Problem 41 - Basic College Mathematics

Question

Multiply. $$ \frac{4}{5} \cdot \frac{3}{28} \quad[2.6 \mathrm{a}] $$

Step-by-Step Solution

Answer

\( \frac{3}{35} \)

Step 1: Multiply the Numerators

To multiply two fractions, start with the numerators. Multiply the numerator of the first fraction by the numerator of the second fraction.\(4 \times 3 = 12\)

Step 2: Multiply the Denominators

Next, multiply the denominators of the two fractions. Multiply the denominator of the first fraction by the denominator of the second fraction.\(5 \times 28 = 140\)

Step 3: Form the New Fraction

Combine the results from the previous steps to form the new fraction. The numerator from Step 1 and the denominator from Step 2 will form the new fraction.\( \frac{12}{140} \)

Step 4: Simplify the Fraction

Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number. The GCD of 12 and 140 is 4.\( \frac{12 \text{ ÷ } 4}{140 \text{ ÷ } 4} = \frac{3}{35} \)

Key Concepts

Numerator

The numerator is the top part of a fraction. In our exercise, the numerators are 4 and 3 for the fractions \(\frac{4}{5}\) and \(\frac{3}{28}\) respectively. When multiplying fractions, you multiply the numerators together. This step is straightforward:
\[4 \times 3 = 12\]
So, the numerator of the resulting fraction becomes 12. Remember, the numerator shows how many parts of the whole you are considering.

Denominator

The denominator is the bottom part of a fraction. It tells us into how many equal parts the whole is divided. In our example, the denominators are 5 and 28 for the fractions \(\frac{4}{5}\) and \(\frac{3}{28}\). To multiply fractions, we also multiply the denominators:
\[5 \times 28 = 140\]
Thus, the denominator of the resulting fraction is 140. Understanding the denominator is crucial for knowing the size of the parts into which the whole is divided.

Simplify Fractions

Simplifying a fraction means making it as simple as possible. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD). For our fraction \(\frac{12}{140}\), we simplify it by finding the GCD of 12 and 140 and dividing both the numerator and denominator by this GCD.
The GCD of 12 and 140 is 4. So, we divide both by 4:
\[\frac{12 \text{ ÷ } 4}{140 \text{ ÷ } 4} = \frac{3}{35}\]
Our simplified fraction is \(\frac{3}{35}\). Simplifying fractions makes them easier to work with and understand.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. To find the GCD of two numbers, such as 12 and 140, you can list their factors:

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 140: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140

The largest common factor between 12 and 140 is 4. Therefore, the GCD is 4. Using the GCD to simplify fractions results in a fraction that is as simple as possible, such as \(\frac{3}{35}\). This concept is key for making fractions easier to use.