Difficulty 10 Medium 12 Medium Easy SMP CCSS 2 CCSS M None 9 How many ordered

Difficulty 10 medium 12 medium easy smp ccss 2 ccss m

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Difficulty: 10: Medium 12: Medium Easy SMP-CCSS: 2 CCSS-M: None 9
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How many ordered pairs of positive integers ( M, N ) satisfy the equation M 6 = 6 N ? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10 2012 AMC 10B, Problem #10— “Cross multiply and consider divisors.” Solution Answer (D): Multiplying the given equation by 6 N gives MN = 36 . The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Each of these divisors can be paired with a divisor to make a product of 36. Hence there are 9 ordered pairs ( M, N ) . Difficulty: Medium SMP-CCSS: 7 CCSS-M: 4.OA4, A-APR.7 10
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A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible? (A) 729 (B) 972 (C) 1024 (D) 2187 (E) 2304 2012 AMC 10B, Problem #11— 2012 AMC 12B, Problem #8— “Consider how many choices of desserts are possible on Saturday and Thursday.” Solution Answer (A): There are 3 choices for Saturday (anything except cake) and for the same reason 3 choices for Thursday. Similarly there are 3 choices for Wednesday, Tuesday, Monday, and Sunday (anything except what was to be served the following day). Therefore there are 3 6 = 729 possible dessert menus. OR If any dessert could be served on Friday, there would be 4 choices for Sunday and 3 for each of the other six days. There would be a total of 4 · 3 6 dessert menus for the week, and each dessert would be served on Friday with equal frequency. Because cake is the dessert for Friday, this total is too large by a factor of 4. The actual total is 3 6 = 729 . Difficulty: 10: Medium Hard 12: Medium Hard SMP-CCSS: 2 CCSS-M: None 11
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Point B is due east of point A . Point C is due north of point B . The distance between points A and C is 10 2 meters, and BAC = 45 . Point D is 20 meters due north of point C . The distance AD is between which two integers? (A) 30 and 31 (B) 31 and 32 (C) 32 and 33 (D) 33 and 34 (E) 34 and 35 2012 AMC 10B, Problem #12— “What type of triangle is 4 ABC ?” Solution Answer (B): Note that ABC = 90 , so 4 ABC is a 45 45 90 triangle. Because hypotenuse AC = 10 2 , the legs of 4 ABC have length 10. Therefore AB = 10 and BD = BC + CD = 10 + 20 = 30 . By the Pythagorean Theorem, AD = p 10 2 + 30 2 = 1000 . Because 31 2 = 961 and 32 2 = 1024 , it follows that 31 < AD < 32 . A B C D 10 10 10 2 20 Difficulty: Medium SMP-CCSS: 5 CCSS-M: G-SRT.8 12
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It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it? (A) 36 (B) 40 (C) 42 (D) 48 (E) 52 2012 AMC 10B, Problem #13— 2012 AMC 12B, Problem #9— “Assign variables to the rate of walking and the rate of the moving escalator, then compare times and rates for the same distance.” Solution Answer (B): Let x be Clea’s rate of walking and r be the rate of the moving escalator. Because the distance is constant, 24( x + r ) = 60 x . Solving for r yields r = 3 2 x . Let t be the time required for Clea to make the escalator trip while just standing on it. Then rt = 60 x , so 3 2 xt = 60 x . Therefore t = 40 seconds.
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