Problem 22 - Prealgebra

Question

Find each product. Use an area model if necessary. $$\frac{5}{9} \cdot \frac{8}{25}$$

Step-by-Step Solution

Answer

The product is \( \frac{8}{45} \).

Step 1: Write the fractions

We start with the multiplication problem, which is to find the product of the two fractions \( \frac{5}{9} \) and \( \frac{8}{25} \).

Step 2: Multiply the Numerators

To multiply fractions, you start by multiplying the numerators. So, multiply 5 and 8: \( 5 \times 8 = 40 \).

Step 3: Multiply the Denominators

Next, multiply the denominators of the fractions. So, multiply 9 and 25: \( 9 \times 25 = 225 \).

Step 4: Write the Product as a Fraction

Place the product of the numerators over the product of the denominators to form the new fraction: \( \frac{40}{225} \).

Step 5: Simplify the Fraction

Look for the greatest common divisor (GCD) of 40 and 225 to simplify the fraction. Both 40 and 225 can be divided evenly by 5. After dividing, we have: \( \frac{40 \div 5}{225 \div 5} = \frac{8}{45} \).

Step 6: Result

The simplified product of \( \frac{5}{9} \) and \( \frac{8}{25} \) is \( \frac{8}{45} \).

Key Concepts

Fraction Simplification

Simplifying fractions is a useful skill that makes fractions easier to work with. When you simplify a fraction, you reduce it to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD). This makes the fraction easier to understand and compare to other fractions.

Let's take our example \[ \frac{40}{225}.\]Both 40 and 225 can be divided by 5, their GCD. By dividing both numbers by 5, you'll reach:

• 40 divided by 5 equals 8
• 225 divided by 5 equals 45

So, the simplest form of the fraction is:\[ \frac{8}{45}.\]This simplified format represents the same proportion of a whole but is easier to compare and work with in other calculations.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is a helpful concept in mathematics, particularly when simplifying fractions. The GCD is the largest number that divides two numbers without leaving a remainder.

Finding the GCD is essential for reducing fractions to their simplest form. For example, to simplify the fraction \[\frac{40}{225},\] we need the GCD of 40 and 225. Here's a brief look at finding the GCD:

• List the factors of each number:
  • Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
  • Factors of 225: 1, 3, 5, 9, 15, 25, 45, 75, 225
• Identify the common factors: 1 and 5
• Select the greatest common factor: 5

Using the GCD, divide both the numerator and denominator to simplify the fraction, just like we did in the fraction simplification section.

Numerators and Denominators

Understanding numerators and denominators is fundamental when working with fractions.

A fraction consists of two parts: the numerator and the denominator. The numerator is the top number and indicates how many parts of the whole we have, whereas the denominator is the bottom number and tells us the total number of equal parts into which the whole is divided.For instance, in the fraction \[\frac{5}{9},\] 5 is the numerator and 9 is the denominator. This fraction tells us that we have 5 parts out of a total of 9 equal parts.
When multiplying fractions, such as \[\frac{5}{9} \times \frac{8}{25},\] we multiply the numerators and denominators:

• Numerators: 5 times 8 equals 40
• Denominators: 9 times 25 equals 225

Thus, we form a new fraction: \[\frac{40}{225}.\]Understanding these components is key to performing operations with fractions.

Area Model for Fractions

The area model is a visual way to understand fractional multiplication. It represents fractions as proportions of actual areas, which aids in grasping the multiplication process intuitively.

Imagine a rectangle, divided into smaller sections according to the denominators of the fractions. Each section represents a unit fraction of the whole. Let's consider the multiplication of \[\frac{5}{9} \times \frac{8}{25}.\]

• Divide the rectangle into 9 equal parts horizontally, representing the denominator 9.
• Shade 5 parts, showing \[\frac{5}{9}.\]
• Divide the same rectangle into 25 equal parts vertically, corresponding to the denominator 25.
• Shade 8 parts vertically, indicating \[\frac{8}{25}.\]
• Where the shaded parts overlap, count the number of overlapping sections. This is the product: \[\frac{40}{225}.\]

This model shows visually why the result's numerator (40) and denominator (225) come from multiplying the numerators and denominators.