Is This Number Rational? Math Quiz

Hey math whizzes! Today, we're diving deep into the fascinating world of rational numbers. You know, those numbers that can be expressed as a simple fraction $\frac{p}{q}$, where 'p' and 'q' are integers and 'q' isn't zero? It sounds straightforward, but sometimes, especially with a mix of operations, it can get a little tricky to spot. We've got a question for you that’s sure to get those brain cells firing. We're going to look at four different expressions, and your mission, should you choose to accept it, is to identify which of the following is rational. Get ready to put your knowledge to the test, because understanding rational numbers is a fundamental building block in mathematics, and nailing this will give you a serious confidence boost. Let's break down each option, shall we?

Option A: $3 \cdot \pi$

Alright guys, let's kick things off with option A: $3 \cdot \pi$. So, what do we know about $\pi$? This famous mathematical constant, approximately 3.14159..., is the ratio of a circle's circumference to its diameter. Now, the crucial piece of information here is that $\\pi$ is an irrational number. This means it cannot be expressed as a simple fraction $\\frac{p}{q}$. Its decimal representation goes on forever without repeating. When you multiply an irrational number (like $\\pi$) by a non-zero rational number (like 3), the result is always an irrational number. Think about it: if $3 \cdot \pi$ were rational, say equal to $\\frac{a}{b}$, then $\\pi$ would be equal to $\\frac{a}{3b}$, which would make $\\pi$ rational. But we know that's not true! Therefore, $3 \cdot \pi$ is definitely not rational. So, toss this one out, and let's move on.

Option B: $\frac{2}{3}+9.26$

Next up, we have option B: $\frac{2}{3}+9.26$. This one looks a bit more promising, doesn't it? We're adding two numbers together. Let's analyze each part. The first part, $\\frac{2}{3}$, is clearly a rational number because it's already in the form $\\frac{p}{q}$ where p=2 and q=3. Now, what about 9.26? This is a terminating decimal. Any terminating decimal can be written as a fraction. For example, $9.26$ can be written as $\\frac{926}{100}$. Since $\\frac{926}{100}$ is a ratio of two integers, it's also a rational number. The sum of two rational numbers is always a rational number. So, when we add $\\frac{2}{3}$ and $9.26$ (which is $\\frac{926}{100}$), the result will be rational. To be super sure, let's convert $9.26$ to a fraction and add them: $\\frac{2}{3} + \\frac{926}{100} = \\frac{2 \\cdot 100}{3 \\cdot 100} + \\frac{926 \\cdot 3}{100 \\cdot 3} = \\frac{200}{300} + \\frac{2778}{300} = \\frac{2978}{300}$. This is a fraction of two integers, so it's rational! Bingo! Option B is looking good, guys.

Option C: $\sqrt{45}+\sqrt{36}$

Let's take a look at option C, shall we? We've got $\\sqrt{45}+\\sqrt{36}$. Here, we're adding two square roots. The key to figuring this out is to simplify each term. First, let's tackle $\\sqrt{36}$. This is pretty straightforward: the square root of 36 is exactly 6. Since 6 can be written as $\\frac{6}{1}$, it's a rational number. Now, let's look at $\\sqrt{45}$. Can we simplify this? Yes, we can! We can rewrite 45 as $9 \\cdot 5$. So, $\\sqrt{45} = \\sqrt{9 \\cdot 5} = \\sqrt{9} \\cdot \\sqrt{5} = 3\\sqrt{5}$. Now, here's the catch: $\\sqrt{5}$ is an irrational number. This is because 5 is not a perfect square. When we multiply an irrational number ($\\sqrt{5}$) by a non-zero rational number (3), the result ($3\\sqrt{5}$) is still irrational. So, option C is the sum of an irrational number ($3\\sqrt{5}$) and a rational number (6). The sum of an irrational number and a rational number is always irrational. Therefore, $\\sqrt{45}+\\sqrt{36}$ is not rational. We're getting closer to the answer, but not quite there yet.

Option D: $14 . \overline{3}+5.78765239$

Finally, let's analyze option D: $14 . \overline{3}+5.78765239$. This involves adding two decimal numbers. The first number, $14.\overline{3}$, is a repeating decimal. A crucial rule in mathematics is that all repeating decimals are rational numbers. The notation $14.\overline{3}$ means $14.3333...$, and it can be converted into a fraction. To convert $14.333...$ to a fraction, let $x = 14.333...$. Then $10x = 143.333...$. Subtracting the first from the second, we get $10x - x = 143.333... - 14.333...$, which simplifies to $9x = 129$. So, $x = \\frac{129}{9}$, which simplifies to $\\frac{43}{3}$. This is clearly a rational number. The second number, $5.78765239$, is a terminating decimal. As we discussed earlier, any terminating decimal is a rational number because it can be written as a fraction (in this case, $\\frac{578765239}{100000000}$). The sum of two rational numbers is always a rational number. Therefore, $14.\overline{3}+5.78765239$ is rational.

Conclusion: The Rational Choice!

So, after breaking down each option, we've found our winner! We examined $3 \cdot \pi$, realizing that multiplying by $\\pi$ always results in an irrational number. Then, we looked at $\\frac{2}{3}+9.26$, a sum of two rational numbers, which is indeed rational. We also investigated $\\sqrt{45}+\\sqrt{36}$, which simplified to an irrational number plus a rational number, resulting in an irrational number. Finally, we tackled $14 . \overline{3}+5.78765239$, the sum of a repeating decimal (rational) and a terminating decimal (rational), which confidently confirmed it as rational. Therefore, the correct answer to 'Which of the following is rational?' is Option D. Great job if you got it right, guys! Keep practicing, and you'll become rational number pros in no time!