A natural number, when increased by 12, equals 160 times its reciprocal. What is the number?

Introduction / Context:
This problem involves a natural number, its reciprocal and a simple algebraic equation. The statement says that when the number is increased by 12, the result is equal to 160 times the reciprocal of the number. Such questions are typical in algebra and aptitude exams, testing your ability to translate a word statement into an equation and solve a quadratic equation that results from the presence of a reciprocal.

Given Data / Assumptions:

• The number is a natural number, say n.
• The reciprocal of n is 1 / n.
• n + 12 = 160 * (1 / n).
• We must find the value of n.

Concept / Approach:
We first convert the verbal statement into an algebraic equation: n + 12 = 160 / n. To eliminate the fraction, we multiply both sides by n, which leads to a quadratic equation in n. Solving this quadratic using the quadratic formula or factorisation gives two roots, but only the positive natural number root is acceptable for this question. Recognising that natural numbers are positive integers helps us discard invalid solutions.

Step-by-Step Solution:
Step 1: Let the natural number be n. Step 2: According to the problem, n + 12 = 160 * (1 / n) = 160 / n. Step 3: Multiply both sides by n to clear the denominator: n * (n + 12) = 160. Step 4: Expand the left side: n^2 + 12n = 160. Step 5: Bring all terms to one side to form a quadratic equation: n^2 + 12n − 160 = 0. Step 6: Compute the discriminant: D = 12^2 − 4 * 1 * (−160) = 144 + 640 = 784. Step 7: The square root of 784 is 28, so the solutions are n = (−12 ± 28) / 2. Step 8: Compute the two roots: n = (−12 + 28) / 2 = 16 / 2 = 8, and n = (−12 − 28) / 2 = −40 / 2 = −20.

Verification / Alternative check:
Since n is a natural number, it must be positive, so we discard n = −20. We take n = 8. Now verify it in the original equation. The reciprocal of 8 is 1 / 8. 160 times 1 / 8 is 160 / 8 = 20. The number 8 increased by 12 is 8 + 12 = 20. Both sides match, confirming that n = 8 is correct. This double check ensures that the quadratic equation was solved correctly and the correct root was chosen.

Why Other Options Are Wrong:
Option 20: If n = 20, then n + 12 = 32, while 160 / n = 8, so the two sides do not match.
Option 12: For n = 12, n + 12 = 24 and 160 / 12 is not equal to 24.
Option 4 or 16: Substituting either of these values will not satisfy the equation n + 12 = 160 / n when checked.

Common Pitfalls:
Students sometimes forget to multiply both sides by n or mishandle the expansion, leading to an incorrect quadratic equation. Another common issue is ignoring the requirement that the number is a natural number and accepting negative roots. Always re-read the question to check domain restrictions (natural, integer, non-zero, etc.) and verify the solution by substituting back into the original equation.

Final Answer:
The natural number that satisfies the condition is 8.