Problem 77 - 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. $$\frac{5}{10} \times \frac{3}{10}$$

Step-by-Step Solution

Answer

The result is 0.15.

Step 1: Write the Fractions as Decimal

Before multiplying, understand that \(\frac{5}{10}\) and \(\frac{3}{10}\) can be written in decimal form as 0.5 and 0.3, respectively.

Step 2: Multiply the Numerators

To multiply the fractions, start by multiplying the numerators. \(5 \times 3 = 15\).

Step 3: Multiply the Denominators

Now, multiply the denominators. \(10 \times 10 = 100\).

Step 4: Combine the Results

Combine the results of the numerator and the denominator to form a new fraction: \(\frac{15}{100}\).

Step 5: Simplify the Fraction

Simplify the fraction \(\frac{15}{100}\) to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor, which is 5. So \(\frac{15 \div 5}{100 \div 5} = \frac{3}{20}\).

Step 6: Convert Back to Decimal

Finally, convert the simplified fraction \(\frac{3}{20}\) back into decimal form by performing the division: \(3 \div 20 = 0.15\).

Key Concepts

Fraction Multiplication

When we multiply fractions, it's important to focus on their numerators and denominators. Think of the numerators as the top parts of the fractions and the denominators as the bottom parts. The simple rule for fraction multiplication is: multiply the numerators together and then multiply the denominators together.
Let's take a look at an example:

• You have \(\frac{5}{10}\) \(\times\) \(\frac{3}{10}\).
• First, multiply the numerators: \(5 \times 3 = 15\).
• Then, multiply the denominators: \(10 \times 10 = 100\).
• Your new fraction is \(\frac{15}{100}\).

This is a straightforward process, and understanding it makes multiplying fractions a breeze! Just remember to handle numerators and denominators separately before combining them back into a single fraction.

Converting Fractions to Decimals

Converting fractions to decimals is a valuable skill that helps you work with numbers in different formats. To convert a fraction, you simply divide the top number (numerator) by the bottom number (denominator). This will give you a decimal equivalent.
Let's use \(\frac{5}{10}\) and \(\frac{3}{10}\) for practice:

• \(\frac{5}{10}\) is easy to convert because \(5 \div 10 = 0.5\).
• Similarly, \(\frac{3}{10}\) converts to \(3 \div 10 = 0.3\).

This practice ensures you're comfortable seeing numbers as both fractions and decimals, which is very useful in daily math tasks. It also simplifies multiplication, as working with decimals can sometimes feel more intuitive than fractions.

Simplifying Fractions

Simplifying fractions can make math much easier to manage. A simplified fraction is one where the numerator and denominator have no common factors other than 1. This is also known as the fraction's lowest terms.
To simplify a fraction like \(\frac{15}{100}\), find the greatest common divisor (GCD) of the numerator and the denominator.

• The GCD of 15 and 100 is 5.
• Divide both the numerator and the denominator by their GCD: \(15 \div 5 = 3\) and \(100 \div 5 = 20\).
• You’ll get the fraction \(\frac{3}{20}\), which can't be simplified further.

Simplifying fractions helps keep your solutions neat and makes subsequent calculations easier. Plus, it's a habit that enhances your number sense, an invaluable skill in mathematics.