Problem 78 - 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. $$\frac{5}{100} \times \frac{3}{1,000}$$

Step-by-Step Solution

Answer

The product of \( \frac{5}{100} \times \frac{3}{1,000} \) is \( \frac{3}{20,000} \).

Step 1: Understand the Fractions

We are multiplying two fractions: \( \frac{5}{100} \) and \( \frac{3}{1,000} \). Fractions are numbers made up of a numerator (top number) and a denominator (bottom number). In a multiplication problem with fractions, we multiply the numerators together and the denominators together.

Step 2: Multiply the Numerators

Multiply the numerators of the fractions. Here, the numerators are 5 and 3. Multiply these together: \( 5 \times 3 = 15 \).

Step 3: Multiply the Denominators

Next, multiply the denominators of the fractions. Here, the denominators are 100 and 1,000. Multiply these together: \( 100 \times 1,000 = 100,000 \).

Step 4: Form the New Fraction

Using the products of the numerators and the denominators, form a new fraction. The new fraction becomes: \( \frac{15}{100,000} \).

Step 5: Simplify the Fraction

To simplify \( \frac{15}{100,000} \), find the greatest common divisor (GCD) of 15 and 100,000. The GCD is 5. Divide both the numerator and the denominator by 5: \( \frac{15 \div 5}{100,000 \div 5} = \frac{3}{20,000} \). This is the simplified fraction.

Key Concepts

Understanding Numerator and Denominator in Fractions

Fractions are an essential part of mathematics. Each fraction consists of two main parts: the numerator and the denominator.

• The numerator is the top number. In the example of \( \frac{5}{100} \), the numerator is 5. It represents how many parts of the whole are being considered.
• The denominator is the bottom number. For \( \frac{5}{100} \), the denominator is 100. It tells us into how many equal parts the whole is divided.

When multiplying fractions, like in the exercise \( \frac{5}{100} \times \frac{3}{1,000} \), understanding the roles of the numerator and denominator is crucial. You need to multiply the numerators together and the denominators together to form a new fraction. Always ensure to multiply the numerators first and then the denominators. This will give you a new fraction that represents the product of the two. It's like combining the two fractions by their core parts.

Simplifying Fractions

Once you have multiplied the numerators and denominators to form a new fraction, the next crucial step is simplifying it. Simplifying makes a fraction as simple as possible and more comfortable to work with.The fraction you get after multiplication might look large or complex. For example, if you multiply \( \frac{5}{100} \) by \( \frac{3}{1,000} \), you get \( \frac{15}{100,000} \). This fraction can be daunting without simplification. Simplifying involves dividing both the numerator and the denominator by a common number. Always start by finding the greatest common divisor (GCD) of the numbers, as it's the largest number that can equally divide both the numerator and the denominator.After finding the GCD, divide both parts of the fraction by this number. This reduces the fraction while keeping its value the same. For the fraction \( \frac{15}{100,000} \), dividing by their GCD, which is 5, simplifies it to \( \frac{3}{20,000} \). Learning to simplify fractions not only helps in math problems but makes understanding quantities in real-life scenarios a lot easier.

The Role of the Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is a key concept in simplifying fractions. It's the greatest number that can evenly divide both the numerator and the denominator without leaving any remainder.To find the GCD, you can list out the factors of both the numerator and the denominator. The largest factor that appears in both lists is the GCD. Let's take the example of \( \frac{15}{100,000} \). Finding the GCD:

• List factors of 15: 1, 3, 5, 15
• List factors of 100,000: 1, 2, 4, 5, 10, 20, ... (and so on up to the full list)

Here, the common factors are 1 and 5, with 5 being the greatest. So, the GCD is 5.Using the GCD enables you to divide both the numerator and the denominator so the fraction assumes simplest form but represents the same quantity.In math problems, particularly involving fractions, knowing how to find and use the GCD improves your efficiency and understanding.