \rightarrow 50} = \frac{1}{50}\)

Key Concepts

Numerator and Denominator

Fractions consist of two parts: the numerator and the denominator. The numerator is the top number and represents how many parts you have. The denominator is the bottom number and shows the total number of equal parts in a whole.
For example, in the fraction \(\frac{3}{20}\), 3 is the numerator and 20 is the denominator. Understanding this structure is crucial as it lays the foundation for fraction operations.
In the multiplication process, you multiply the numerators with each other and the denominators with each other. This maintains the ratio represented by the fractions.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This process involves expressing a fraction in such a way that the numerator and denominator are as small as possible while still representing the same value. Simplification makes fractions easier to understand and work with.
The steps to simplify a fraction are:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• Rewrite the fraction using the results from the division.

For example, if you have \(\frac{6}{300}\), you first find the GCD of 6 and 300, which is 6. Then, divide both the numerator and denominator by 6 to get \(\frac{1}{50}\). This fraction is now simplified because 1 and 50 have no common divisors other than 1.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can evenly divide both the numerator and the denominator of a fraction. Finding the GCD is essential for simplifying fractions.
To find the GCD of two numbers, you can use several methods, one of the most common being the Euclidean algorithm:

• Divide the larger number by the smaller number.
• Take the remainder and divide the smaller number by this remainder.
• Repeat the process until the remainder is 0. The last non-zero remainder is the GCD.

For example, to find the GCD of 6 and 300:

• 300 ÷ 6 = 50 with no remainder.
• Here, 6 is the GCD as it perfectly divides both 6 and 300.

Understanding and using the GCD allows you to simplify fractions effectively and ensures your solutions are as clear as possible.