Which of these numbers is half of a square number? Circle your answer: 8.5, 12.5, 16.5, or 20.5.
So we want to know which of these numbers is, so equals, half of a square number or a number squared. Say we call it 𝑦 squared. Well, since we don’t know which number it is, let’s just write down 𝑥. So this tells us what stipulations 𝑥 must follow. So 𝑥 must be equal to one-half times a square number. Now that’s kind of hard to think about. So maybe let’s look at what 𝑦 needs to be like.
So let’s multiply both sides of the equation by two. This way, the one-half cancels and we have that two 𝑥 is equal to 𝑦 squared. So this means our square number must be equal to two times one of these answers. So if we take each of these numbers and multiply it by two, one of them should be equal to the square of a number.
First, we have 8.5 times two. That’s 17. Is 17 equal to the square of a number? No, it’s not. How about 12 and a half times two? Well, 12.5 times two is 25. Is 25 equal to the square of some number? Yes, it is. If we were to square-root 25, we would find that the number that’s been squared would be five. So 12.5 should be our answer.
However, let’s go through the other two options and make sure that they don’t work. 16.5 times two is 33. And 33 is not equal to the square of some number. And lastly, 20.5 times two is equal to 41. And 41 is not equal to the square of a number.
Now we also could’ve thought about this a different way. We could’ve taken our equation and taken the square root of both sides, meaning our number would be equal to taking each of our answers, multiplying by two. And we should be able to take the square root of that. And once again, the only one that we can actually take the square root and get a nice solid answer is the square root of 25. It’s five. So once again, the number that’s equal to half of a square number is 12.5.