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Math Counts Test Review 2010 Chapter 7. A copy of the Sunday Times costs $0.95 more than a copy of the Sunday News. Ms. Jones buys both papers for $7.25. How much does the Sunday News cost? Let X be the cost of Sunday News X + (X + 0.95) = 7.25 2 X = 7.25 – 0.95 = 6.3 X = 3.15 11. Joshua has taken four tests. His median score is 80 points and the difference between his greatest score and his least score is 12 points. What is the maximum possible value of the mean of his scores? Let A,B, C, D be the test scores. A B C D (B + C)/2 = 80, D – A = 12, C D & A B Largest possible D = 80 + 12 = 92 with A = 80 Largest mean = (80*2+92+80)/4 = 40 + 23 + 20 = 83 17. A grandfather clock chimes at the start of each hour. Beginning precisely at six o’clock the grandfather clock chimes 6 times with the final chime beginning exactly 10 seconds after six o’clock. If the chimes are uniformly spaced, how long after 12 o’clock does the twelfth chime begin? Time in between chimes: 10/5 = 2 seconds At 12 o’clock, there are 12 chimes The time before the 12th chime: 11*2 = 22 18. A bag contains exactly six balls; two are red and four are green. Sam randomly selects one of the six balls and puts it on the table. Then he randomly selects one of the five remaining balls. What is the probability that the two selected balls are of different colors? Express your answer as a common fraction. Total # of possible selection: 6 * 5 # of desired outcome: 2 (r) x 4 (g) + 4(g) x 2(r) Probability = (8 + 8) / 30 = 16/30 = 8/15 Alternatively: P(r)(g) + P(g)(r) = 2/6 * 4/5 + 4/6 * 2/5 = 8/15 23. In rectangle ABCD, side AB measures 6 units and side BC measures 3 units, as shown. Points F and G are on side CD with segment DF measuring 1 unit and segment GC measuring 2 units, and lines AF and BG intersect at E. What is the area of triangle AEB? Note that EFG EAB FG = 6 - 1 - 2 = 3, AB = 6 Let H be the height of EFG H / (H + 3) = 3/6 = ½ 2H = H + 3 H = 3 Area of EAB = 6 *( H + 3)/2 = 18 27. If f(x) = 5x – 3, g(x) = 3x2 + 1 and h(x) = f(x) + g(x), what is the sum of the x-values for which h(x) = 0? Express your answer as a common fraction. h(x) = 3x2 + 1 + 5x – 3 = 3x2 + 5x – 2 = 0 Let X1, X2 be the roots, by sum of the root, We have: X1 + X2 = -5/3 28. Two real numbers have an average of 7. The average of their squares is 54. What is the product of the two numbers? (X + Y) / 2 = 7 X + Y = 14 ----- (1) (X2 + Y2) /2 = 54 X2 + Y2 = 108 --- (2) From (1), (X + Y)2 = X2 + 2XY + Y2 = 196 --- (3) (3) – (2): 2 XY = 196 – 108 = 88 Hence XY = 88/2 = 44 29. In triangle ABC, AB = AC and D is a point on AC so that BD bisects angle ABC. If BD = BC, what is the A measure, in degrees, of angle A? D By AB = AC, ABC = ACB B By BD = BC, BDC = ACB Hence ABC BCD We get: CBD = A Since BD bisects ABC, A = ½ ABC We have: 5 * A = 180 A = 180/5 = 36 C X + 43 = m * m ----- (1) X + 16 = n * n ------ (2) (1) – (2) m*m – n*n = 27 (m + n) (m – n) = 27 Since m, n (m > n) are integers, m+n & m – n are also positive integers Possible answers are: m+n = 27, m-n =1 -- (3) or m+n = 9, m-n =3 -- (4) From (3), m = 14, n = 13 mn = 14*13=182 From (4), m = 6, n = 3 mn = 18, but wrong X G3. The distribution of the 37 test scores in a math class is given in the stem and leaf plot where 5|6 represents 56 points. What percent of the scores are at most 5 points from the value of the median? Express your answer to the nearest whole number. There are total of 37 scores Median is the 19th -- 78 There are 8 scores that are between 73 and 83: Answer: 8/37 = 0.216 = 22 G4. If a + b + c + d = 11, 2a + 3c = 19, b + 4d = 22, 4a + d = 14 and 5b + 3c = 5, what is the value of d? Listing out the equations: a + b + c + d = 11 ------- (1) 2a + 3c = 19 ------- (2) b + 4d = 22 ------- (3) 4a + d = 14 ------- (4) 5b + 3c = 5 ------- (5) (2) + (3) + (4) + (5), we have: 6a + 6b + 6c + 5d = 19+22+14+5 = 60 ------- (6) 6 * (1) – (5): d = 66 – 60 = 6 G6. A shopper notes that a box of brand A cornflakes costs 50% more than a box of brand B and weighs 25% more than a box of brand B. According to this information, for equal weights, the cost of brand A is what percent more than the cost of brand B? Let Ca – cost of A, Cb – cost of B Let Wa – weight of A, Wb – weight of B Unit-Cost-A = Ca /Wa = (150% Cb)/(125% Wb ) = 6/5 Cb/Wb Difference in unit-cost: Ca /Wa - Cb/Wb = 6/5 Cb/Wb - Cb/Wb = 1/5 Cb/Wb Percentage of the difference = (1/5 Cb/Wb )/ (Cb/Wb) = 1/5 = 20% G8. How many ordered pairs of positive integers satisfy the equation, 1/X + 1/Y = 1/6 with x < y? From 1/x + 1/y = 1/6, we get: (x+y)/xy = 1/6 6 x + 6 y = xy xy – 6x – 6y = 0 xy - 6x – 6y + 36 – 36 = 0 (x – 6) (y – 6) = 36 Since X, Y are positive integers, x-6, y-6 are integers (x – 6) and (y – 6) are integer factors of 36 Since X < Y, we have (x-6) < (y-6) Possible values for (x-6): 1, 2, 3, 4, Answer: 4 (7, 42), (8, 24), (9, 18), (10, 15) M5. A set of five positive integers has a mean, median and range of 7. How many distinct sets whose members are listed from least to greatest could have these properties? Note: {4, 6, 7, 7, 11} is one such set to include. By 7 be the mean, median, 7 must be in the set a 7 b d c Let a, b, c, d, be differences of the other 4 numbers to 7, as shown. We have a + b = c + d, and a + d = 7 We have a + b = c + d, and a + d = 7 If a >= 5 d <=2 c + d <= 4 < a + d no fit answers If a = 4, d = 3 c = 1, 2, 3 b = 0, 1, 2 3 answers If a = 3, d = 4 b = 1, 2, 3 c = 0, 1, 2 3 answers If a = 2, d = 5 a + b <= 2+2 = 4 < 5 no fit answers Answer: 6 M7. What is the greatest common factor of 20! and 200,000? 200000 = 26 * 55 20! = 5*10*15*20 * 2*4*8*12*14*16*… Common factors: 26 * 54 = 200000/5 = 40000 M8. Harry has 49 coins totaling $1.00. Harry has only pennies, nickels, dimes and/or quarters. What is the smallest number of dimes he could have? Start with 100 pennies - $1.00 Need to remove 51 coins Every replace of 1 dime removes 9 coins Every replace of 1 nickel removes 4 coins Every replace of 1 quarter removes 24 coins We can’t remove 51 coins with zero or even number of dimes as nickels and quarters remove even # of coins If use 1 dime, we need to remove 51 – 9 = 42 coins, which can’t be done with multiple 4 & 24 If use 3 dimes, we need to remove 51 – 27 = 24 coins – can be achieved with 1 quarter OR 6 nickels Answer: 3 M9. A bag contains 3 red balls, 4 green balls, and 5 yellow balls. If balls are drawn one at a time without replacement, what is the probability that the first yellow ball is drawn on the eighth draw? Express your answer as a common fraction. There are total of 7 balls that are not of color yellow. Prob. = 7/12 * 6/11 * 5/10 * 4/9 * 3/8 * 2/7 * 1/6 * 1 = 7/12 * 6/11 * 5/10 * 4/9 * 3/8 * 2/7 * 1/6 * 1 = 7/12 * 6/11 * 5/10 * 4/9 * 3/8 * 2/7 * 1/6 * 1 = 1/3 * 1/11 * 1/3 * 1/8 = 1/792