Problem 37 - Basic College Mathematics

Question

Multiply and simplify. $$ \frac{4}{10} \cdot \frac{5}{10} $$

Step-by-Step Solution

Answer

The simplified product is \(\frac{1}{5}\).

Step 1: Multiply the Numerators

To multiply fractions, first multiply the numerators (the top numbers) together. Given \(\frac{4}{10} \cdot \frac{5}{10}\), multiply: \[4 \cdot 5 = 20\]

Step 2: Multiply the Denominators

Next, multiply the denominators (the bottom numbers) together. Multiply: \[10 \cdot 10 = 100\]

Step 3: Form the New Fraction

Combine the results from Step 1 and Step 2 to form the new fraction: \(\frac{20}{100}\)

Step 4: Simplify the Fraction

Simplify the fraction \(\frac{20}{100}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 20 and 100 is 20. Thus, \[\frac{20}{100} = \frac{20\div20}{100\div20} = \frac{1}{5}\]

Key Concepts

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This helps in making calculations easier and understanding results better. To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD). For example, the fraction \(\frac{20}{100}\) can be simplified by dividing both parts by 20, giving us \(\frac{1}{5}\). Simplifying fractions allows us to work with the most basic representation of the fraction, which is particularly useful in both mathematical and real-world applications. Simplification is essential, as it often provides clearer insights and makes further calculations more manageable.

Numerators and Denominators

Understanding numerators and denominators is crucial in working with fractions. In the fraction \(\frac{4}{10}\), the numerator is 4 and the denominator is 10. The numerator represents how many parts we have, while the denominator shows the total number of equal parts in a whole. When multiplying fractions like \(\frac{4}{10} \cdot\frac{5}{10}\), we multiply the numerators together (4x5) and the denominators together (10x10), resulting in \(\frac{20}{100}\). Understanding this concept simplifies various operations in arithmetic and helps grasp more advanced topics in math.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder. Finding the GCD is a vital step in simplifying fractions. For example, in simplifying \(\frac{20}{100}\), we identify that the GCD of 20 and 100 is 20. We then divide both the numerator and the denominator by this GCD to get \(\frac{1}{5}\). Calculating the GCD can be done through various methods, including listing out the factors of each number or using the Euclidean algorithm. Gaining proficiency in finding the GCD not only aids in simplifying fractions but also enhances overall number sense and problem-solving skills.