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

Introduction / Context:
This problem connects basic algebra with fractions. We are given the sum and product of two numbers and asked to find the sum of their reciprocals. Instead of solving for the individual numbers directly, we can use a useful identity for reciprocals in terms of sum and product. This approach is efficient and avoids unnecessary quadratic equation solving, which is especially helpful in timed aptitude tests.

Given Data / Assumptions:

- Let the two numbers be a and b. - Sum of the numbers is a + b = 12. - Product of the numbers is a * b = 35. - We need to find 1/a + 1/b.

Concept / Approach:
There is a direct algebraic relationship between the reciprocals of numbers and their sum and product. Specifically, 1/a + 1/b can be rewritten as (a + b) / (a * b). Because we already know both a + b and a * b, we can substitute these values directly into the identity. This eliminates the need to find the actual values of a and b.

Step-by-Step Solution:
Step 1: Start with the expression for the sum of reciprocals: 1/a + 1/b. Step 2: Combine the two fractions over a common denominator: 1/a + 1/b = (b + a) / (a * b). Step 3: Recognize that b + a is equal to a + b, which is given as 12. Step 4: Recognize that a * b, the product of the numbers, is given as 35. Step 5: Substitute into the identity: 1/a + 1/b = (a + b) / (a * b) = 12 / 35. Step 6: Simplify the fraction if possible. In this case, 12 and 35 have no common factors other than 1, so 12/35 is already in simplest form.

Verification / Alternative check:
As a check, we can find the actual numbers. They satisfy the quadratic equation t^2 − 12t + 35 = 0. Solving, t = 5 or t = 7. Then 1/5 + 1/7 = (7 + 5) / 35 = 12/35, which matches our algebraic result. This confirms that the identity and substitution were applied correctly.

Why Other Options Are Wrong:
Option B (1/35) would correspond to 1/(a * b) rather than (a + b)/(a * b). Option C (35/8) is larger than 4 and clearly inconsistent with reciprocals of positive numbers whose sum is 12. Option D (7/32) does not arise from any natural combination of the given sum and product and fails the direct identity check.

Common Pitfalls:
Some learners attempt to find a and b by solving the quadratic and may make arithmetic mistakes. Others forget the identity for reciprocals and try to guess. Remembering that 1/a + 1/b = (a + b)/(a * b) saves time and reduces the chance of error, especially when the sum and product are provided explicitly.

Final Answer:
The sum of the reciprocals of the two numbers is 12/35, which corresponds to option A.