Bases and Rates 8/22/11 (from random tests that I do not care to disclose at 11:00 at night)

1. Bases and Rates8/22/11 (from random tests that I don’t care to disclose at 11:00 pm but I still want to credit them) • Write 4410 in base 5. Write 6610 in base 11. Write 100 in hexadecimal notation. Write A3B (hexadecimal) in base 10. • What is the last digit of the product (in base 10) 10012 x 13435? • What is the largest base 10 number that can be expressed as a three digit base 5 number? • Given 96 = 531,441, write 531,440 in base 9. • Balanced tertiary is a system in which numbers are written as 1 (negative 1), 0, and 1 with the base 3. Look on board for example. Calculate (1100)(11) + (111) and write the answer in balanced tertiary. • Bob can save the world in 5 hours. THE SUSHINATOR can save the world in 3 hours. How long will it take the two of them to save the world together (even though we all know THE SUSHINATOR can do it better)?

2. Bases and Rates8/22/11 (from random tests that I do not care to disclose at 11:00 at night) • Ms. Harris and Mr. Dimsdale paint a room for 4 hours. Then, Mr. Michael comes and helps them finish it 2 hours later. If he hadn’t come, it would have taken Ms. Harris and Mr. Dimsdale 5 more hours. How long would it take Mr. Michael alone? • Cool problem. Two cats, each traveling 10 ft/sec , run toward each other from 500 ft apart. As they run, a flea flies from the whiskers of one cat to the other’s at 25 ft/sec. (Crazy, isn’t it?) The flea flies back and forth until it is crushed when the cats collide. How far did the flea fly? • From AOPS. A canoeist paddled upstream for 2 hours, then downstream for 3. The rate of the current was 2 mph. When he stopped, the canoeist realized he was 20 miles downstream from his starting point. How many hours will it take him to get back to the starting point?

3. Key • Write 4410 in base 5. Write 6610 in base 11. Write 100 in hexadecimal notation. Write A3B (hexadecimal) in base 10. Make your chart, table, etc. How many 25’s fit into 44? Hopefully, you’re thinking 1, so subtract (1 x 25) from 44. How many 5’s fit into 19? …. 3! Fill that in. If you used up (25 + 15) of 44, how much do you have left? That’s right. Four. Fill that in and you have your answer, 1345  6011 Do the same process again. How many 11’s go into 66? What’s left? 6416 Make sure you know that hexadecimal is base 16… It’ll pop up every once in a while. So do the same process but backwards. How many 16’s are there in 100? How much of the 100 is left? 10 x 162 + 3 x 161 + B x 160 = 2560 + 48 + 11 = 261910 Here’s where the problem is with base 16… The highest “digit” is 15, but how do we represent that? As usual, when we can use numbers, we use letters, so 10 through 15 is written as A through F.

4. Key Last digit problems are pretty common at math competitions, especially in the ciphering section… All you have to do is remember that for last digit problems, ONLY THE LAST DIGIT MATTERS. • What is the last digit of the product (in base 10) 10012 x 13435? Basically, do NOT convert all these numbers to base 10 and then multiply… Do that as a last resort, and only if you have a wild boar and a dinosaur chasing after you. What you should do is find the last digit of each of the numbers in base 10 and then multiply. What does 4 x 6 end in? The product is 24, but it ends in 4. What about 464626783276427344 x 2346384627364376? Yeah, that ends in 4 too, no matter what. So let’s try using that rule now… What does that number end in? You have 8 + 1 = 9…. That was pretty easy. Here’s where it gets a little trickier. Writing this number in base 10 is really annoying, so let’s try our last digit trick… You have one 125, three 25’s, four 5’s and three 1’s. Have you noticed that 5 multiplied by any odd number ends in a 5 and that if multiplied by an even number, the product ends in a 0? So let’s see what you have (last-digit wise)… 5 + 5 + 0 + 3 = 13. We’re multiplying the numbers right? So what’s the last digit of 9 x 13? You got it. 7

5. Key Here’s a question that may seem a bit confusing if you don’t watch out for math team wording. The people who write questions for tests are often trying to trick you… So let’s look at this question. What does it mean? The question is asking for the largest base 10 number that can be written as a 3-digit base 5 number. The first number you might be think is 999. NO. Sorry, I didn’t mean to be that harsh, but remember, what’s the largest 3 digit number in base 5? Think about our number system (base 10)… What’s the highest digit you can get to? Congrats if you were thinking 9!! So now take that information and apply it… What’s the highest digit possible in base 5? Yeah, it’s 4! So what should the largest 3 digit base 5 number be? Look below for the answer… • What is the largest base 10 number that can be expressed as a three digit base 5 number? 4 x 52 + 4 x 51 + 4 x 50 = 100 + 20 + 4 = 12410 • 12410

6. Key This problem seems really complicated, but just draw out your table and think it through… Where does 96 show up in a base 9 table?? • Given 96 = 531,441, write 531,440 in base 9. Now we need to subtract 1 from that number… But remember, the highest digit in base 9 is an 8. (Notice how that fact comes up a lot?) 8 8 0 8 8 8 10 1 0 0 0 0 0 0 Method 1 – The Long Way: Wherever you would normally put a 9, put an 8! - 1 8 8 8 8 8 8 Method 2 – Using Another Example: What if you did 10,000 – 1 (in base 10)? You would get 9,999, right? When you’re in base 9, you use 8’s instead of 9’s. Wherever you would normally put a 9, put an 8! The two methods are very similar, so it doesn’t matter which one you pick. No matter what, your answer is… 888,8889

7. Key • Balanced tertiary is a system in which numbers are written as 1 (negative 1), 0, and 1 with the base 3. Look on board for example. Calculate (1100)(11) + (111) and write the answer in balanced tertiary. Example: 1011 = (1)33 + (0)32 + (-1)31 + (1)30 = 25 For this problem, I’d suggest converting to base 10, doing the calculations, and then converting back to balanced tertiary  1100 = (1)33 + (-1)32 + (0)31 + (0)30 = 18 11 = (-1)31 + (1)30 = -2 111 = (1)32 + (-1)31 + (1)30 = 7 Final result: (18)(-2) + 7 = -29 How do you convert back to balanced tertiary… I’d use a table. We’re trying to get a negative number, so try placing -1 (1) somewhere… I’d pick 33 because well, it’s the closest… You’re already at -27. But how do you get -2? 2 is not a number you can use in this system… What about a combination of -3 + 1?

8. Key 1 15 1 1 1 1 1 1 1 1 8 1 S 15 8 S B 3 B 3 B 5 5 S The above equations aren’t really necessary to solve the problem, but hopefully you’ll understand them… Basically I multiplied the # of hours to save the world by the amount of world I saved in 1 hour. Do you see why that equals 1? Let’s have B = 5 hours. That’s how long it takes for Bob to save the world. Let’s have S = 3 hours because that’s how long it takes for me to save the world. And can we all agree that we are saving one world? • Bob can save the world in 5 hours. THE SUSHINATOR can save the world in 3 hours. How long will it take the two of them to save the world together (even though we all know THE SUSHINATOR can do it better)? 5( ) = 1 = 3( ) = 1 = Now let’s combine our talents. We don’t know how long it takes… We’ll call that ‘x’. Now we need to multiply by how many worlds we can save in an hour… What is that? Just add the two fractions above!! And how many worlds are we saving? One. Now you have your equation, so just solve!! x( )= 1 x( + )= 1 x( + )= 1 x = hours

9. Key 1 1 1 1 1 1 1 M D H D H D H Remember, word problem = equations. Don’t stress out when you see one. Break it into pieces figure what everything means and fit some sort of variable to it… Here’s a good example: • Ms. Harris and Mr. Dimsdale paint a room for 4 hours. Then, Mr. Michael comes and helps them finish it 2 hours later. If he hadn’t come, it would have taken Ms. Harris and Mr. Dimsdale 5 more hours. How long would it take Mr. Michael alone? 2 hours for all three of them working together. Ms. Harris and Mr. D work for 4 hours. The fractions in parentheses tell you how much of a room they finish in an hour when they are working together… Refer to the previous problem for an explanation. 4( + ) + 2( + + )= 1 9( + ) = 1 It would have taken Ms. Harris and Mr. D 9 hours in total if Mr. Michael hadn’t shown up… There’s a problem… you don’t know the values of ANY of the variables! So what do you do?? Go to the next slide for the answer 

10. Key 1 1 2 1 2 1 1 2 1 1 1 1 1 1 1 1 1 M M M M H D H D H D 9 D 3 9 9 H 3 Math Team = “Thinking out of the box” – Jen Palamuso • Ms. Harris and Mr. Dimsdale paint a room for 4 hours. Then, Mr. Michael comes and helps them finish it 2 hours later. If he hadn’t come, it would have taken Ms. Harris and Mr. Dimsdale 5 more hours. How long would it take Mr. Michael alone? Let’s see… We were at the part where we didn’t know any of the variables. Here’s a neat math team trick… You will ALMOST NEVER find 1 variable at a time. Usually, you’re going to have to substitute a combination… So look at the two equation. What parts are the same? 4( + ) + 2( + + )= 1 I hope you’re thinking that Ms. Harris and Mr. D seem a little repetitive… + = Solve for the sum… Now you have a value you can plug in for the two fractions!! 9( + ) = 1 4( ) + 2( + )= 1 Hey, cool! Now you only have 1 variable!! Solve for M!!!!! + = 1 M = 6 hours =

11. Key If you are like me, you were probably really confused when you first read this problem… but no worries! Remember to break apart the problem!! What are you looking for? A distance. What do you have? A whole bunch of speeds. Let’s get to work! • Cool problem. Two cats, each traveling 10 ft/sec , run toward each other from 500 ft apart. As they run, a flea flies from the whiskers of one cat to the other’s at 25 ft/sec. (Crazy, isn’t it?) The flea flies back and forth until it is crushed when the cats collide. How far did the flea fly? d = rt That is pretty much all you need to know. So these cats are each traveling at 10 ft/s, right? It would take each cat 50 seconds to run the whole 500 meters, but remember, they both are covering that distance, so essentially, it takes half the time: 25 seconds. Does that make sense? Now look at the huge equation. You have the time the flea is flying, and you have the rate, or the speed… What do you do next? MULTIPLY!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (25 s)(25 ft/s) = 625 m Note: It didn’t take much to solve this problem… I even had enough room to put an equation in 199 point font. :P

12. Key • From AOPS. A canoeist paddled upstream for 2 hours, then downstream for 3. The rate of the current was 2 mph. When he stopped, the canoeist realized he was 20 miles downstream from his starting point. How many hours will it take him to get back to the starting point? Here’s the key to these problems: You have to account for the current. So follow along as we define these variables… x = speed of canoeist y = speed of current = 2 Pretty straightforward… So what is the speed upstream and downstream? Why? Let’s think about it… If you’re going upstream, the current slows you down, but if you’re going downstream, it speeds you up, so you add the two values. x – y = speed upstream x + y = speed downstream I prefer drawing out the situation… Let’s break the problem apart and write the equation! 2(x – y) – 3(x + y) = -20 2hrs Why negatize the 3 and the 20? They’re in different directions… Always consider everything relative to the starting point. Now, all you have to do is plug in 2 for y and solve for x! 3hrs 20 mi 2(x – 2) – 3(x + 2) = -20 …. x = 10! But wait! You’re not done! You have to find how long it will take for him to get back… Remember, he’s traveling upstream… so use that speed! (20 is positive here because you’re not going back and forth – just one direction so you don’t need to keep track ) t(10 – 2) = 20 …. t = 2.5 hours!