Problem 71 - 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. $$\frac{3}{100} \cdot \frac{17}{100}$$

Step-by-Step Solution

Answer

\( \frac{51}{10000} \)

Step 1: Understand the Problem

We need to multiply two fractions: \( \frac{3}{100} \) and \( \frac{17}{100} \). When multiplying fractions, multiply the numerators and the denominators separately.

Step 2: Multiply the Numerators

Find the product of the numerators of the fractions. Here, we multiply 3 by 17: \( 3 \times 17 = 51 \). Thus, the numerator of the resulting fraction is 51.

Step 3: Multiply the Denominators

Find the product of the denominators of the fractions. Here, we multiply 100 by 100: \( 100 \times 100 = 10000 \). Thus, the denominator of the resulting fraction is 10000.

Step 4: Combine the Results

Write the results from Steps 2 and 3 as a fraction: \( \frac{51}{10000} \).

Step 5: Simplify the Fraction

Check if the fraction \( \frac{51}{10000} \) can be simplified. Here, 51 and 10000 do not have any common factors other than 1, so the fraction is already in its simplest form.

Key Concepts

Multiplying Fractions

When multiplying fractions, each fraction consists of a numerator and a denominator, and the process involves multiplying these parts separately. Here's how:

• Multiply the numerators to get the numerator of the result. For example, if you multiply \( \frac{3}{100} \) by \( \frac{17}{100} \), you multiply 3 by 17 to get 51.
• Multiply the denominators to get the denominator of the result. In the same example, you multiply 100 by 100, resulting in 10000.

After doing this, you combine the results to form a single fraction: \( \frac{51}{10000} \). A key point to remember is that fractions simplify multiplication because you only deal with two pairs of numbers: numerators and denominators.

Numerator and Denominator

A fraction consists of two parts: the numerator, which is the top number, and the denominator, which is the bottom number. Understanding these parts is crucial when you're working with fractions.

• The numerator represents how many parts of a whole you have. For example, in \( \frac{3}{100} \), the number 3 is the numerator.
• The denominator indicates the total number of equal parts that make up the whole. So, in \( \frac{3}{100} \), 100 is the denominator, meaning the whole is divided into 100 equal parts.

To successfully multiply fractions, you simply multiply numerators together and denominators together as discussed in the previous section. It’s important not to cross-multiply or add the denominators.

Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form. A fraction is in its simplest form when the numerator and the denominator are the smallest possible integers, with only 1 as their common factor.To simplify a fraction, follow these steps:

• Identify any common factors of the numerator and denominator. If there’s a common factor, divide both numbers by this factor to reduce the fraction.
• If no common factor (other than 1) exists, the fraction is already simplified.

For instance, the fraction \( \frac{51}{10000} \) was checked for common factors, and since 51 and 10000 share no common factors other than 1, it remains \( \frac{51}{10000} \). Simplifying is an important step as it helps in reducing the complexity of fractions, making them easier to use in further calculations.

Whole Numbers and Fractions

Whole numbers and fractions might seem different, but they can often work together seamlessly in calculations. Here’s how they relate:

• Whole numbers can be expressed as fractions. For example, the whole number 3 can be written as \( \frac{3}{1} \).
• When multiplying a whole number by a fraction, treat the whole number as a fraction with a denominator of 1. This makes the process similar to multiplying two fractions.

For example, when multiplying the fraction \( \frac{17}{100} \) by the whole number 3, you would think of it as \( 3 \times \frac{17}{100} \) or \( \frac{3}{1} \times \frac{17}{100} \), multiplying to \( \frac{51}{100} \). Understanding this relationship simplifies many math problems, as it provides a way to incorporate whole numbers into fraction operations.