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Solve, what ratio with two integers is equal to the ratio \(1 + \sqrt{5}\) to \(2\)?

I am getting 4:3 using conjugates, the actual question says 'nearly equal' and estimates that 8:5 is correct, but I believe that 4:3 is more accurate.

The ratio is not exactly equal to any ratio involving integers, which is why the question asks for an approximation. Since root(5) is greater than root(4), or 2, then 1 + root(5) must be greater than 3, and the ratio in question should be slightly greater than 3 to 2. Since 4 to 3 is less than 3 to 2, it should not be the right answer.
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What answer are you getting if you use conjugates?

There's no reason to use conjugates here, but if you did, you'd find:

root(5) + 1 to 2 = [root(5) + 1] * [root(5) - 1] to 2*[root(5) - 1] = 5 - 1 to 2*root(5) - 2 = 2 to root(5) - 1

It's harder to get an estimate now than it was in the original question, which is why it doesn't make much sense to use conjugates here. Still, even with a rough estimate of root(5) here, you can see that the answer should be somewhere between 3 to 2 and 2 to 1.
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As integers increase, one can find a so near ratio. root 5 is something near 2,23. So equation comes near 1,62. One can say this 81/50. That is to say, something missing in question.