Explanatory Answer

Method of solving this CAT Quantitative Ability Question from Number System question

Correct Answer : 2704

Given x + y = 102, we need to find the minimum possible value of the expression 2601(1 + 1/x)(1 + 1/y).

To find the minimum value, we can use the AM-GM inequality. According to the inequality:

(a + b) / 2 ≥ √(ab)

For any positive real numbers a and b.

Applying this inequality to the given expression, we have:

[(1 + 1/x) + (1 + 1/y)] / 2 ≥ √[(1 + 1/x)(1 + 1/y)]

Simplifying further:

(2 + (1/x + 1/y)) / 2 ≥ √((1 + 1/x)(1 + 1/y))

We know that x + y = 102, so we can rewrite the expression as:

(2 + (1/x + 1/(102 - x))) / 2 ≥ √((1 + 1/x)(1 + 1/(102 - x)))

To find the minimum value, we need to maximize the right-hand side. Since the minimum value of the square root occurs when its argument is minimized, we need to minimize the expression (1 + 1/x)(1 + 1/(102 - x)).

To find the minimum value of (1 + 1/x)(1 + 1/(102 - x)), we can use calculus by taking the derivative and setting it to zero:

d/dx [(1 + 1/x)(1 + 1/(102 - x))] = 0

Solving this equation is a bit complicated and involves higher-order polynomials. Instead, we can use trial and error or approximation techniques to find the minimum value.

By trying different values of x, we can find that when x = 51, the expression (1 + 1/x)(1 + 1/(102 - x)) is minimized.

Substituting x = 51 into the expression, we have:

(1 + 1/51)(1 + 1/(102 - 51)) = (1 + 1/51)(1 + 1/51) = (52/51)(52/51) = 2704/2601

Therefore, the minimum possible value of the expression 2601(1 + 1/x)(1 + 1/y) is 2704.