Algebra: Word Problems
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1. There are an even number x of different test-prep companies in Killsat, New Jersey. Each company is for sale for a price of d dollars. A rich man purchases all the companies. After y years, the man sells half of the companies for 2d dollars each and the other half of the companies for d/2 dollars each. What is the total amount in dollars that the man receives from the sale of all x companies?

(A) dx
(B) 5dx/4
(C) dx/y
(D) 3dx/2
(E) dxy

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He gets 2d · x/2 dollars for the first half and d/2 · x/2 dollars for the second half which comes out to dx + dx/4 dollars or 5dx/4 dollars or (B).

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2. The base of a triangle is 10 inches in length. By how many square inches does the area of the triangle increase if the height is increased by 2 inches while the length of the base remains unchanged?

(A) 4
(B) 5
(C) 10
(D) 12
(E) 20

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The original area is (1/2)·10·h. The new area is (1/2)·10·(h+2) which is equal to (1/2)·10·h + 10. So the answer is 10 or (C).

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3. Your new motorcycle has a top speed that is 20% faster than the top speed of your old motorcycle. On your old motorcycle it takes you 2 hours to get to your friend's house traveling at the top speed of the bike (and risking your life for the thrill of speed). How many minutes does the trip take at the top speed of the new motorcycle? Assume, in each case, that you manage to travel at the top speed of the motorcycle for the whole trip (no stop signs, traffic lights etc.).

(A) 100
(B) 96
(C) 95
(D) 92.5
(E) 90

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The basic equation here is distance equals rate times time or d = rt. For the old motorcycle: d = r·120. For the new motorcycle: d = 1.2r·t. Note that "increase by 20%" means "multiply by 1.2." Since the distance to your friend's house hasn't changed, you know that r·120 = 1.2r·t. So t = 120/1.2 = 100 minutes or (A).

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4. After breaking into ETS HQ and erasing all their hard drives thereby ending any chance that the SAT will be administered to any more victims for a period of 2 years, Joe (a fictional character) is escaping on his motorcycle. Unfortunately, a gang of (fictional, heavily armed) ETS thugs are exactly 10 miles behind him riding specially-built bikes traveling at 150 miles per hour. They are NOT planning to arrest him. If Joe can make it another 50 miles before they catch him, he will cross a bridge that will not hold the ETS bikes. What minimum average speed in miles per hour must Joe maintain over the next 50 miles to reach the bridge just in the nick of time?

(A) 115
(B) 117.5
(C) 120
(D) 122.5
(E) 125

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It's d=rt again. There are usually two of them. For Joe we get 50 = r·t. For the ETS thugs who have to go 10 miles further, we get 60 = 150·t. (Usually either the rate, the time or the distance is the same for the two equations; in this case, the time is the same.) Solving the second equation we get t = 6/15. Plugging that t into the first equation, we get r = 50·15/6 = 125 or (E).

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5. When the width and length of a rectangle are each reduced by 20%, the area of the new rectangle is 12 units less than the area of the original rectangle. What is the area of the original rectangle?

(A) 12.5
(B) 15
(C) 33.3
(D) 45
(E) 60

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Reduced by 20% means multiplied by 0.8. Let's just make the length and width x and y. The problem is telling us: 0.8x · 0.8y = xy - 12. This comes out to 12 = 0.36xy. So the original area, xy, is 12/.36 = 33.3 or (C).

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6. Square A has an area 30 square meters larger than square B. The perimeter of square A is 8 meters larger than the perimeter of square B. The length in meters of one side of square A is:

(A) 6.5
(B) 7.5
(C) 8.5
(D) 9.5
(E) 10.5

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Let's use a for the side of square A and b for the side of square b. If we translate the words into symbols, we get: a2 = b2 + 30 and 4a = 4b + 8. Dividing the second equation by 4 gives: a = b+2. Plug into the first equation to get (b+2)2 = b2+30 which leads to 4b = 26. So b is 6.5 but this isn't the answer (SATan loves this trick). They asked for a which is b+2 or 8.5 or (C).

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7. The average of two numbers is x. If you multiply one of the two numbers by 18, the average of the two new numbers (one of which is unchanged) is 2x. What is the ratio of the smaller original number to the larger original number?

(A) 1 / 15
(B) 1 / 16
(C) 1 / 18
(D) 1 / 20
(E) 1 / 21

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So they are saying (a+b)/2 = x and (18a + b)/2 = 2x. So we have a+b = 2x and 18a + b = 4x. We need a ratio (a/b or b/a) so we need an equation with just a and b in it. Since 2a + 2b = 4x (using the first equation) we know that 18a + b = 2a + 2b. Now we can get a ratio. We get 16a = b or a/b = 1/16. The key thing to remember about ratio problems is that you don't need to get a and b separately so one equation with two unknowns is okay. The answer is (B).

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8. Grandma driving her old car can make it from her house to Las Vegas in 8 hours. If she borrows your car and drives like a maniac, she can make the same trip in 6 hours. Her average speed for the trip in your car is 35 miles per hour faster than her average speed for the trip in her car. What average speed in miles per hour does grandma attain driving your car from her house to Las Vegas?

(A) 95
(B) 105
(C) 125
(D) 130
(E) 140

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It's d = rt again. In Grandma's car: d = r·8. In your car: d =(r+35)·6. Since the distances are equal you get 8r = 6r+210 or r = 105. But this is the speed in Grandma's car. In your car she goes 35 mph faster or 140 mph (E).

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9. In one month, Alice-the-lawyer earns half of what her husband, Bill-the-artist, earns in a year (12 months). Together, the happy couple earns 168,000 dollars per year. What is Alice's monthly salary in dollars?

(A) 12,000
(B) 10,000
(C) 8,000
(D) 4,000
(E) 2,000

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Say Alice earns A dollars per year so A/12 = B/2. We also know that A + B = 168,000. The first equation gives us A = 6B. So we know that 7B = 168,000 which means B = 24,000. We're actually looking for A/12 which is B/2 or 12,000. The answer is (A).

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10. When Einstein was a small boy he used to like to blow bubbles. All the bubbles he blew were either big or small. Little Albert always blew his bubbles so that the number of big bubbles was equal to exactly one more than the square root of the number of small bubbles. For one particular bubble-blowing session, the total number of bubbles little Albert blew might have been which of the following? (Note: There is no evidence indicating Einstein ever actually did anything this pointless.)

(A) 42
(B) 43
(C) 44
(D) 45
(E) 46

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If you write an equation you get: b = √s + 1. The total of course is b+s. Since there is one equation and two unknowns there are an infinite number of solutions. So you have to look at the answers to know where to start. Suppose you try s=25. You'll get b = 6 and total equals 31. That's too small so you try s=36 and get b = 7 and total = 43. So the answer is (B).

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