Problem 70 - Prealgebra

Question

To understand how to multiply decimals, we need to understand multiplication with whole numbers, fractions, and mixed numbers. The following problems review these concepts. $$\frac{5}{10} \cdot \frac{6}{10}$$

Step-by-Step Solution

Answer

The product of \( \frac{5}{10} \cdot \frac{6}{10} \) is \( \frac{3}{10} \).

Step 1: Review the Problem

We need to multiply two fractions: \( \frac{5}{10} \cdot \frac{6}{10} \). Both fractions have a denominator of 10.

Step 2: Multiply Numerators

Multiply the numerators of both fractions together. Here, the numerators are 5 and 6. \( 5 \times 6 = 30 \).

Step 3: Multiply Denominators

Multiply the denominators of both fractions together. Here, the denominators are both 10. \( 10 \times 10 = 100 \).

Step 4: Form the New Fraction

Combine the results from Step 2 and Step 3 to form a new fraction: \( \frac{30}{100} \).

Step 5: Simplify the Fraction

To simplify \( \frac{30}{100} \), find the greatest common divisor (GCD) of 30 and 100, which is 10. Divide both the numerator and the denominator by this GCD.\( \frac{30}{10} = 3 \) and \( \frac{100}{10} = 10 \).So, \( \frac{30}{100} \) simplifies to \( \frac{3}{10} \).

Key Concepts

Multiplying Fractions

When it comes to multiplying fractions, the good news is that it's simpler than adding or subtracting them. The main idea is to multiply straight across both numerators (top numbers) and denominators (bottom numbers).

• Start by multiplying the numerators together. This means you'll take the top number of one fraction and multiply it with the top number of the other fraction.
• Do the same with the denominators, multiplying the bottom numbers of each fraction.

For example, if you have the fractions \( \frac{5}{10} \) and \( \frac{6}{10} \), you would multiply the numerators \( 5 \times 6 = 30 \) and the denominators \( 10 \times 10 = 100 \) to get \( \frac{30}{100} \).

After you have formed this new fraction, you proceed by simplifying it, which means reducing it into a cleaner, more concise form.

Simplifying Fractions

Simplifying, or reducing, fractions involves making a fraction as simple as possible. This means ensuring that the numerator and the denominator have no common factors other than 1. A simplified fraction is easier to understand and use in further calculations.

Here's how to simplify a fraction:

• First, determine the greatest common divisor (GCD) of the numerator and the denominator. This is the largest number that divides both numbers without a remainder.
• Next, divide both the numerator and the denominator by their GCD.

For example, with the fraction \( \frac{30}{100} \), you first find the GCD of 30 and 100. This number is 10.

• Divide the numerator by 10, \( \frac{30}{10} = 3 \), and the denominator by 10, \( \frac{100}{10} = 10 \).

Now, the fraction \( \frac{30}{100} \) is simplified to \( \frac{3}{10} \). Simplification always results in a fraction with the same value, just in a smaller, more straightforward form.

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept in simplifying fractions. It refers to the highest number that can divide two integers without leaving a remainder.

Finding the GCD involves a few steps:

• List all the factors of each number. For instance, when dealing with the numbers 30 and 100, you list their factors.
• The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.
• Identify the largest factor common to both numbers. Between 30 and 100, the largest common factor is 10.

Knowing how to find the GCD allows you to simplify fractions efficiently, ensuring you are working with the simplest form possible, which can be particularly handy in mathematics and real-life applications.