Comparing Expressions: 8 × 4/7 Vs. 8

Alright, let's dive into this mathematical comparison! We need to figure out whether $8 \times \frac{4}{7}$ is greater than, less than, or equal to $8$. And we also need to understand why that's the case by looking at the fraction $\frac{4}{7}$. Get your thinking caps on, guys; we're about to break this down step by step!

Understanding the Problem

First, let’s rewrite the expression we're dealing with. We have $8 \times \frac{4}{7} ? 8$. The question mark is where we need to put our comparison symbol: >, <, or =. The key here is understanding what happens when you multiply a number by a fraction.

Key Concept: Multiplying by a Fraction

When you multiply a number by a fraction, you're essentially taking a fractional part of that number. Think of it like cutting a pizza. If you have 8 slices and you only want $\frac{4}{7}$ of them, you're not going to end up with more than 8 slices, right? You'll end up with less.

Let's consider the fraction $\frac{4}{7}$. To understand its impact, let’s compare it to 1.

• If the fraction is equal to 1 (like $\frac{7}{7}$), multiplying by it would keep the number the same (e.g., $8 \times 1 = 8$).
• If the fraction is greater than 1 (like $\frac{8}{7}$), multiplying by it would make the number larger (e.g., $8 \times \frac{8}{7} > 8$).
• If the fraction is less than 1 (like $\frac{4}{7}$), multiplying by it would make the number smaller (e.g., $8 \times \frac{4}{7} < 8$).

Analyzing the Fraction 4/7

So, what about $\frac{4}{7}$? Well, 4 is less than 7, which means $\frac{4}{7}$ is less than 1. This is super important because it tells us exactly what will happen when we multiply 8 by it. Since $\frac{4}{7}$ is less than 1, multiplying 8 by $\frac{4}{7}$ will result in a value less than 8.

Calculating 8 × 4/7 (Optional)

Just to prove our point, let’s actually calculate $8 \times \frac{4}{7}$. To do this, we multiply 8 by the numerator (4) and keep the denominator (7):

$8 \times \frac{4}{7} = \frac{8 \times 4}{7} = \frac{32}{7}$

Now, $\frac{32}{7}$ is an improper fraction (the numerator is larger than the denominator). Let's convert it to a mixed number to get a better sense of its value. 32 divided by 7 is 4 with a remainder of 4. So, $\frac{32}{7} = 4\frac{4}{7}$.

$4\frac{4}{7}$ is clearly less than 8. You can see that we started with 8, multiplied it by a fraction less than 1, and ended up with a number less than 8. This confirms our initial understanding.

Determining the Correct Symbol

Okay, so we know that $8 \times \frac{4}{7}$ is less than 8. That means the correct symbol to use is “<” (less than).

Therefore, the complete expression is:

$8 \times \frac{4}{7} < 8$

Filling in the Blanks

Now, let's fill in the blanks in the original statement:

Since $\frac{4}{7}$ is less than 1, multiplying by $\frac{4}{7}$ makes the value smaller.

In summary: When you multiply by a fraction less than one, you're essentially taking a part of the whole, which results in a smaller value. In our case, $\frac{4}{7}$ is less than one, so multiplying 8 by $\frac{4}{7}$ gives you a value less than 8. Therefore, $8 \times \frac{4}{7} < 8$.

Additional Examples to Solidify Understanding

To really hammer this home, let's look at a couple more examples.

Example 1: 12 × 1/2 ? 12

Here, we're multiplying 12 by $\frac{1}{2}$. Since $\frac{1}{2}$ is less than 1, the result will be less than 12.

$12 \times \frac{1}{2} = 6$

So, $12 \times \frac{1}{2} < 12$.

Example 2: 5 × 3/2 ? 5

In this example, we're multiplying 5 by $\frac{3}{2}$. Notice that $\frac{3}{2}$ is greater than 1 (it's equal to 1.5). Therefore, the result will be greater than 5.

$5 \times \frac{3}{2} = \frac{15}{2} = 7.5$

So, $5 \times \frac{3}{2} > 5$.

Example 3: 9 x 7/7 ? 9

What happens when we multiply 9 by $\frac{7}{7}$? Well, $\frac{7}{7}$ is equal to 1, so the value doesn't change!

$9 \times \frac{7}{7} = 9 \times 1 = 9$

Thus, $9 \times \frac{7}{7} = 9$.

By considering these examples, you can start to see a pattern. The relationship between the fraction you're multiplying by and the number 1 determines whether the resulting value is larger, smaller, or the same as the original number.

Why This Matters: Real-World Applications

Understanding how fractions affect multiplication isn't just about solving math problems; it has real-world applications. Think about discounts, for example. If a store offers a 25% discount on an item, you're essentially multiplying the original price by $\frac{75}{100}$ (or 0.75), which is less than 1. Therefore, the discounted price will always be less than the original price. This concept is used in many areas, from cooking to construction. If a recipe calls for half the ingredients, you're multiplying each ingredient amount by $\frac{1}{2}$. Similarly, architects and engineers use fractions extensively when scaling building plans or calculating material requirements.

Conclusion: Mastering Fraction Multiplication

So, there you have it! When comparing $8 \times \frac{4}{7}$ and 8, the correct symbol is “<” because $\frac{4}{7}$ is less than 1, and multiplying by a fraction less than 1 makes the value smaller. By understanding the core concepts and practicing with different examples, you can master fraction multiplication and confidently tackle similar problems. Keep practicing, and you'll become a math whiz in no time, guys!