Question 24

Given $$a = 2025$$, $$b = 2024$$, the numerical value of $$\left(a + b - \frac{4ab}{a+b}\right) \div \left(\frac{a}{a+b} - \frac{b}{b-a} + \frac{2ab}{b^2 - a^2}\right)$$ is

Correct Answer: 1

Solution

The first bracket equals $$\frac{(a+b)^2 - 4ab}{a+b} = \frac{(a-b)^2}{a+b}$$. Taking $$(a+b)(b-a)$$ as the common denominator, the second bracket equals $$\frac{a(b-a) - b(a+b) + 2ab}{(a+b)(b-a)} = \frac{-(a-b)^2}{(a+b)(b-a)} = \frac{a-b}{a+b}$$. Dividing the first by the second gives $$a - b = 2025 - 2024 = 1$$.

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