general-solutions

# general-solutions - SMT 2013 General Test Solutions...

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SMT 2013 General Test Solutions February 2, 2013 1. Robin goes birdwatching one day. He sees three types of birds: penguins, pigeons, and robins. 2 3 of the birds he sees are robins. 1 8 of the birds he sees are penguins. He sees exactly 5 pigeons. How many robins does Robin see? Answer: 16 Solution: 1 - 2 3 - 1 8 = 5 24 of the birds that Robin sees are pigeons. Therefore, Robin sees exactly 5 5 24 = 24 birds, so 2 3 · 24 = 16 robins. 2. Jimmy runs a successful pizza shop. In the middle of a busy day, he realizes that he is running low on ingredients. Each pizza must have 1 lb of dough, 1 4 lb of cheese, 1 6 lb of sauce, and 1 3 lb of toppings, which include pepperonis, mushrooms, olives, and sausages. Given that Jimmy currently has 200 lbs of dough, 20 lbs of cheese, 20 lbs of sauce, 15 lbs of pepperonis, 5 lbs of mushrooms, 5 lbs of olives, and 10 lbs of sausages, what is the maximum number of pizzas that Jimmy can make? Answer: 80 Solution: The limiting factor is the cheese. With only 20 lbs of cheese, the most pizzas that can be made is 20 · 4 = 80 . 3. Queen Jack likes a 5-card hand if and only if the hand contains only queens and jacks. Consid- ering all possible 5-card hands that can come from a standard 52-card deck, how many hands does Queen Jack like? Answer: 56 Solution: There are a total of 8 queens and jacks, each of which is distinguishable from the others. Thus, the number of hands that Queen Jack likes is ( 8 5 ) = 56 . 4. What is the smallest number over 9000 that is divisible by the first four primes? Answer: 9030 Solution: The first four primes are 2, 3, 5, and 7, so the number must be a multiple of 2 · 3 · 5 · 7 = 210. The least multiple of 210 that is greater than 9000 is 210 · 43 = 9030 . 5. A rhombus has area 36 and the longer diagonal is twice as long as the shorter diagonal. What is the perimeter of the rhombus? Answer: 12 5 Solution: Let d be the length of the shorter diagonal and 2 d the length of the longer diagonal. If d 1 , d 2 are the lengths of the diagonals of a rhombus, its area will be d 1 · d 2 2 . Thus, we have d 2 = 36 and d = 6. Therefore, the side length of the rhombus is 3 2 + 6 2 = 3 5, so the perimeter is 12 5 . 6. Nick is a runner, and his goal is to complete four laps around a circuit at an average speed of 10 mph. If he completes the first three laps at a constant speed of only 9 mph, what speed does he need to maintain in miles per hour on the fourth lap to achieve his goal? Answer: 15 Solution: Let d be the length of one lap in miles. Then he needs to complete the four laps in 4 d 10 = 2 d 5 hours. He has already spent 3 d 9 = d 3 hours on the first three laps, so he has 2 d 5 - d 3 = d 15 hours left. Therefore, he must maintain a speed of 15 mph on the final lap.

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SMT 2013 General Test Solutions February 2, 2013 7. A fly and an ant are on one corner of a unit cube. They wish to head to the opposite corner of the cube. The fly can fly through the interior of the cube, while the ant has to walk across the faces of the cube. How much shorter is the fly’s path if both insects take the shortest path possible?
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