Alan

"So the process is, calculate the third unknown side using pythagoras and then tan a = opposite side / closest side. "

This is correct.  

The "closest" side is called the "adjacent" side.

The "opposite" side is just that.

The longest side is called the "hypotenuse".

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If N = total number, then 420 represents (100 - 35 - 45)% (=20%) of the total, so 20%N = 420 or 0.2N = 420

N = 420/0.2 or N = 2100

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There are two possibilities:

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"For the last bit you just have to show that 9m/19 is not an integer for  0<m<19 can't you just say that 9 and 19 are relatively prime so 9m/19 will only be an integer if m is a multiple of 19.  Since m cannot be a multiple of 19, 9m/19 is not an integer. That is, 9m is not divisable by 19"

Yes.  Much neater Melody.

If you really mean 5.0909 then you just get 50909/10000 as a fraction.  However, if you mean 5.0909.... where the 09s go on forever, then, because 1/11 = 0.09090909.... it can be written as 5+(1/11) or 56/11

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If the complex number is in rectangular form:   z = a + i*b then the polar form is:

z = r*e^iθ, where r = √(a² + b²)  and  θ = tan^-1(b/a)

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It seems to me that there must be an infinite number of inverses of 35 ...

True, but the question specifies that the answer to be supplied must be a nonnegative integer that is less than 83.

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The small difference between my result and Melody's is that I've assumed the the total includes the interest paid at the end of the last month; whereas Melody's assumes that in the last month only the $200 deposited at the beginning of the month is added in.

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Multiply the fraction 1/26 by 100 and add a % sign.

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