Exchangeable operators

Algebra Level 3

One day, Samuel was not completely paying attention in his math class, and his teacher decided to call him out to do some calculation.

He was told to find the sum of 2 integers. However, he misheard and calculated the ratio of these 2 integers instead. Miraculously, he still got the correct answer!

If Samuel did his calculation correctly, what is the final answer that he found?

-1 -2 -3 -4

Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

Viki Zeta
Mar 13, 2017

a + b = a b a b + b 2 = a a ( b − 1 ) = − b 2 Assume b > 0 , a < 0 , a ≠ b ⟹ a ( b − 1 ) = 1 × − b 2 If a = 1, then b will have a non-integer solution ( b − 1 ) = 1 b = 2 Use that in first equation 2 a + 4 = a a = − 4 Therefore the soution is (a,b) = (-4, 2) a + b = \dfrac{a}{b} \\ ab + b^2 = a \\ a( b - 1) = -b^2 \\ \text{Assume }b>0, a < 0, a \ne b\\ \implies a (b-1) = 1 \times -b^2 \\ \text{If a = 1, then b will have a non-integer solution} \\ (b-1) = 1 \\ b = 2 \\ \text{Use that in first equation} \\ 2a + 4 = a \\ a = -4 \\ \text{Therefore the soution is (a,b) = (-4, 2)}

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