Units: Prices, Percents, 2-dimensional, 3 dimensional ($%2D3D)
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1. A politician puts $1000 into an account at Bank of Anything For You (BOAFY) that earns 50% interest per year. If no withdrawals are made, approximately how much money will be in the account at the end of the four years?

(A) $4000
(B) $5000
(C) $6000
(D) $8000
(E) $10000

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To increase by 50%, multiply by 1.5. Increasing by 50% four years in a row means multiplying by (1.5)4 which is (3/2)4 = 81/16 which is just over 5. The answer is (B).

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2. Another politician makes an initial deposit of $1000 at BOAFY. Every year, the amount of money in the account increases by x percent. No money is withdrawn. At the end of 10 years there is $12000 in the account. Which equation is correct?

(A) 1 + (12x/100) = 1012
(B) 1 + (10x/100) = 1210
(C) 1 + (x/100)10 = 12
(D) ((x+1) / 100)10 = 12
(E) (1 + (x/100))10 = 12

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Increasing by x percent means multiplying by (1+x/100). This increase happened every year for 10 years so you multiplied the original amount of money by (1+x/100)10 and wound up with 12 times what you started with. The answer is (E).

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3. Joe the sucker has $32 left after playing a gambling game in which you are guaranteed to lose exactly one-third of your money every time you play. Joe played 5 times before he wised up. The first time he played, he played with all the money in his pocket. Each time after the first, he played with whatever money he had left. How much money did Joe start with?

(A) $160
(B) $162
(C) $243
(D) $333
(E) $441

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If you lose a third of your money, you multiply by 2/3. If Joe started with x dollars and played 5 times he would have (2/3)5 · x left and we are told this is 32. Solving for x we get 243 or (C).

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4. An unscrupulous store wants to have a Christmas sale in which everything is marked 20% off. However, they don't want to actually reduce any of their prices. So, early in November, they mark up everything in the store by x percent so that when they take 20% off the day after Thanksgiving, everything will be back to its original price. What is x?

(A) 10
(B) 15
(C) 20
(D) 25
(E) 30

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With mark ups and mark downs, never subtract; always multiply. Taking 20% off means multiplying by 0.8 = 4/5. If P is the original price then the evil store owners want the following equation to be true: (Mark Down) · (Mark Up) · P = P. Since the Mark Down is 4/5, the Mark Up has to be 5/4 which is 1.25 or a 25% increase. The answer is (D). Note that the dollar amounts of the Mark Up and Mark Down are the same but the percentages are different.

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5. Farmer Brown divided his k acre farm into two parcels, A and B. Parcel A was k/3 acres and he sold x percent of it. He sold y percent of parcel B. What percent of the whole farm was sold?

(A) 100(x+2y)/k
(B) 100(x+2y)/(3k)
(C) (x+2y)/(3k)
(D) (x+2y)/3
(E) (x+3y)/2

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Fraction is percent/100; amount is fraction times total. Percent of parcel A sold is x; fraction sold is x/100.
Acres (amount) of parcel A sold = (x/100) · (k/3).
Acres (amount) of parcel B sold = (y/100) · (2k/3).
For the whole farm, fraction sold = (acres sold)/(total acres) = (kx/300 + 2ky/300)/k.
Multiply the fraction by 100 to get percent and note that the k's cancel. The answer is (x+2y)/3 or (D).

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6. The radius of each of the 4 tires on a car is 30 cm. If the car travels at 72 km per hour, how many revolutions per minute are made by each tire? (1 km = 1000 meters; 1 meter = 100 cm)

(A) 2000/π
(B) 1000/π
(C) 500/π
(D) 250/π
(E) 200/π

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We are given "meters/hours" = 72000 and "revolutions/meters" = 1/2π(.3) and we know that "hours/minutes" = 1/60. Multiply these together to cancel "meters" and "hours" and get "revolutions/minutes" = 2000/π. Note that "meters/revolutions" is always 2πR and you can always turn it upside down if you need to. The answer is (A).

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7. The inside of a small hollow cube has a total surface area of 24 square centimeters. A larger hollow cube has inside edges that measure 8 centimeters in length. Wanda drills a hole in the top face of each cube. She fills the small cube with water and then pours the water from the small cube into the large cube without spilling. Wanda repeats this process until the large cube is filled with water. How many times (total) does Wanda fill the small cube with water?

(A) 8
(B) 27
(C) 64
(D) 125
(E) 216

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Each face of the small cube has an area of 4 (since there are six faces and 24/6=4). So the small cube must be 2 by 2 by 2 and so its volume is 8 cubic centimeters. The volume of the big cube is 8·8·8 so the ratio of the volumes is 8·8 = 64 and the answer is (C).

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8. The total surface area of a cube is x square meters. The volume of the cube is x cubic meters. What is the value of x?

(A) 216
(B) 125
(C) 64
(D) 6
(E) 3

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The surface area of a cube is 6 times the area of each face since there are six faces. So the surface area is 6x2. The volume is of course x3. If you set these equal you get x=6. The volume is 63 or 216 so the answer is (A).

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9. A one liter can of paint is used by the legendary Peter the Perfect Painter to perfectly cover exactly 100 square feet of non-porous wall perfectly evenly. All the paint is used. Approximately how thick in millimeters is the layer of paint? (1 liter = 1000 cubic centimeters; 1 inch = 2.5 centimeters (approx.); 1 centimeter = 10 millimeters)

(A) 0.1
(B) 0.2
(C) 0.25
(D) 0.5
(E) 1.0

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It's easiest to assume the wall is 10 by 10. Ten feet is 120 inches. Multiply by 2.5 to get 300 cm. So the wall is 300 times 300 equals 90,000 square centimeters. If x is the thickness of paint then 90,000x = 1000 because 1 liter is 1000 cubic centimeters and 90,000x is the number of cubic centimeters used (area of 90,000 times thickness of x is the volume of paint used). So x = 1/90 cm which is about 0.01 cm or 0.1 mm. The answer is (A).

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10. A 625 square mile area is represented (to scale) on a 100 square inch map. What is the scale of the map?

(A) 1 inch = 20 miles
(B) 1 inch = 10 miles
(C) 1 inch = 6.25 miles
(D) 1 inch = 5 miles
(E) 1 inch = 2.5 miles

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If you assume the map is a square then it is 25 by 25 miles or 10 inches by 10 inches. So an inch is 2.5 miles. The answer is (E). If they don't tell you the shape of the map then it can't affect the answer and so you are safe to pick the simplest shape possible.

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