Welcome to The Puzzler. Every week, there are 2 new puzzles related to my passion: math, logic, and thinking. The first puzzle will be The Puzzler Quik, meant for those who crave something fun-sized. The second puzzle will be The Puzzler Think, meant for those who love to ponder. The answers will be posted in next week’s column. Don’t forget to submit your answer for a potential shout-out in the next edition of the Puzzler.

Puzzler Quik

50 people(including you) enter the giveaway That ThePuzzlr is hosting. If 2 prizes will be given out, what is the probability that you will be one of the winners? Add the numerator and denominator of your answer!

Puzzled?

Here is a hint to help you get going in the right direction:

Want to enter the giveaway? Enter here!

Hint 2:

No hint this time!

Puzzler ThinK

Bill Gates is considering a business investment. The investment has a 30% chance that his anticipated returns would be -$9,000, a 45% chance that his anticipated returns would be $15,000, and a 25% chance that his anticipated returns would be $30,000.

Find Bill’s expected return from this investment.

Puzzled?

Here is a hint to help you get going in the right direction:

Don’t know about Expected Value, open the next hint for a major hint!

Hint 2:

Watch the video on Expected Value at https://youtu.be/TCFoRx2R2ew. It has an explanation to this question!

ANSWERS TO LAST WEEK’S PUZZLERS

Last Week's Puzzler Quik

Ariana from Wyoming was the winner of this shout-out!

From the list above and the note about only whole-dollar amounts we note that if Mushrooms & Onions are $2 together they must be $1 each. From there it is simple math to figure out the following.

Onions $1
Mushrooms $1
Pepperoni $3
Black Olives $2
Extra Cheese $4
Broccoli $2
Hot Peppers $3
Pineapple $4

Last Week's Puzzler Think

Consider the numbers $\frac{1}{30}, \frac{2}{30}, \ldots, \frac{300}{30}$ . We want to find the sum of those for which the numerator is relatively prime to $30$ (in particular, note that this step will eliminate the final term $10$ ). Between $1$ and $300$ inclusive, there are $150$ multiples of $2$ , $100$ multiples of $3$ , $60$ multiples of $5$ , $50$ multiples of $6$ , $30$ multiples of $10$ , $20$ multiples of $15$ , and $10$ multiples of $30$ . By the inclusion-exclusion principle, there are $150+100+60-50-30-20+10=220$ numbers between $1$ and $300$ that are not relatively prime to $30$ , so there are $300-220=80$ numbers which are relative prime to $30$ .

Note that $k$ is relatively prime to $30$ if and only if $300-k$ is relatively prime to $30$ . Therefore, for the $80$ numbers less than $10$ that are of the form $\frac{k}{30}$ for which $k$ is relatively prime to $30$ , we can partition them into $40$ pairs $\Big(\frac{k}{30}, \frac{300-k}{30}\Big)$ , where each pair sums to $10$ . The answer is $40 \cdot 10 = \framebox{400}$ .