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Park Forest Math Team Self-study Packet Meet #2 try-outs are November 21 Meet #2 is December 5 at Mt. Nittany from 4:30-6:30 Problem Categories for this Meet: 1. 2. 3. 4. 5. Mystery: Problem solving Geometry: Area and perimeter of polygons Number Theory: Divisibility GCF, LCM, prime factorization Arithmetic: Fractions, terminating and repeating decimals, percents Algebra: Word problems with 1 unknown; working with formulas; reasoning in number sentences Meet #2 – Algebra Ideas you should know: ! Common Fraction: ! Money Answers: 23¢ “What is One-Quarter of a dollar, minus two cents?” $0.23 ! Square Root of a Product: $.23 Not 0.23¢ Not 23 “What is the square root of 14x21 x 6?” ! Slow hard way – “Multiply it out first”: 14x21=294, 294x6=1764, “Um, Are we allowed to use calculators? How are we supposed to do this?” " Faster way – “Factor it first”: From question: 14 21 6 Factor: 2x7 3x7 2x3 Regroup factors: 2x2 3x3 7x7 =3 =7 Now do ! Answer: 2x3x7=42 ! “Five consecutive multiples of 5 have a sum of 250 …” problems: 40 45 50 55 60 Average=50 If 5 numbers add to 250, the average is 50. That’s also the middle number, so the five numbers are 40,45,50,55,60. Often the problem will ask for the 2nd number times the 4th number (45 x 55 here). Meet #2, Algebra ! “Four times the sum of a number and one is two more than seven times an amount one less than the number. Find the number.” So confusing! Make N be the number you don’t know, then translate the words to Algebra: Four times the sum of a number and one is two more than 4 x ( N + 1) -2 = Seven times an amount one less than the number 7 Or: x ( N - 1) 4 (N+1) – 2 = 7 (N-1) Find the number. Solve for N. First, distribute, then combine like terms: 4N + 4 – 2 = 7N – 7 4–2+7 9 (Add 7 – 4N to each side) = 7N – 4N = 3N 3=N Then check this answer in the original problem! “Four times (3+1) is two more than 7(3-1)” Meet #2, Algebra Yes, 14=14 ! Relatively Prime (review): Having no common factors besides 1. 100 and 99 are relatively prime, since the only prime factors of 100 are 2 and 5, and 99 has 3 and 11. Are 6 and 10 relatively prime? No, both share 2 as a factor. Are 27 and 111 relatively prime? No, both share 3 as a factor. Are 35 and 66 relatively prime? Yes, 35 is 5x7, 66 is 2x3x11. How many natural numbers less than 10 are relatively prime to 10, counting 1 as relatively prime to everything? Answer: 4: 1,3,7,9 How many whole numbers less than 17 are relatively prime to 17? P3 ! Subscripts like P3 : pronounced “P sub three” If P = {2, 3, 5, 7, 11, 13, 17, 19, 23, …} then P1 = 2 P2 = 3 P3 = 5 If Pn=19, what is n? P3 just means the 3rd P in a list. For example, if M is the set of how much money Anna, Bridget, and Caroline have, MAnna (pronounced M sub Anna) is how much money just Anna has. If T is the set of multiples of 3: T = {3, 6, 9, 12, …} then T2=6, and Tn=3n Answer to Pn=19: n=8. ! Time for some real problems from previous meets. Meet #2, Algebra Category 5 Algebra Meet #2, November, 2002 1. A plumber charged a flat fee of $50 to come to the house, plus an hourly rate for the time he spends working at the house. If the plumber stayed for three and a half hours and charged a total of $274, what is his hourly rate? Express your answer in dollars. 2. Five less than three times the sum of a number and eleven is equal to seventythree minus four times the sum of the number and six. What is the number? 2 3. The formula for the surface area of a torus (a donut) is S = 2πr ⋅ 2πR= 4π rR, where r is the radius of a cross section of the ring and R is the radius from the center of the hole to the center of the ring. (See diagram below.) Imagine a perfect chocolate frosted donut with r = 0.5 inches and R = 1.5 inches. If exactly half of this donut is covered with chocolate frosting, how many square inches of the surface of the donut is covered with frosting? Use 3.14 for ! and round your answer to the nearest tenth. Answers 1. _______________ 2. _______________ 3. _______________ Solutions to Category 5 Algebra Meet #2, November, 2002 Answers 1. 64 2. 3 3. 14.8 1. From the details given, we can write the following equation: 3.5x + 50 = 274, where x is the hourly rate. Subtracting 50 from both sides of the equation, we get: 3.5x = 224. Dividing both sides of the equation by 3.5, we get x = 64. The plumber’s hourly rate is $64. 2. The verbal sentence translates to the following equation: 3(x +11)− 5 = 73− 4(x + 6) . Distributing the multiplication over each addition, we get 3x + 33− 5 = 73− 4x − 24 and then 3x + 28= 49− 4x . Adding 4x and subtracting 28 from each side of the equation, this becomes: 7x = 21. x = 3 is the solution to this equation, so the number must be 3. 3. Substituting 0.5 for r and 1.5 for R in the formula S = 4π 2rR, we get S = 4⋅ π 2 ⋅ 0.5⋅1.5. Since 4⋅ 0.5⋅ 1.5 = 2⋅ 1.5 = 3, the surface of the whole 2 donut is S = 3⋅ π and the surface of half the donut 1 3 S= ⋅π 2 2 would be 2 . Using 3.14 for !, we get: 3 3 ⋅ 3.142 = ⋅ 9.8596= 3⋅ 4.9298= 14.7894 2 2 . Rounding this to the nearest tenth, we can say that about 14.8 square inches of the surface of the donut is covered in chocolate frosting. Category 5 Algebra Meet #2, November 2004 1. Five consecutive multiples of 11 have a sum of 1155. What is the product of the second and the fourth of these five numbers? 2. At a school play, tickets for students cost $3 and tickets for adults cost $5. If 620 tickets were sold and a total of $2660 was made, how many adults bought tickets for the play? abc can be used to calculate the radius of a circle that 4 ⋅ ATriangle is circumscribed around a triangle, where a, b, and c are the side lengths of the triangle and ATriangle is the area of the triangle. The formula 3. The formula R = ATriangle = s (s − a )(s − b)(s − c) can be used to calculate the area of a triangle, where a, b, and c are again the side lengths of the triangle and s is the semiperimeter (half the perimeter). Find the radius of the circle that circumscribes a triangle with side lengths 13, 14, and 15 units. Express your result in lowest terms as a common fraction, not a mixed number. b a Answers 1. _______________ 2. _______________ 3. _______________ R www.Imlem.org c Solutions to Category 5 Average team got 9.17 points, or 0.8 questions correct Algebra Meet #2, November 2004 Answers 1. 53240 2. 400 3. 65 8 1. If five consecutive multiples of 11 have a sum of 1155, then their average is 1155 ÷ 5 = 231, which is also the middle number. The five multiples of 11 are: 209, 220, 231, 242, and 253. The product of the second and forth of these is 220 × 242 = 53240. 2. If the same number of tickets had been sold to students and adults, the average ticket price would have been $4 and the total ticket sales would have been $4 × 620 = $2480. Since the total ticket sales was $2660, we know that there were more adults than students. The difference $2660 – $2480 = $180, tells us that there were 180 more adult tickets than student tickets. Subtracting these excess adults tickets from the 620 tickets, we get 440 tickets that were sold equally to adults and students. Thus there were 220 student tickets sold and 220 + 180 = 400 adult tickets. 3. First we need to plug the values 13, 14, and 15 into the equation for the area of the triangle. The semiperimeter is s = (13 + 14 + 15) ÷ 2 = 42 ÷ 2 = 21. ATriangle = 21(21−13)(21−14 )(21−15) = 21⋅ 8 ⋅ 7 ⋅ 6 = 2 4 ⋅ 32 ⋅ 7 2 = 22 ⋅ 3 ⋅ 7 = 84 Now we can use the first formula to find the radius of the circumscribed circle. 13⋅14 ⋅15 13 ⋅ 5 65 R= = = . 4 ⋅ 84 4⋅2 8 www.Imlem.org Category 5 Algebra Meet #2, December 2006 1. Use the five equations below to find the value of A. A 2 = 92 − B B=C+5 D 4 D = 3E 4 2C = E 5 = 32 2. The formula for the area of an equilateral triangle with side length s is s2 3 A= . A regular hexagon can be subdivided into six equilateral triangles as 4 shown in the figure below. If the area of a regular hexagon is 600 3 square centimeters, how many centimeters are in the side length of the hexagon? 3. The sum of three consecutive multiples of 29 is equal to the sum of four consecutive multiples of 9. If the smallest of the four multiples of 9 is 117, what is the value of the greatest of the three multiples of 29? Answers 1. _______________ 2. _______________ 3. _______________ www.imlem.org Solutions to Category 5 Algebra Meet #2, December 2006 Answers 1. 9 (or -9) 2. 20 3. 203 1. Starting with the last equation and working our way back up, we find that E = 2, D = 48, C = 6, B = 11, and A = 9. -9 is also an acceptable answer, as (-9)2=81. 2. First we will equate the area we are given with six times the formula for the area of an equilateral triangle. Then we will solve for the side length s. !s2 3$ 600 3 = 6# & " 4 % Dividing both sides of the equation by 6 and by 3 , we get s2 100 = . Next, we multiply both sides of the equation by 4, 4 which gives us 400 = s 2 . Since 202 is 400, the side length of the hexagon must be 20 centimeters. 3. The four consecutive multiples of 9 must be 117, 126, 135, and 144. Their sum is 522. Dividing 522 by 29, we get 18. Students with correct answer in a Since 5 + 6 + 7 = 18, we can figure out that the three cluster of 6 schools: consecutive multiples of 29 must be 5 × 29 = 145, 6 × 29 = 174, and 7 × 29 = 203, which is the greatest. 1. 34/36 2. 23/36 3. 25/36 . www.imlem.org Category 5 Algebra Meet #2, December 2008 1. The sum of 5 consecutive odd numbers is 105. What is the largest of the 5 numbers? 2. Shandra and Terri are sisters who were both given the same amount of money by their mother. Shandra was able to triple her money doing chores, while Terri spent 6 of her dollars. Shandra now has 4 times as much money as Terri. How many dollars did Terri’s mother give her? 3. The sum of the first n natural numbers is known as the nth triangular number. "$"%&' The formula for the nth triangular number is !" # . The sum of the ( first n cubic numbers is equal to the nth triangular number squared. If the sum of the first n cubes is 6084, what is the value of n? Answers 1. _______________ 2. _______________ 3. _______________ Solutions to Category 5 Algebra Meet #2, December 2008 1. By calling the middle of the five odd numbers !, we can use the equation below to find the middle number "! # $% & "! # '% & ! & "! & '% & "! & $% (105 )! ( *+) , ! ( '* If the middle number is 21, then the five numbers are 17, 19, 21, 23, 25. Answers 1. 25 2. 24 3. 12 2. Calling the amount of money Terri and Shandra started with -, we can use the equation .- ( $"- # /% , .- ( $- # '$ , - ( '$. 3. The sum of the first n cubes is equal to "01 %2 , so the first thing we need to figure out is the square root of 6084. We can estimate this by noticing that 3+2 ( /$++ so we know the square root is less than 80. Since the number ends in 4 its square root must end in 2 or 8. So the square root of 6084 is either 72 or 78 (since we know it is an integer). 722 = 5184 and 782 = 6084. We now know that 1"156% 01 ( 43 ( , *)/ ( 7"7 & *% , *' 8 *. 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