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2. Cheese, bologna, and peanut butter sandwiches were made for a picnic in a ratio of 5 to 7 to 8. If a total of 120 sandwiches were made, how many bologna sandwiches were made?
3. A sink contains exactly 12 liters of water. If water is drained from the sink until it holds exactly 6 liters of water less than the quantity drained away, how many liters of water were drained away?
4. Company C sells a line of 25 products with an average retail price of $1,200. If none of these products sells for less than $420, and exactly 10 of the products sell for less than $1,000, what is the greatest possible selling price of the most expensive product?
5. In a certain game, each player scores either 2 points or 5 points. If n players score 2 points and m players score 5 points, and the total number of points scored is 50, what is the least possible positive difference between n and m?
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1. B The family pays $800 per year for the plan, plus (100 percent minus 80 percent) or 20 percent of the first $1,000 in medical expenses, while the insurance company pays 80 percent of the first $1,000, or $800. It must pay an additional $200 to match what the family pays out. Since the $200 comes after the first $1,000 in expenses, it must represent 100 percent of additional expenses. Therefore, there must have been $1,000 plus $200 or $1,200 in medical expenses altogether. 2. D We’re told that cheese, bologna, and peanut butter sandwiches are made in the ratio of 5 to 7 to 8. Every time they make 5 cheese sandwiches, they also have to make 7 bologna and 8 peanut butter. So there must be 5x cheese sandwiches (and we don’t know what x is at this point), 7x bologna sandwiches, and 8x peanut butter. How many bologna sandwiches were made? Well, the number of bologna sandwiches must be a multiple of 7. But only choice D is a multiple of 7. In other words: 5x + 7x = 8x = 120 20x = 120 x = 6 7(6) = 42 3. E The key to solving this one is to focus on the quantity of water drained away, which we will call x. We’re told that x liters of water are drained away, and x — 6 liters are left. So x (liters taken away) plus x — 6 (liters left) equals 12 (total liters in the sink). Therefore 2x — 6 = 12, and x = 9. 4. D Since 25 products sell at an average of $1,200, to buy one of each we’d have to spend 25 x $1,200 = $30,000. We want to find the greatest possible selling price of the most expensive product. They way to maximize this price is to minimize the prices of the other 24 products. Ten of these products sell for less than $1,000, but all sell for at least $420. This means that we can have 10 sell at $420. That leaves 14 more that sell for $1,000 or more. So, in order to keep minimizing, we’ll price these at $1,000. That means that, out of the $30,000 we know it will take to purchase one of each item, only 10($420) + 14($1,000) = $18,200 is needed in order to purchase the 24 cheap items. The final most expensive item can thus cost as much as $30,000 — $18,200 = $11,800. 5. B The quickest solution is to pick numbers for n and m. Since n = 1 and m = 1 would amount to 7 points, and since we want to minimize the difference between n and m, and since 50/7 is just a bit more than 7, we’ll start with values near 7. The key is to discover what values for n, when multiplied by 2 points, will leave a muliple of 5 as the remaining points. The solution turns out to be 5 for n (10 points), which allows for 8 for m (40 points). That’s a total of 50 points, and the positive difference between the two values is only 3.