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1. The sum of two numbers is 75 and their difference is 25. The product of the two numbers is :

A. 1350

B. 1250

C. 125

D. 1000

Solution:

Let the numbers be a and b
According to the question,

\begin{aligned} a + b = 75 \\ a - b = 25 \\ ∵ {(a + b)}^{2} - {(a - b)}^{2} = 4 a b \\ \Rightarrow 75^{2} - 25^{2} = 4 a b \\ \Rightarrow 4 a b = (75 + 25) (75 - 25) \\ [∵ a^{2} - b^{2} = (a + b) (a - b)] \\ \Rightarrow 4 a b = 100 \times 50 \\ \Rightarrow a b = \frac{100 \times 50}{4} \\ \Rightarrow a b = 1250 \end{aligned}

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