Weekend Challenge - May Day edition

 NOT A REAL PLANE ON FIRE. Source.
Appropriately for May Day, this question will cause you to send distress signals. Prize this week: you get to strut around all weekend telling people how much smarter you are than they, and they have to humbly agree. You may force up to three strangers to "kiss the ring."
Note: Figure not drawn to scale.

In the figure above, P is the center of a circle, and Q and R lie on the circle. If the area of the sector PQR is 25Ï€3 and the length of arc QR is 5Ï€2, what is the measure of ∠RPQ?

UPDATE: props and ring kisses to Elias, who answered on Facebook, and to JD, who did his thing right here on the site (and spelled out the solution quite elegantly). Side note: I'd love to be able to integrate those two commenting systems -- is that possible?

Solution below the cut:

It's crunch time, and I am duly crunched.

I know things have seemed quiet around here lately, and for that I'm sorry. The May SAT, as you're well aware I'm sure, is 8 days away, and I've been very much in demand in my real life. It has, regrettably, cut a bit into my blogging time. That said, I have still been busy behind the scenes, and I thought I'd point out a few things that I've posted to the site that you might have missed, and let you know what I'm working on.

1. PWN the SAT Q&A (qa.pwnthesat.com). I've been looking for a way to make this site a bit more interactive, and this is a step in that direction. It's a tumblr site (I'm completely inspired here by The YUNiversity, AKA Broseidon - god of the broceans, whose tumblr is a must-read for meme-ified grammar tips, and who knocked my socks off when he posted the image at the top of this post) where you can ask me questions about the SAT. Debbie of the amazing Perfect Score Project got things started with a great question, my response to which I'll reproduce here since it's very relevant to how you should be spending your time this week if you're taking the May test:
Remember that in general, you can improve your score 100 points by making one fewer mistake per multiple choice section. Two fewer mistakes per section? You’re looking at about 200 points. So you don’t need to cover everything at once in the next week to have an effect on your score.

In fact, you might be better off concentrating on a few of your weakest areas (especially if they’re common question types — right triangles, for example) than trying to study everything.

If you’re looking to identify some common weak areas, try this drill. Note that, although it’s 20 questions, it’s MUCH harder than a typical 20 question section on the SAT, so don’t try to limit yourself to 25 minutes. The answer key will point you in the direction of some tips specific to the kinds of mistakes you’re making. Iron a few of those out, and you’re well on your way! Good luck!
It's an experiment, and I haven't even settled on a design yet, but hopefully it'll be another way for us to have a conversation about this beast of a test. Get at me.
2. The Math Section. In an effort to make this site more useful, I've organized all my math posts to date in an outline, and posted it at the top of the page so it's always easy to find. I'm getting a lot of use out of it myself, so hopefully you will too. On a related note, I've also been going back through some of the older posts and trying to bring them up to snuff. I began this site on a whim a few months ago, but it has evolved over that time into something I really enjoy and care about. Some of the older posts don't yet reflect the level of quality I'm going for now, so I've been working back through them. I appended a few good practice questions to the circles post this week, for example.

Keep probability questions as simple as you possibly can. Please.

Disclaimers: 1) Probability problems are some of the SAT's most difficult, but they're also some of the most rare. There's a pretty decent chance you won't see a very hard question like this on your test, so prioritize your prep time; don't worry too much about this stuff until you've really nailed the basics. Ironically, this is my most involved post to date, but it corresponds to the smallest point potential. 2) As you surely know if you've done probability in school, as involved as this is it really only scratches the surface of a concept that can go VERY far down the rabbit's hole. I'm only covering the kind of stuff you might see on the SAT. If you're looking for a more complete treatment of probability, try this. 3) I'm assuming some rudimentary knowledge of combinations here.

The basics (if you're comfy with the basics, skip to the big reveal.)
The probability of an event is equal to the number of ways that event can occur divided by the total number of possible outcomes.

So the probability that you will be chosen at random to represent your 30-person class in the hot dog eating contest is 1/30. Likewise, the probability that your frenemy Ashley will be chosen for the contest is 1/30.

What if I asked you about the probability that EITHER you OR Ashley would be chosen? Well, now the event we're concerned with happens in 2 of the 30 possible outcomes. It would be satisfied if you were picked, or if Ashley was. Therefore, the probability is 2/30, which simplifies to 1/15. When there are multiple, mutually exclusive ways an event can occur, ADD the probabilities of each way to get your overall probability. Again, this only works if you're talking about events that are mutually exclusive. In other words, it only works if both events can't happen at the same time. Example: if you buy lottery tickets, since only one ticket can win, every ticket adds to your probability of winning.

This is important, so make sure you're solid. Hover your mouse over the following examples to ensure this won't leak out of your brain.
• What's the probability of rolling an even number on a standard 6-sided die?
• What's the probability of rolling a number less than 6 on a standard 6-sided die?
• What's the probability of rolling a prime number on a standard 6-sided die?
• What's the probability of flipping a coin and getting either heads or tails?
• What's the probability of flipping two coins and having one or the other come up heads?
OMG this is so easy! Well buckle up, kids. Things are about to get more interesting.

Weekend Challenge - Chocolate bunny edition

Prize this week: you get to decide, once and for all, whether Marshmallow Peeps or Cadbury Creme Eggs are the better candy. No longer will disagreements be chalked up to "difference of opinion." Once you issue your decree, it will be a matter of record. Minor dissent will be tantamount to outright prevarication. Think you can handle all that power?

Right, the question. Here we go.
Four adjacent offices are to be assigned to four employees at random. What is the probability that Scooter and The Big Man (two of the employees) will be placed next to each other?

UPDATE: Props to Elias, who nailed it on Facebook. He's going with whichever candy has the least packaging (a man after my own sustainable heart). I'm thinking that's the Creme Egg.

Solution below the cut...

I haven't gone dark.

If you visit regularly, you've maybe noticed that there's only been one post so far this week. I'm sorry for that. I've been concentrating on trying to put together good Reading Comp passages and questions for you, and honestly it's really difficult to do it right. Also, Portal 2 came out on Tuesday. That said, there will still be a weekend challenge question up later today, and hopefully I'll be back to posting more regularly next week.

I've also been meaning to thank a few people for their kind words and links, and I might as well do it now since this post is as yet devoid of any useful information.

Practice reading for the main idea.

 Jesse Lacey of Brand New. Found this here.
It's important, on the SAT reading section, to be able to nail down the main idea of a passage, even if you're not sure what every single word means. There's no quick remedy for this if you're struggling; you're just going to have to practice. A lot. If the only reading you do is on practice tests, you're dooming yourself to failure. You're going to have to start reading everything you can get your hands on, and reading it actively; asking questions as you go and making sure you're understanding the author's argument and point of view.

Have a look at the following passage*, and see if you can answer the questions below it.
The following passage is about "emo," a genre of popular music.
1. Based on the passage as a whole, which of the following scenarios from the sports world is most analogous to emo music?

(A) A recreational soccer league requires teams to have equal numbers of male and female players.
(B) Body checking is not allowed in women's ice hockey, but such forceful physical contact is an essential part of men's ice hockey.
(C) Television ratings for the men's college basketball tournament exceed those for the corresponding women's tournament.
(D) Professional football has many female fans, despite the fact that it is played only by men and its marketing is aimed almost exclusively at men.
(E) Most golf courses contain separate "ladies' tees" so that female golfers don't have to drive the ball as far as male golfers.

2. The author of the passage would most likely agree with which of the following statements?

(A) Emo bands have male and female fans.
(B) Singers in emo bands are immature.
(C) Music critics should not take emo seriously.
(D) Many emo bands would be better if they had more female members.
(E) Most people who like emo music are lonely.

3. Lines 8-11 ("However ... men") serve primarily to

(A) explain a population's distaste for emo music
(B) highlight a surprising fact
(C) condemn a style of songwriting
(D) dispute a previous argument
(E) suggest a solution to a problem

4. According to the passage, with which of the following characteristics of emo music do female fans identify most strongly?

(A) Its "androcentrism" (line 1)
(B) The "women in emo bands" (line 2)
(C) The "'lonely boy's aesthetic'" (lines 4-5)
(D) Its "litany of one-sided songs" (line 6)
(E) Its "expression of emotional devastation" (line 13)

Answers and explanations below the cut...

Weekend challenge for 4/15

Same prize as last weekend for whoever gets this first (either on the blog, or on Facebook; you're competing against each other here). Any $5 album from Amazon.com. Ready? A small town has 3 theaters in it. Last Saturday night, the average age in the first theater was 29, and the average age in the second theater was 24. If the overall average theatergoer on that night was 36 years old and the ratio of attendees for the three theaters was 2:3:10, respectively, then what was the average age of the attendees in the third theater? Good luck, folks. I'll post the answer and contact the winner (if there is one) on Monday. Remember: you can't win if I can't contact you. UPDATE: Congrats to Codi, who nailed it on Facebook. Answer below the cut. Absolute values are rare, PWNable.  Absolute rock. Source. Disclaimer: this is really minor stuff as far as how often it appears on the SAT, so if you're looking for quick tips to really raise your score, I suggest you start elsewhere. This kind of question is pretty rare. I trust you already know the very basics of absolute value: that |5| = 5, and |-5| = 5, etc. If you don't, leave this page open and view a quick tutorial here before continuing. Ok, all caught up? Let's do this. Absolute values and inequalities Remember that |x| = 5 means that x = 5, or x = -5. You can draw similarly simple conclusions with inequalities. If I told you that |y| < 3, and y is an integer, then what are the possible values for y? There aren't many: y could equal 2, 1, 0, -1, or -2. In other words, y has to be less than 3, and greater than -3. Like so: |x| < 5 < 5 and x > -5 -5 < x < 5 Here's an example of the kind of question you'll usually get on the SAT: 1. In order to be considered "good for eating" by the La'Urthg Orcs of Kranranul, a human must weigh between 143 and 181 pounds. Which of the following inequalities gives all the possible weights, w, that a human in Kranranul should NOT want to be? (A) |w - 143| < 38 (B) |w - 162| < 19 (C) |w + 38| < 181 (D) |w - 181| < 22 (E) |w - 162| < 24 Answer, explanation, and so much more below the cut: Keep track of your units, and you'll be fine on ratio questions.  Source. So here's the thing with ratios and proportions on the SAT: they're really easy. No, seriously, where are you going? Come back! They're easy, I swear. All you have to do is keep very close track of your units, and you'll be good to go. That means when you set up a proportion, actually write the units next to each number. Make sure you've got the same units corresponding to each other before you solve, and you're home free. Pass Go, collect your$200, and spend it all on Lik-M-Aid Fun Dip.

So uh...let's try one?
1. A certain farm has only cows and chickens as livestock. The ratio of cows to chickens is 2 to 7. If there are 63 livestock animals on the farm, how many cows are there?

(A) 13
(B) 14
(C) 16
(D) 18
(E) 49
The SAT writers would love for you to set up a simple proportion here and solve:

$\large&space;\inline&space;\dpi{300}&space;\fn_cm&space;\frac{2}{7}=\frac{x}{63}$

Hooray! = 18! That's answer choice (D)! NOT SO FAST, SPANKY.

Backsolve, or figure out a much more difficult way to solve these backsolve problems.

It's important to be ever-cognizant of the fact that on a multiple choice test, one of the 5 answers has to be right. Because of this, it's sometimes possible to answer a question correctly by starting at the end, and ending at the start. Most in the prep world call this "backsolving," and it's even more powerful on the SAT because the SAT will always put numerical answer choices in order, making it even easier for us to do this efficiently. Pattern recognition, folks: it's super important.

Anyway, let's have a look at a backsolve example. The setup for this question comes from a contest winner; I apologize in advance if it's too macabre (or just plain weird) for you.
1. Rex is a dinosaur who eats kids who eat nothing but corn. He eats 1/4 of the kids on his island on Monday, 13 kids on Tuesday, and half as many as he ate on Monday on Wednesday. If there are 22 kids left on the island on Thursday, and no kids came to the island or left in a way other than being eaten in that time period, how many kids were on the island before Rex's rampage?

(A) 56
(B) 64
(C) 68
(D) 74
(E) 75

You're young, so if you pull your hair out it'll probably grow back.

Hey all. Here's your weekend challenge question. The prize this weekend: any \$5 album from the Amazon mp3 store. I can only give you the prize if I can get in touch with you (using your email or Disqus/Google/Yahoo/Twitter/Facebook account), so please don't be completely anonymous if you want the prize.

(m + n + p + 180)(q - b - d - 2e - a - c + r) = 3x + y

Based on the figure and equation above, what is x in terms of y?
Good luck, and have a great weekend! I'll post the answer Monday.

UPDATE: solution below the cut.

Figure drawn to scale? Guesstimate that ish.

Here's an important thing to remember: all figures on the SAT are drawn to scale unless indicated otherwise. In other words, if it doesn't say "Note: figure not drawn to scale," underneath it, it is drawn to scale. Most figures on the SAT are drawn to scale, which means it's a good idea to guesstimate whenever possible.

Guesstimating could mean actively trying to eyeball relative angle measures, areas, or segment lengths, or it could mean sliding pieces of the diagram around in your mind. You might still end up doing some math because guesstimating doesn't lead you all the way to an answer. But it's important that you not waste the opportunity when a diagram is drawn to scale. Let's dig right into an example:
1. The figure above depicts two intersecting diameters of two concentric circles of radius 6 and 10. If the diameters are perpendicular, what is the area of the shaded regions?

(A) 32Ï€
(B) 50Ï€
(C) 58Ï€
(D) 64Ï€
(E) 74Ï€

If you've ever sat down and taken a practice (or real) SAT, you've come across shaded region questions. They're among the most iconic question types on the test, so much so that you may find that the memory of them remains with you long after your SAT taking days have passed. True story: I had a roommate in college that used to talk in his sleep sometimes, and one time I woke up in the middle of the night to hear him plaintively moaning about shaded regions.

Should you let yourself get intimidated by a shaded region questions? ABSO-EFFING-LUTELY NOT.
Say I told you that the area of the entire blob shape in the figure above was 15, and then asked you for the area of the shaded region. It'd be cake, right?

If I know you like I think I do, you'd probably say something like: "Thank you for insulting my intelligence with this asinine question; it's 5."

All you did was recognize that since areas just add up, and you know that the unshaded areas add up to 10, the shaded region has to make up the rest of the total area. If the total area is 15, and the unshaded part is 10, then the shaded one has to be 5. Easy, yes?

So it is with all shaded region questions:

Let's try an example, shall we?
1. In the figure above, P is the center of the circle and also the intersection of the two right triangles. If the radius of the circle is 3, what is the area of the shaded region?

(A) Ï€
(B) 6Ï€
(C) 9Ï€ - 9
(D) 9Ï€ - 6
(E) 9Ï€ - 3

Weekend challenge question

Pretty easy question here, so I'll attach a prize of fairly small value, but that might still be fun: the first person to comment with the answer gets to name a character in a future word problem on this site, and what they do/sell/wear/eat. For example: "Rita is the diaper changer at a daycare center that feeds the kids nothing but corn." Of course, I have no way of contacting an anonymous commenter, so you're going to have to identify yourself. Cool? Let's goooo.

Four kids are in a room; their average age is 8 years old. Then an adult enters the room, and the average age becomes 16. A tense conversation quickly escalates, culminating in one of the children screaming "You're not even my real dad!" and leaving the room, but the average age in the room stays the same. A few minutes later, another kid leaves the room in search of a sandwich. If the last kid to leave was 4 years old, then the adult is how much older than the average age of the people remaining in the room, awkwardly staring at each other?