Reciprocal condition — A positive number decreased by 4 equals 21 times its reciprocal. Find the number and verify your result.

Introduction / Context:This equation mixes a number with its reciprocal, a common aptitude theme. Converting the sentence directly into algebra and clearing the fraction yields a quadratic with small integer roots. Choosing the positive, context-appropriate root solves the problem neatly.

Given Data / Assumptions:

• Let the positive number be n.
• Condition: n - 4 = 21 / n.
• n > 0 (explicitly positive).

Concept / Approach:Multiply both sides by n to remove the fraction and create a quadratic. Then factor or use the quadratic formula. After solving, check sign and verify the original relation to exclude extraneous roots.

Step-by-Step Solution:Start: n - 4 = 21 / n.Multiply by n: n^2 - 4n = 21.Rearrange: n^2 - 4n - 21 = 0.Factor: (n - 7)(n + 3) = 0 → n = 7 or n = -3.n must be positive → n = 7.

Verification / Alternative check:Plug n = 7: 7 - 4 = 3 on the left; 21 / 7 = 3 on the right. Equality holds, confirming the solution.

Why Other Options Are Wrong:

• 3 / 5 / 9 / 21: These do not satisfy n - 4 = 21 / n when substituted. Only 7 works.

Common Pitfalls:Dropping the negative solution without justification; arithmetic errors when multiplying by n; mis-factoring the quadratic.

Final Answer:7