Comparing the Product to the Factors

Oct 13, 2025 by windailys.com 38 views

Multiplying Fractions: Understanding the Product of $\frac{2}{3}$ and a Fraction Less Than One

When working with fractions, it's essential to understand how they interact with each other when multiplied. In this article, we'll explore the concept of multiplying $\frac{2}{3}$ by a fraction less than one and explain how the product compares to the factors.

Understanding Fractions

A fraction is a way to represent a part of a whole. It consists of two parts: the numerator (the number on top) and the denominator (the number on the bottom). For example, the fraction $\frac{2}{3}$ represents 2 parts out of 3 equal parts.

Multiplying Fractions

When multiplying fractions, we multiply the numerators together and the denominators together. For example, to multiply $\frac{2}{3}$ by $\frac{1}{4}$ , we would multiply the numerators (2 and 1) and the denominators (3 and 4) separately.

$\frac{2}{3} \times \frac{1}{4} = \frac{2 \times 1}{3 \times 4} = \frac{2}{12}$

Comparing the Product to the Factors

Now that we've multiplied $\frac{2}{3}$ by a fraction less than one, let's compare the product to the factors. In this case, the product is $\frac{2}{12}$ , and the factors are $\frac{2}{3}$ and $\frac{1}{4}$ .

The Product is Less Than the Factors

When we multiply $\frac{2}{3}$ by a fraction less than one, the product is always less than the factors. This is because we are multiplying a fraction by a number less than 1, which results in a smaller fraction.

Why is the Product Less Than the Factors?

The product is less than the factors because we are essentially "shrinking" the fraction $\frac{2}{3}$ by a factor of $\frac{1}{4}$ . This means that we are reducing the size of the fraction by a quarter, resulting in a smaller fraction.

Example 1: Multiplying $\frac{2}{3}$ by $\frac{1}{2}$

Let's multiply $\frac{2}{3}$ by $\frac{1}{2}$ to see how the product compares to the factors.

$\frac{2}{3} \times \frac{1}{2} = \frac{2 \times 1}{3 \times 2} = \frac{2}{6}$

In this case, the product is $\frac{2}{6}$ , and the factors are $\frac{2}{3}$ and $\frac{1}{2}$ . As we can see, the product is less than the factors.

Example 2: Multiplying $\frac{2}{3}$ by $\frac{1}{3}$

Let's multiply $\frac{2}{3}$ by $\frac{1}{3}$ to see how the product compares to the factors.

$\frac{2}{3} \times \frac{1}{3} = \frac{2 \times 1}{3 \times 3} = \frac{2}{9}$

In this case, the product is $\frac{2}{9}$ , and the factors are $\frac{2}{3}$ and $\frac{1}{3}$ . As we can see, the product is less than the factors.

In conclusion, when you multiply $\frac{2}{3}$ by a fraction less than one, the product is always less than the factors. This is because we are multiplying a fraction by a number less than 1, which results in a smaller fraction. By understanding how fractions interact with each other when multiplied, we can better grasp the concept of fractions and make more informed decisions in mathematics.

Key Takeaways

When multiplying fractions, we multiply the numerators together and the denominators together.
The product of two fractions is always less than the factors when one of the fractions is less than one.
Multiplying a fraction by a number less than 1 results in a smaller fraction.

Practice Problems

Multiply $\frac{2}{3}$ by $\frac{1}{5}$ and compare the product to the factors.
Multiply $\frac{2}{3}$ by $\frac{1}{4}$ and compare the product to the factors.
Multiply $\frac{2}{3}$ by $\frac{1}{6}$ and compare the product to the factors.

Answer Key

$\frac{2}{15}$ , the product is less than the factors.
$\frac{2}{12}$ , the product is less than the factors.
$\frac{2}{18}$ , the product is less than the factors.
Frequently Asked Questions: Multiplying Fractions

Q: What happens when you multiply a fraction by a fraction less than one?

A: When you multiply a fraction by a fraction less than one, the product is always less than the factors. This is because you are essentially "shrinking" the fraction by a factor of the second fraction.

Q: Why is the product less than the factors?

A: The product is less than the factors because you are reducing the size of the fraction by a factor of the second fraction. This means that the product is smaller than the original fraction.

Q: Can you give an example of multiplying a fraction by a fraction less than one?

A: Let's multiply $\frac{2}{3}$ by $\frac{1}{2}$ .

$\frac{2}{3} \times \frac{1}{2} = \frac{2 \times 1}{3 \times 2} = \frac{2}{6}$

In this case, the product is $\frac{2}{6}$ , and the factors are $\frac{2}{3}$ and $\frac{1}{2}$ . As we can see, the product is less than the factors.

Q: What if the second fraction is greater than one?

A: If the second fraction is greater than one, the product will be greater than the factors. For example, let's multiply $\frac{2}{3}$ by $\frac{2}{1}$ .

$\frac{2}{3} \times \frac{2}{1} = \frac{2 \times 2}{3 \times 1} = \frac{4}{3}$

In this case, the product is $\frac{4}{3}$ , and the factors are $\frac{2}{3}$ and $\frac{2}{1}$ . As we can see, the product is greater than the factors.

Q: Can you give another example of multiplying a fraction by a fraction less than one?

A: Let's multiply $\frac{2}{3}$ by $\frac{1}{4}$ .

$\frac{2}{3} \times \frac{1}{4} = \frac{2 \times 1}{3 \times 4} = \frac{2}{12}$

In this case, the product is $\frac{2}{12}$ , and the factors are $\frac{2}{3}$ and $\frac{1}{4}$ . As we can see, the product is less than the factors.

Q: What if the numerator and denominator of the second fraction are the same?

A: If the numerator and denominator of the second fraction are the same, the product will be equal to the first fraction. For example, let's multiply $\frac{2}{3}$ by $\frac{3}{3}$ .

$\frac{2}{3} \times \frac{3}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}$

In this case, the product is $\frac{6}{9}$ , which is equal to the first fraction $\frac{2}{3}$ .

Q: Can you give an example of multiplying a fraction by a fraction greater than one?

A: Let's multiply $\frac{2}{3}$ by $\frac{2}{1}$ .

$\frac{2}{3} \times \frac{2}{1} = \frac{2 \times 2}{3 \times 1} = \frac{4}{3}$

In this case, the product is $\frac{4}{3}$ , and the factors are $\frac{2}{3}$ and $\frac{2}{1}$ . As we can see, the product is greater than the factors.

Q: What if the numerator and denominator of the first fraction are the same?

A: If the numerator and denominator of the first fraction are the same, the product will be equal to the second fraction. For example, let's multiply $\frac{2}{2}$ by $\frac{3}{4}$ .

$\frac{2}{2} \times \frac{3}{4} = \frac{2 \times 3}{2 \times 4} = \frac{6}{8}$

In this case, the product is $\frac{6}{8}$ , which is equal to the second fraction $\frac{3}{4}$ .

In conclusion, multiplying fractions can be a complex topic, but by understanding the rules and examples, you can become more confident in your ability to multiply fractions. Remember, when multiplying fractions, you multiply the numerators together and the denominators together. The product of two fractions is always less than the factors when one of the fractions is less than one.