If 𝑥 over 10 is equal to 40 over 𝑥, where 𝑥 is in the natural numbers, find the value of 𝑥.
So we have this equation where we’re going to solve for 𝑥. And we’re told that 𝑥 is in the natural numbers. The natural numbers are counting numbers, the positive ones: one, two, three, four, five, six, seven, eight, and so on.
So let’s begin solving for 𝑥. To solve, we can cross-multiply. It’s a way of getting rid of our denominators. So if we want 𝑥 to be away or gone from the denominator, we can multiply to both sides of the equation. So it would cancel on the right. And we’re left with 𝑥 times 𝑥, which is what we had circled.
So we said 𝑥 times 𝑥 equal to 40 times 10 because now we’ll be getting rid of the 10 on the denominator. Multiplying this side by 10, so it cancels. And whatever we do to one side, we must do to the other. So we multiply the other side by 10, 40 times 10. 𝑥 times 𝑥 is equal to 𝑥 squared. And 40 times 10 is equal to 400.
So to isolate 𝑥, we need to get rid of the squared. And the inverse of squaring a number would be to square root a number. The square root of 𝑥 squared would be 𝑥. And the square root of 400, well 400 is 20 times 20. So it’s a perfect square. So the square root of 400 is 20.
Now when we take a square root, we have positive 20 and we have negative 20 because 20 squared is equal to 20 times 20, which is 400. But we also know that negative 20 squared is negative 20 times negative 20, which is positive 400. So we have to include the positive and the negative. However, we have that stipulation where 𝑥 is in the natural numbers. So it must be positive. So we can actually disregard the negative 20.
So the value of 𝑥 that we would want would be positive 20.