Problem 67 - Prealgebra

Question

Multiply. $$\frac{3}{2}\left(\frac{2}{3}\right)$$

Step-by-Step Solution

Answer

The product is 1.

Step 1: Identify the Operation

You need to multiply the fractions \( \frac{3}{2} \) and \( \frac{2}{3} \). To multiply two fractions, multiply the numerators together and the denominators together.

Step 2: Multiply the Numerators

Multiply the numerators: \( 3 \times 2 = 6 \).

Step 3: Multiply the Denominators

Multiply the denominators: \( 2 \times 3 = 6 \).

Step 4: Simplify the Fraction

The product of the fractions is \( \frac{6}{6} \). Simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6: \( \frac{6}{6} = 1 \).

Key Concepts

Fraction Simplification

Simplifying fractions is an important part of dealing with fractions. It's like making the fraction as simple as possible, kind of like cleaning your room to make it less cluttered. When we simplify fractions, we divide both the numerator and the denominator by their greatest common divisor (GCD), the largest number both can be evenly divided by.
For example, if you're left with a fraction like \( \frac{6}{6} \), both the numerator and the denominator are the same, so they're already as simple as they can get. Their greatest common divisor is 6. Dividing both by 6 gives us \( \frac{1}{1} \), or simply 1.

• Simplification makes fractions easier to understand and use.
• Always check for a common number that can evenly divide both parts of the fraction.
• When the GCD is found, divide both numerator and denominator by this number.

By following these steps, you can simplify any fraction you come across, keeping math neat and manageable.

Numerator Multiplication

In the process of multiplying fractions, the numerators get multiplied first. The numerator is the top number of the fraction. It's like the star of the show giving the fraction its identity.
When multiplying, you take each fraction’s numerator and multiply them together. With the exercise problem, you had to multiply 3 (from \( \frac{3}{2} \)) by 2 (from \( \frac{2}{3} \)), giving you the result of 6.

• Take the numerators of both fractions.
• Multiply those numerators together for a new numerator.

This step is straightforward, and it follows basic multiplication rules. Be careful with each step to avoid mistakes in your calculations.

Denominator Multiplication

Just like with the numerators, the denominators also need to be multiplied. The denominator is the bottom number of a fraction—it's like the foundation of a building, holding everything up and stable.
In the given problem, after multiplying the numerators, you multiply the denominators: 2 (from \( \frac{3}{2} \)) by 3 (from \( \frac{2}{3} \)). This results in a product of 6.
To multiply denominators:

• Identify the denominators of both fractions.
• Multiply them to form a new denominator.

This becomes the denominator of your new fraction. Like stacking blocks, everything stays in balance when done right—just remember not to rush, staying precise in your work.