The sum of two numbers is 12 and their product is 35. What is the sum of the reciprocals of these two numbers?

Introduction:
This question explores the relationship between the sum and product of two numbers and the sum of their reciprocals. It is a standard algebraic identity that is very useful in many aptitude problems involving symmetric expressions in two variables.

Given Data / Assumptions:

• Two numbers, say a and b, have a sum a + b = 12.
• Their product is a * b = 35.
• We are asked to find 1/a + 1/b.

Concept / Approach:
Use the identity for the sum of reciprocals of two non-zero numbers:
1/a + 1/b = (a + b) / (a * b)Since we already know a + b and a * b, we can directly compute the sum of the reciprocals without solving for a and b individually.

Step-by-Step Solution:
Step 1: Recall the identity.1/a + 1/b = (a + b) / (a * b)Step 2: Substitute given values.a + b = 12a * b = 351/a + 1/b = 12 / 35Step 3: Simplify if possible.12/35 is already in simplest form (no common factor between 12 and 35).

Verification / Alternative check:
If you want, you can find the actual numbers. They satisfy t^2 − 12t + 35 = 0. The roots are t = 5 and t = 7 (since 5 + 7 = 12 and 5 * 7 = 35). Then 1/5 + 1/7 = (7 + 5) / 35 = 12/35, confirming the identity-based result.

Why Other Options Are Wrong:
1/35 is far too small; 35/8 and 7/32 are unrelated to the ratio of sum to product here. 47/35 does not arise from any correct manipulation of the given data. Only 12/35 matches the derived expression (a + b)/(a * b).

Common Pitfalls:
Some students attempt to find the two numbers first every time, which is not necessary. Others mistakenly compute (a * b)/(a + b) instead of (a + b)/(a * b). Memorizing the simple identity saves time and prevents sign and inversion errors.

Final Answer:
The sum of the reciprocals of the two numbers is 12/35.