A natural number, when increased by 12, becomes equal to 160 times its reciprocal. Find the number.

To solve the problem, let's denote the natural number as x. According to the question, when this number is increased by 12, it becomes equal to 160 times its reciprocal. We can express this relationship mathematically as:

• Equation: x + 12 = 160 imes rac{1}{x}

Next, we will eliminate the fraction by multiplying both sides of the equation by x (assuming x is not zero):

• x(x + 12) = 160

This simplifies to:

• x² + 12x - 160 = 0

Now, we can use the quadratic formula to find x. The quadratic formula is given by:

• x = rac{-b ext{±} ext{√}(b² - 4ac)}{2a}

Here, a = 1, b = 12, and c = -160. Plugging these values into the formula:

• Discriminant: b² - 4ac = 12² - 4(1)(-160) = 144 + 640 = 784

Now we find the roots:

• x = rac{-12 ext{±} ext{√}(784)}{2 imes 1} = rac{-12 ext{±} 28}{2}

Calculating the two possible values for x:

• x = rac{16}{2} = 8 (valid natural number)
• x = rac{-40}{2} = -20 (not a natural number)

Thus, the natural number we were looking for is 8.

Quick takeaway: Whenever you're faced with a problem involving relationships between a number and its reciprocal, setting up an equation is a powerful first step!