3 questions

[cheeyounglee]

s = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - 1/8 + 1/9 - 1/10
Column A: s
Column B: 1/2
Answer: A

If one number is chosen at random from the first 1,000 positive integers, what is the probability that the number chosen is a multiple of both 2 and 8?
B) 1/8

Two different integers are to be selected from a set of 10 different integers in which half of the integers are even and half are odd. How many of the 45 possible selections consist of one even and one odd integer?
E) 25

Thank you!

[jen]

s = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - 1/8 + 1/9 - 1/10
Column A: s
Column B: 1/2


Hi Chee,

In the first question, you don't really need to calculate s.

s = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - 1/8 + 1/9 - 1/10

Simplify just the first two terms. 1 - 1/2 = 1/2, so:

s = 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - 1/8 + 1/9 - 1/10

Now group the rest: adding 1/3 and subtracting 1/4 makes the number LARGER. Then adding 1/5 and subtracting 1/6 still makes the number LARGER.

Every pair of two terms results in a net INCREASE in s.

So, A is constantly getting bigger and bigger than 1/2.

[jen]

If one number is chosen at random from the first 1,000 positive integers, what is the probability that the number chosen is a multiple of both 2 and 8?

In answer to your second question, a "multiple of 2 and 8" is really just a multiple of 8, since 2 is already included in 8.

So, 1 in every 8 numbers is a multiple of 8.

Since 8 goes into 1,000 evenly, the answer is just 1/8.

(If 1,000 were instead some number that weren't a multiple of 8, we'd have a remainder to contend with -- for instance, if 1,000 were 15, the answer would be 1/15, since the list would actually only contain 8 and wouldn't reach up to the next multiple of 8, which is 16).

This question would be a lot more interesting if it said something like "a multiple of both 7 and 11." Since 7 and 11 don't share any factors, the smallest multiple of 7 and 11 is 77, so you'd be counting how many multiples of 77 are between 1 and 1,000.

(That's more information than you probably needed here!)

[jen]

Two different integers are to be selected from a set of 10 different integers in which half of the integers are even and half are odd. How many of the 45 possible selections consist of one even and one odd integer?

If there are five even and five odd numbers, then you can pair them up in 5 x 5 = 25 different ways.

Any time you're picking ONE out of a number of things and then ONE out of some other number of things, you can just multiply -- for instance, a man has 2 pairs of pants, 3 shirts, and 5 hats. If he wears one of each, how many different outfits can he make? (It's just 2 x 3 x 5 = 30).

Good luck on your test!
Jennifer