Units: Algebra
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1. It takes G gallons of water to fill 6 hot-tubs. If you have N gallons of water but use 80% of it to water the lawn, how many hot-tubs can you fill using the remaining water?

(A) 6N/(5G)
(B) 6G/(5N)
(C) 30N/G
(D) 6GN/5
(E) 5GN/6

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You've got 6/G "tubs/gallons." You've also got N/5 "gallons" (4/5 was used to water the lawn). You want "tubs." Look at the units. To get "tubs" you multiply "tubs/gallons" by "gallons" so that "gallons" cancels and you are left with what you want. So the answer is (6/G) times (N/5) which is 6N/(5G). The answer is (A). Note that you can say 6/G "tubs/gallons" OR G/6 "gallons/tubs" whichever is convenient.

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2. Joe (a fictional character) wants to set off some stinkbombs during the administration of the SAT. He can buy n stinkbombs for d dollars and the per-stinkbomb price is independent of the number of stinkbombs purchased. How much will it cost in dollars for Joe to buy n3 stinkbombs?

(A) (d/n)3
(B) d3
(C) n2d
(D) d/n3
(E) d2

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They give you "dollars/stinkbombs" = d/n. They also tell you "stinkbombs" = n3. You want "dollars." Multiply "dollars/stinkbombs" by "stinkbombs" so that "stinkbombs" cancel and you are left with "dollars." The answer is (d/n) times n3 which. Cancel one of the n's and you get dn2 or (C).

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3. Joe and Blow ran an 800 meter race on a long, straight track. Joe won the race, crossing the finish line after 120 seconds at which point Blow stopped and yelled obscenities. If Joe's average speed for the race was 2 meters per second faster than Blow's average speed, how many meters did Blow run before giving up?

(A) 760
(B) 680
(C) 560
(D) 480
(E) 240

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Every second Joe adds 2 meters to his lead. After 120 seconds, Joe is ahead by 240 meters. So Blow ran 800-240 = 560 meters. The answer is (C).

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4. One Jiffy Lube Dude can do x oil changes per year. The town of Carsville has y cars. If all oil changes are performed at Jiffy Lube and all cars get exactly 3 oil changes per year, how many Jiffy Lube Dudes are needed to take care of all the cars in Carsville?

(A) 3xy
(B) 3y/x
(C) 3x/y
(D) x/(3y)
(E) 1/(3xy)

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Assume we are talking about 1 year. You are given "Dudes/Oil Changes" = 1/x. You are also given "Cars" = y. Finally, you are given "Cars/Oil Changes" = 1/3. You want everything to cancel and leave you with "Dudes." First turn the last one upside down to get "Oil Changes/Cars" = 3/1. Now multiply everything together. You get "Dudes" = (1/x) times y times (3/1) which is 3y/x. The answer is (B).

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5. Every time a question is answered incorrectly or left blank by someone taking the SAT, Satan laughs with delight and Hell gets hotter. The rate of temperature increase in Hell is d degrees for every x questions not answered correctly. One year, N people took the SAT and the average test-taker answered p percent of Q questions correctly. By how many degrees did Hell get hotter that year due to the SAT?

(A) dNQ/x · (1 – p/100)
(B) dNQx · (1 – p/100)
(C) dNQx · (100 – p)
(D) NQpx / (100d)
(E) dNQp / (100x)

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Always convert percents to fractions. The fraction answered correctly is p/100. The fraction not answered correctly is 1 — p/100. The number not answered correctly by the average test-taker is Q·(1—p/100). The total number not answered correctly is N·Q·(1—p/100). We are given "degrees/question not answered correctly" = d/x. The number of degrees by which Hell gets hotter is (d/x) · N·Q·(1—p/100). The answer is (A).

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6. In Piano City, there are n pianos that are all tuned exactly once a year. There are t piano tuners in the city. A tuner is paid d dollars for every 5 pianos tuned. What is the average income in dollars per year earned by a tuner for tuning pianos in Piano City?

(A) dnt / 5
(B) 5d / (nt)
(C) dn / (5t)
(D) 5dt / n
(E) 5nt / d

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Assume 1 year in Piano City. You are given "pianos" = n and "tuners" = t and "dollars/pianos" = d/5. You want "dollars/tuner" so you need "pianos" to cancel and you need "tuners" on the bottom. Note that plurals don't matter: "tuners" and "tuner" are different grammatically by the same mathematically. First turn "tuners" upside down so you get "1/tuners" = 1/t. Now multiply n·(1/t)·d/5. to get the answer (C). You can also just say that each tuner tunes n/t pianos per year and makes d/5 dollars for each piano and then say "it is obvious the answer is C." Sometimes, the mechanical approach works best, othertimes you want to think about it.

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7. A bicycle has two wheels with radius r meters. Ryder rides the bicycle 5 times around a circular track with radius R meters. How many rotations are made by each wheel?

(A) R / r
(B) 5R / r
(C) 2πR / (5r)
(D) 5R / (2πr)
(E) 5πR / r

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You have the equation for the circumference of a circle memorized so you know that Ryder rides a total distance of 10πR meters and that "meters/rotations" = 2πr. Turn this upside down to get "rotations/meters" = 1/2πr. Now multiply to cancel meters and get "rotations" = 5R/r. The answer is (B).

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Questions 8-10: Some college students who have trouble adjusting to "daily life without mommy" respond by drinking alcohol until they vomit. The technical term used by mental health professionals to refer to this behavior is "stupid." The technical term for people who drink more than 6 ounces of hard liquor per hour at a party is "moronic." People who are moronic are not considered stupid (they are beyond stupid) although they may vomit.

8. There were x moronic people at a party with a total of N people. Thirty percent of the people who were not moronic were stupid. What percent of the people at the party were stupid?

(A) 30 · (1 – x/N)
(B) 70 · (1 – x/N)
(C) (30N + 70x) / N
(D) 0.3 · (N + x)
(E) 0.7 · (N + x)

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The number who were not moronic is N–x. Since 30% of these were stupid, the number of stupid people was 0.3·(N–x). The fraction of stupid people is 0.3 · (N–x) / N. The percent of stupid people is 100 · 0.3 · (N–x) / N which is the same as 30·(1–x/N) or (A).

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9. At a party, the number of stupid people plus the number of moronic people was 72. All of these individuals vomited and 20 of them experienced projectile vomiting. The number of moronic projectile vomiters was 4 times the number of stupid projectile vomiters. If ten percent of stupid people experienced projectile vomiting, what percent of moronic people experienced projectile vomiting?

(A) 40
(B) 50
(C) 60
(D) 75
(E) 90

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There must be 16 moronic projectile vomiters (mpv's) and 4 stupid projectile vomiters (spv's) because we are given that the total is 20 and that one is 4 times more than the other. Since the 4 spv's are 10% of the total stupids, there must have been 40 stupid people at the party. That leaves 32 moronic people so 16/32 = 50% of the morons experienced projectile vomiting.

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10. At a party with N people there were no moronic people. One third of the people at this party didn't drink any alcohol. There were x bottles of hard liquor each containing y ounces of liquor. If the party lasted three hours, what is the maximum number of bottles that could have been consumed at the party?

(A) 12Nx/y
(B) 12N/y
(C) 6N/y
(D) 6Ny/x
(E) 6Nx/y

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The maximum liquor consumption would occur if two-thirds of the people (2N/3 people) were stupid and if each one of them drank 18 ounces in the three hours (remember, the maximum consumption rate for stupid people is 6 ounces per hour). So we've got "people" = 2N/3 and "ounces/person" = 18/1 and "bottles/ounces" = 1/y. Multiply these to cancel everything but bottles and get 12N/y or (B).

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