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 information given, when this number is increased by 12, it equals 160 times its reciprocal. We can express this relationship with the following equation:

Setting Up the Equation

The equation can be written as:

x + 12 = 160 * (1/x)

Clearing the Fraction

To eliminate the fraction, we can multiply both sides of the equation by x (assuming x is not zero):

x(x + 12) = 160

Expanding the Equation

Now, let's expand the left side:

x² + 12x = 160

Rearranging the Equation

Next, we rearrange the equation to set it to zero:

x² + 12x - 160 = 0

Factoring the Quadratic

Now we need to factor the quadratic equation. We look for two numbers that multiply to -160 and add up to 12. These numbers are 20 and -8:

(x + 20)(x - 8) = 0

Finding the Solutions

Setting each factor to zero gives us:

• x + 20 = 0 → x = -20 (not a natural number)
• x - 8 = 0 → x = 8 (valid solution)

Final Answer

The natural number we are looking for is 8.