A natural number, when increased by \(12\), equals \(160\) times its reciprocal. Find the number.
8
Step 1: Let the natural number be \(x\). Since it is a natural number, \(x > 0\).
Step 2: The reciprocal of \(x\) is \(\dfrac{1}{x}\). The question says: "When the number is increased by 12, it equals 160 times its reciprocal." So, the equation is:
\(x + 12 = 160 \times \dfrac{1}{x}\)
or, \(x + 12 = \dfrac{160}{x}\).
Step 3: Remove the fraction by cross multiplying:
\(x(x + 12) = 160\)
\(x^2 + 12x = 160\)
\(x^2 + 12x - 160 = 0\).
Step 4: This is a quadratic equation. Solve it using the quadratic formula:
Formula: \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(ax^2 + bx + c = 0\).
Here, \(a = 1, b = 12, c = -160\).
Step 5: Substitute values:
\(x = \dfrac{-12 \pm \sqrt{12^2 - 4(1)(-160)}}{2}\)
\(x = \dfrac{-12 \pm \sqrt{144 + 640}}{2}\)
\(x = \dfrac{-12 \pm \sqrt{784}}{2}\)
\(x = \dfrac{-12 \pm 28}{2}\).
Step 6: Now calculate both values:
Step 7: Since \(x\) must be a natural number (positive), we take \(x = 8\).
Final Answer: The required natural number is 8.