Here are the scores as of this morning: Srikhar, 100 points; Caroline, 140 points. Keep up the good work, and don’t forget to check your answers!
Solutions to yesterday’s quiz.
11. We simply write an equation, and solve. Remember to raise both sides to the reciprocal power when you have a fractional exponent. The answer is 243.
12. Drawing a Venn diagram to help you visualize the problem is a good idea. Let’s also think of the ninth graders as a hat with names in it. We know there are 408 different names in the hat, but there are actually 164 + 208 + 99 = 471 slips of paper in it. That means there are 471 – 408 = 63 names that are repeated at least once. Also remember that 11 names are in the hat three times, so we can subtract that from the number of repeated names to get 52 names that are in the hat at least twice. The answer, then, is 52/408 or 13/102.
13. To solve this problem, we need to know the area of the circle that Polly can reach and the area of the hexagonal yard. The area of the circle is , or square feet. We know the yard has an apothem of 30 feet, so imagine that as the height of the six equilateral triangles that make up a hexagon. Since equilateral triangles can be divided into two 30-60-90 triangles, we can use that special right triangle rule to determine that the side length of the hexagon is feet. That means the area of each triangle is , and the area of the entire hexagon is 6 times that. Dividing the two areas, we get an answer of 90.7%. Be sure to round properly, it might be counted wrong otherwise.
14. Like Srikhar observed the other day with Robert’s college math problem, sometimes the best way to solve these boggling problems is to start with smaller examples. By listing the remainders of various powers of 2 when divided by 7, you’ll find a pattern of 2, 4, 1, 2, 4, 1,… This way, it’s easy to see that the 2014th term of this sequence is 2.
15. The intersection of the four red circles with the blue square together comprise one red circle. Since the radius of the red circles is 1/4 the large circle, we know that the area will be 1/16 of the total area. Then, we must consider the area of the one green circle, whose radius is 1/8 the large circle. Therefore, its area is 1/64 of the large circle. Adding 1/16 and 1/64, we get 5/64.
16. Today, Madame Mathematique told her French students that if their class average was 93 or above on their last test, they would get a creme glacee party. The 18 students who have already taken the test had an average of 91, but 7 students still have to make up the test. What is the minimum average score those 7 students must make to bring the class average to a 93? Express your answer as a decimal to the nearest tenth. (10 pts.)
17. What is the lowest common multiple of 2x, x2y, and 3xy2? Write your answer as an algebraic expression in terms of x and y. (10 pts.)
18. The word CALCULATOR is spelled out on a refrigerator using magnets. If two letters fall off, what is the probability that they are both vowels? Express your answer as a common fraction. (10 pts.)
19. Bonus. Look at #12’s solution for help with this one. The 30 students in Mrs. Matrice’s class all have either dark hair, green eyes, or freckles. 7 of them have at least two of those traits, and 3 have all three traits. If there are twice as many students with green eyes as students with freckles, and there are four more brown-haired students than freckled students, how many students have freckles? (20 pts.)
20. Bonus. If a term is selected at random from the expansion of (a + b)6, what is the probability that its coefficient is odd? (20 pts.)