The difference between a positive fraction and its reciprocal is 9 20. Find the sum of the fraction and its reciprocal.

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Let's solve this step by step. We have a positive fraction x, and its reciprocal is 1 over x. The problem tells us that the difference between the fraction and its reciprocal equals 9 over 20. We need to find the sum of the fraction and its reciprocal. Since we have the absolute value of x minus 1 over x equals 9 over 20, we can square both sides to eliminate the absolute value. This gives us x minus 1 over x squared equals 9 over 20 squared, which simplifies to 81 over 400. Now we use the algebraic identity: a plus b squared equals a minus b squared plus 4ab. We substitute a equals x and b equals 1 over x. This gives us x plus 1 over x squared equals x minus 1 over x squared plus 4 times x times 1 over x, which simplifies to x plus 1 over x squared equals x minus 1 over x squared plus 4. We substitute our known value: x plus 1 over x squared equals 81 over 400 plus 4. Converting 4 to the same denominator gives us 1600 over 400. Adding these fractions: 81 plus 1600 equals 1681, so we get 1681 over 400. Taking the square root gives us plus or minus 41 over 20. Since x is positive, both x and 1 over x are positive, so their sum must be positive. Therefore, x plus 1 over x equals 41 over 20. We have found our final answer: the sum of the positive fraction and its reciprocal is 41 over 20, which equals 2.05. This elegant solution demonstrates the power of using algebraic identities. Instead of solving a quadratic equation directly, we used the identity a plus b squared equals a minus b squared plus 4ab to find the sum efficiently.