[High School - Precalculus] Skills Review

[insanefreddy926]

Alright, I got this paper with some selected problems on it, and a few of them are a little confusing to me.

#2: A regular hexagon is inscribed within a circle of radius 1 unit. Find the ratio of the area of the circle to the area of the hexagon.
I got 2(pi)sqrt(3)/9 for the answer, but I'm just wondering if I did it right or if I'm completely off.

#7: How many positive integers are factors of 5400?
I got 47 for this one, but I did it "the long way." I was wondering if there was a shorter way of doing this than finding all of the factors and counting them up.

And finally #10: Express 0.31245 (with the "45" at the end repeating) as the ratio of two positive integers.
The decimal to fraction button on my calculator failed at this one.

[dag12]

#2: I got the same answer as you. Good Job? (Assuming I didn't screw up either...)

#7: Start by prime factoring 5400.
5400 = 2^3 * 3^3 * 5^2
So, the factors of 5400 will be combinations of these numbers.
How many combinations can you make?
2 can appear 4 ways [0,1,2, or 3 times], 3 can appear 4 ways, 5 can appear 3 ways.
So, the number of combinations = 4*4*3 = 48.
Perhaps you missed one?
(You know 47 is wrong immediately because only square numbers have an odd number of factors, and 5400 is not a square number)

#10: let n = 0.31245454545...
then 100n = 31.24545454545...
subtract one from the other, 99n = 30.933
n = 3437 / 11000
So, 0.312454545... = 3437 / 11000

[sumzup]

Alright, #2 is right.

#7: The short way involves first getting a prime factorization; after you do so, you see that the prime factorization is 2^3 * 3^3 * 5^2.

Now, think about how you would get all of the factors. You have a choice between choosing 2s, 3s, and 5s. You multiply the number of ways you can pick each one together, and you should get the right answer. Thus, the answer is 4 ways to pick 2s (pick zero 2s, one 2, two 2s, or 3 twos) * 4 ways for 3 * 3 ways for 5 (3 and five have the same logic as 2). So the final answer is 4*4*3 = 48. (ouch, you were one off!)

#10: Suppose 0.31245 (45 repeat) = x. Thus, 312.45 (45 repeat) = 1000x. Thus 312 and 45/99 (I'm sure you know this) = 1000x. Convert to an improper fraction --> 30933/99 = 1000x. Thus, x, the fraction form of the decimal, is equal to 30933/99000 which is also, 3437/1100.

Edit: Whoa, I didn't simplify, and used 100x instead of 1000x. Fixed. Also, damn you dag12 for posting before me.

[insanefreddy926]

Ahhh, I started with the prime factorization for #7 but then had no clue what from there, heh. And I just had no clue at all with #10 but I understand now. Thanks!