Multiplying By 7/8: Finding Rational Numbers
Hey math enthusiasts! Let's dive into a cool little problem. We're asked to figure out which numbers, when multiplied by $ rac{7}{8}$, will give us a rational number as a result. Remember, a rational number is any number that can be expressed as a fraction $ rac{p}{q} $, where p and q are integers, and q isn't zero. Basically, it's a number that can be written as a simple fraction or a terminating or repeating decimal. So, grab your calculators (or your thinking caps) and let's break this down, shall we? This problem isn't just about crunching numbers; it's about understanding the core concepts of rational and irrational numbers. We'll explore each option, see what happens when we multiply it by $ rac{7}{8}$, and decide if the result is rational.
Decoding Rational Numbers
Before we start, let's brush up on what makes a number rational. As mentioned earlier, a rational number is any number that can be written as a fraction. This includes all integers (like -3, 0, 5), fractions (like $ rac{1}{2}$, $ rac{3}{4}$, and decimals that either terminate (like 0.25) or repeat (like 0.333...). On the other hand, irrational numbers are those that cannot be expressed as a simple fraction. They have decimal representations that go on forever without repeating. Famous examples of irrational numbers are $\pi$ (pi) and $\sqrt{2}$ (the square root of 2). Knowing this difference is super important to solve our problem. The key to this problem lies in understanding that we want to end up with a number that can be expressed as a fraction. To get there, let's go through the options one by one, keeping an eye out for how they interact with the $ rac{7}{8}$ multiplication.
Now, let's look at the options:
- A. $\pi$: We know that $\pi$ is an irrational number. When you multiply an irrational number by a rational number (like $ rac{7}{8}$), the result is almost always irrational. So, multiplying $\pi$ by $ rac{7}{8}$ gives us $ rac{7}{8}\pi$, which is still irrational. No dice on this one, guys.
- **B. $\sqrt100}${1}$). Multiplying $ rac{7}{8}$ by 10, we get $ rac{7}{8} * 10 = rac{70}{8}$, which simplifies to $ rac{35}{4}$. Since $ rac{35}{4}$ is a fraction of two integers, it's a rational number. Score one for option B!
- **C. $\sqrt13}${8}$, we get $ rac{7}{8}\sqrt{13}$, which is irrational. Sorry, C, you're out.
- **D. $\sqrt rac{1}{5}}${5}} = rac{\sqrt{1}}{\sqrt{5}} = rac{1}{\sqrt{5}}$. Since $\sqrt{5}$ is irrational, multiplying $ rac{7}{8}$ by $ rac{1}{\sqrt{5}}$ results in $ rac{7}{8\sqrt{5}}$, which is irrational. Not what we're looking for, folks.
- **E. $ rac1}{6}${8}$ by $ rac{1}{6}$ gives us $ rac{7}{8} * rac{1}{6} = rac{7}{48}$. Since both 7 and 48 are integers, and 48 isn't zero, $ rac{7}{48}$ is a rational number. That's a win for option E!
The Answer Revealed
So, after all the calculations and explanations, the two options that result in a rational number when multiplied by $ rac{7}{8}$ are:
- B. $\sqrt{100}$
- E. $ rac{1}{6}$
Congratulations if you got it right! This problem shows how important it is to understand the properties of rational and irrational numbers. Keep practicing, and you'll become a pro at these types of problems in no time. Keep the mathematics spirit alive!
Deep Dive into the Concepts: Rational vs. Irrational
Let's take a moment to really hammer home the difference between rational and irrational numbers. This is the heart of understanding the original question. Rational numbers, as we know, are those that can be expressed as a fraction $ rac{p}{q}$, where p and q are integers, and q is not zero. They can be written as simple fractions or decimals that either terminate or repeat. Examples include 0.5, 0.75, $ rac{2}{3}$ (which is 0.666...), and every whole number (since, like we saw with the square root of 100, they can be easily represented as fractions over 1). This is a broad category, encompassing almost everything you commonly deal with. The important bit? They can be precisely represented. When you write a rational number as a decimal, you know you're getting an exact value, even if the decimal repeats forever.
Now, let's talk about the mysterious irrational numbers. These guys are a bit trickier. They cannot be expressed as a simple fraction $ rac{p}{q}$. Their decimal representations go on forever without repeating. The classic example is $\pi$ (pi), which is approximately 3.14159... but the decimal digits keep going without any pattern. Another common example is the square root of 2, or $\sqrt{2}$, which is about 1.41421... and also has a non-repeating, never-ending decimal. You can't write these numbers down exactly as a decimal; you can only approximate them. The implications of this difference are significant. When you perform calculations with irrational numbers, your answer is often an approximation, not an exact value. This is why our problem asked us to think about how these numbers behave when multiplied by a rational number. You'll almost always get an irrational result unless there is some magic cancellation (like in our $\sqrt{100}$ example).
This distinction is fundamental to number theory and is essential in many areas of mathematics. Understanding it allows you to predict the behavior of numbers and solve problems like the one we tackled. Thinking about how these different kinds of numbers interact with each other is very interesting! When you multiply a rational number by another rational number, you always get a rational number. However, as we have seen, the result is usually irrational when dealing with irrational numbers, which makes this problem a perfect way to test your understanding.
The Square Root Secrets
Let's zoom in on something we saw in the original problem: the square roots. Remember that we had $\sqrt{100}$, which turned out to be rational, and then $\sqrt{13}$, which was irrational. Why the difference? The key is whether the number inside the square root is a perfect square. A perfect square is a number that is the result of squaring an integer (multiplying an integer by itself). So, 100 is a perfect square because $10 * 10 = 100$. The square root of a perfect square is always an integer, which is a rational number. On the other hand, 13 isn't a perfect square. There's no integer that, when multiplied by itself, equals 13. Therefore, the square root of 13 is irrational. This is a crucial concept to grasp! It explains why $\sqrt{100}$ was rational while $\sqrt{13}$ was not. Always check if the number under the radical (the square root symbol) is a perfect square. If it is, the result is a rational number; if not, it's irrational.
Going back to our original problem, we had $\sqrt{\frac{1}{5}}$. It's a bit more complex, but we can break it down. If you simplify the square root of a fraction, you get the square root of the numerator divided by the square root of the denominator. In this case, that means $\frac{\sqrt{1}}{\sqrt{5}}$. Because the square root of 5 isn't a perfect square, it's irrational, making the entire expression irrational. Remember, you should always simplify radicals when you can. This will make it much easier to determine if the result is rational or irrational. Being able to spot perfect squares and understanding how they interact with square roots will serve you well in future math problems. It's a handy trick to have up your sleeve!
Multiplying and the Outcome
Let's circle back to the core of the problem: multiplying by $ rac{7}{8}$. When you multiply a rational number by another rational number, the result will always be rational. This is a fundamental rule of arithmetic. Our option E, $ rac{1}{6}$, is a good example. We multiplied it by $ rac{7}{8}$ and got $ rac{7}{48}$, which is a rational number. No surprises there. The main challenge comes when dealing with irrational numbers. In most cases, multiplying an irrational number by a rational number gives you an irrational result (like when we multiplied $\pi$ or $\sqrt{13}$ by $ rac{7}{8}$). There are exceptions, of course, but you have to think carefully. The exception happens when the irrational term cancels out with another term. For example, if we had something like $(\sqrt{5} + 2) * (\sqrt{5} - 2)$, we'd end up with a rational number because the square roots would cancel. This type of situation is something you'll encounter more of as you study more advanced math concepts. Keep this concept in mind! It is extremely useful when calculating.
So, the key takeaways are:
- Multiplying a rational number by a rational number always gives a rational number.
- Multiplying an irrational number by a rational number usually gives an irrational number.
- Always simplify expressions to see if irrational terms cancel out.
Conclusion: A Rational Choice
So, there you have it, folks! We've successfully navigated the world of rational and irrational numbers, and we've determined which options, when multiplied by $ rac{7}{8}$, result in a rational number. This problem is a great example of how math isn't just about memorizing formulas; it's about understanding the concepts behind them. Always be curious, ask questions, and practice regularly. And most of all, have fun with math! If you're still curious, try creating your own examples of rational and irrational numbers and see what happens when you multiply them. You can also try changing the multiplier (instead of $ rac{7}{8}$, use something else). This will give you more practice and make you even better at recognizing rational and irrational numbers. Remember, practice is key, and every problem you solve brings you closer to mastering the fascinating world of mathematics. Keep up the awesome work!