I hope that after all of you finish Mathcounts, you keep up with what you learned here. You’re all really excited about how cool math is, and that’s something I really enjoy seeing. Because believe it or not, it is the frantic studying, the voracious scrambling for problems, it is that kind of math that really makes you good at it. It’s getting engaged with every problem that meets your eyes. That’s why I’m giving you these problems and going over them with you. And if you have any questions, I would love to discuss them with you in the comments ad nauseam.
This will be the last daily quiz before the competition. Try to get as high a score as you can on it (both the quiz and the competition) – the problems might be hard, but remember that you can always think of a way to make sure your answer is right! Good luck, and have fun at state!
The scores as of this morning are as follows:
Solutions to yesterday’s quiz.
21. A useful tidbit to know before solving this problem is that only perfect squares have an odd number of factors. This is quite simply because factors come in pairs, but in a perfect square one factor is repeated. So by listing the factors of the first several perfect squares, we find that the answer is 144.
22. As you know, the area of a parallelogram is the product of the base and height. We’re given the base, but the height has to be calculated from the other side length of the parallelogram. If you draw the picture, you can see that the 2-inch side is a hypotenuse of a 30-60-90 triangle. Therefore, the height is inches. Now we can find the area, square inches. (Everyone missed this problem, but I think for most of you it was the special right triangle triangle that screwed things up. Remember, 1-2-square root of 3).
23. Like we solved #9 earlier, we need to write an expression that represents the combined rates of both cores. The rate of the first core is 1.6 million computations per second, while the second’s is 1.5 million computations per second. They add up to 3.1 million computations per second. Writing an equation using the desired 8 million computations, , we find that the answer is 2.6 seconds.
24. Let x be the width of the pool, and x + 4 the length of the pool. We need to write an equation representing the area of the pool and deck area: (x + 10)(x + 4 + 10) = 1440. Honestly, at this point you can choose whether you want to solve a quadratic or not. If you don’t, you could list factors of 1440 that have a difference of 4. You’ll come up with 36 and 40, or x = 26. If you do write a quadratic, you’ll get x2 + 24x – 1300 = 0. Then you’ll look for factors of 1300 that have a difference of 24, and come up with 50 and 26. Either way, the side lengths of the pool are 26 and 30, and the perimeter is 112.
25. I sincerely apologize for not doing this before with you guys, I just assumed you knew how to do it. You need to fill out a table with rows for the solution before and after diluting. The first column will represent the percent of hydrochloric acid (HCl), the next the size of the solution, and the next the amount of HCl. Remember that the unknown is the amount of solution that we need of the concentrated HCl.
|Substance||Percent HCl||Solution (mL)||Amt. HCl|
Now, we know that the amount of HCl has to be the same before and after, since Ms. Clements just added water. So setting the two expressions equal to each other, 0.80x = 50 and x = 62.5.
26. Blaise wrote each factor in the prime factorization of 10! on an index card (separately, so two factors of 3 would be written on two index cards). If Caroline selects a card from this deck, what is the probability that she will choose a 3? (10 pts.)
27. Srikhar wants to get from building A to building B in New York City, and the way to get there is through roads that are arranged in a grid pattern. Building B is five blocks east and two blocks north of building A. Provided that the shortest distance to walk is 7 blocks, how many routes can Srikhar take that will be the shortest distance possible? (10 pts.)
28. You just lost all your financial records! Imagine that you have $15,000 in a CD that has been earning 6.1% interest, compounded annually since 2001. (Your parents deposited the money for you at your birth, presumably.) Use your math skills to figure out much money they deposited back then without your financial records. Don’t worry about inflation, by the way.
29. Bonus. This is a rerun of problem 25, since I didn’t teach you guys this before. So Ms. Clements is back at the lab, and she’s using silver nitrate this time. She has a bottle of concentrated silver nitrate (80%) and dilute silver nitrate (10%). How much of the 80% solution does she have to put into the volumetric flask (along with the 10% solution) to get 500 mL of a 40% solution? Express your answer to the nearest tenth of a milliliter. (Hint: fill in the below table using the information you know, and follow the general way we did #25.) (20 pts.)
|Substance||Percent AgNO3||Solution (mL)||Amt. AgNO3|
|10%||500 – x|
30. Bonus. The product of one less than four times a number and three more than the number is 17. What is the sum of the two possible values for the number? Express your answer as a common fraction. (20 pts.)