Problem 61 - Prealgebra

Question

These problems review the four basic operations with fractions from this chapter. Perform the indicated operations. $$\frac{3}{4} \cdot \frac{8}{9}$$

Step-by-Step Solution

Answer

The result of multiplying \( \frac{3}{4} \) and \( \frac{8}{9} \) is \( \frac{2}{3} \).

Step 1: Understand the Problem

We need to multiply two fractions: \( \frac{3}{4} \) and \( \frac{8}{9} \).

Step 2: Multiply the Numerators

Multiply the numerators of both fractions: \( 3 \times 8 = 24 \).

Step 3: Multiply the Denominators

Multiply the denominators of both fractions: \( 4 \times 9 = 36 \).

Step 4: Combine the Results

The product of the fractions is obtained by combining the results from Step 2 and Step 3 into a new fraction: \( \frac{24}{36} \).

Step 5: Simplify the Fraction

Simplify \( \frac{24}{36} \) by finding the greatest common divisor (GCD) of 24 and 36, which is 12. Then divide both the numerator and the denominator by this value: \( \frac{24 \div 12}{36 \div 12} = \frac{2}{3} \).

Key Concepts

Multiplying Fractions

Multiplying fractions is one of the basic operations in fraction arithmetic that lets you find the product of two fractions. This process involves just a few straightforward steps.
To multiply two fractions:

• Multiply the numerators (the top parts of each fraction) together. This will give you the numerator of the answer.
• Multiply the denominators (the bottom parts of each fraction) together. This gives you the denominator of the answer.

Let's look at an example using the fractions \( \frac{3}{4} \) and \( \frac{8}{9} \).
To find the product, you multiply:

• Numerators: \( 3 \times 8 = 24 \)
• Denominators: \( 4 \times 9 = 36 \)

This results in the fraction \( \frac{24}{36} \).
Remember, multiplying fractions is always performed straight across the top for numerators, and straight across the bottom for denominators.
No need to find a common denominator!

Simplifying Fractions

Once you have multiplied your fractions and reached a product, you might be left with a fraction that can be simplified. Simplifying a fraction means reducing it to its smallest or simplest form.
This involves finding the greatest common divisor (GCD) of the numerator and the denominator.

• The numerator is the top part of the fraction.
• The denominator is the bottom part of the fraction.

Let's work through this with the example we have: \( \frac{24}{36} \).To simplify, first find the GCD of 24 and 36. The GCD is the largest number that divides both the numerator and the denominator. In this case, the GCD is 12.
Now, divide both the numerator and the denominator by the GCD:

• \( 24 \div 12 = 2 \)
• \( 36 \div 12 = 3 \)

This leaves you with \( \frac{2}{3} \). Remember that the aim of simplification is to make the fraction as easy to work with as possible.
Always check your work by ensuring both numbers have no other common factors.

Basic Operations with Fractions

Fractions are an essential part of mathematics and understanding them well is crucial. Fractions represent a part of a whole and have two key parts: the numerator and the denominator.
Understanding basic operations with fractions involves:

• Adding fractions: Finding a common denominator before adding numerators.
• Subtracting fractions: Similar to addition, ensure a common denominator.
• Multiplying fractions: As explained, multiply numerators and denominators directly.
• Dividing fractions: Multiply by the reciprocal of the divisor fraction.

Fractions often appear in various types of problems and situations, so mastering these operations is highly beneficial.
Each operation follows specific steps, but they all require careful attention to numerators and denominators.
Developing confidence in these operations will make solving fraction problems quicker and simpler.