Algebra: Fractions
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1. If you pour half of a full small bucket of water into an empty big bucket, the big bucket will be three-tenths full. How many times bigger than the small bucket is the big bucket?

(A) 1.5
(B) 1.67
(C) 3
(D) 3.33
(E) 5

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You want b / s which is the ratio of big (b) to small (s). Don't think too much. The problem says (in words): (1/2)s = (3/10)b. Now multiply both sides by (10/3) and divide both sides by s to get b / s = 5/3 = 1.67 (B).

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2. A small bucket holds 80 percent of the volume of a big bucket. The big bucket is two-thirds full with water. The small bucket is empty. What fraction of the small bucket do you need to fill with water to have just enough liquid to finish filling the big bucket?

(A) 5 / 12
(B) 4 / 15
(C) 7 / 15
(D) 8 / 15
(E) 5 / 6

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Use b and s for the big and small buckets. You are given s = (4/5)b and you need (1/3)b. So get b and multiply by 1/3. You get (1/3)b = (1/3)(5/4)s = (5/12)s. So the answer is 5/12 or (A).

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3. In circle O, central angle AOB measures 72 degrees. The radius of the circle is r. The perimeter of sector AOB consists of line segments AO and BO and arc AB. Which expression gives the perimeter of sector AOB in terms of r?

(A) (π + 10) · r / 5
(B) (π + 8) · r / 4
(C) (π + 6) · r / 3
(D) (π + 2) · r / 5
(E) (2π + 10) · r / 5

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The arc is 1/5 of the circle (72/360). So the arc is 2πr/5. Add r+r = 2r = 10r/5 to the length of the arc, factor out the r and you get the answer (E).

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4. If w = x and z = x/(x+1), then the ratio of w to z is:

(A) x
(B) x+1
(C) x2
(D) x2 / (x+1)
(E) (x+1) / x2

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"Ratio" just means divide. The ratio of w to z is w/z. The only thing that makes this hard is that z is a fraction. To divide by a fraction multiply by the reciprocal ("flip and multiply"). The reciprocal of z is (x+1)/x. Now multiply. The x's cancel and the answer is (x+1) or (B).

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5. Half of the people who showed up to take the SAT one day at the Painisgood test center got nauseous and one-third of the people who got nauseous threw up. One-fifth of the people who did not throw up got bits of vomit on their test papers. One-third of the people with vomit on their test papers decided to leave. All of the people who threw up also left. What fraction of prospective test-takers did not leave?

(A) 2 / 9
(B) 4 / 9
(C) 5 / 9
(D) 2 / 3
(E) 7 / 9

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1/2 get nauseous, 1/6 throw up, 5/6 don't throw up, (1/5)(5/6) = 1/6 get bits of vomit on their test papers, everyone who throws up plus 1/3 of the people with bits of vomit leave the test center so 1/6 + (1/3)(1/6) = 4/18 = 2/9 leave. So 7/9 did not leave (E).

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6. You have a hundred dollars in dollar bills. You are going to divide it up (unequally) among four people. You want everyone to get a whole number of dollars and you don't want any money left over. To fulfill these conditions, you can divide up the money according to which of the following ratios?

I. 4 : 3 : 2 : 1
II. 10 : 5 : 2 : 1
III. 9 : 7 : 3 : 1

(A) I, only
(B) II, only
(C) I and II, only
(D) I and III, only
(E) II and III, only

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If x is one "share" then dividing it 4 : 3 : 2 : 1 means 4x, 3x, 2x, and 1x. The total is 10x which has to be 100. So x is 10. If you try to divide it 10 : 5 : 2: 1 you get 10x+5x+2x+1x = 18x so x is not a whole number of dollars. If you want to do 9 : 7 : 3 : 1, that's no problem. You get x=5. So I and III work and the answer is (D).

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7. Potter and Malfoy fly on their brooms at full speed directly toward each other. Potter's broom is 5 times as fast as Malfoy's, they start 150 meters apart, and they begin flying simultaneously. They collide and Malfoy dies. How far did Malfoy fly?

(A) 125 meters
(B) 120 meters
(C) 36 meters
(D) 30 meters
(E) 25 meters

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Potter flies 5x, Malfoy flies x. You get 6x=150 and x=25. So Malfoy flies 25 meters (E).

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8. The duration of the lunch break for a (fictional) ETS executive is directly proportional to the number of years the executive has been with the company. Executive A has been with ETS for 5 years and takes 2 hour lunch breaks. The lunch breaks of executive B take a whopping 3 hours and 40 minutes. How long has executive B been "working" for ETS?

(A) 7 years 10 months
(B) 8 years 4 months
(C) 9 years 2 months
(D) 10 years 8 months
(E) 11 years 6 months

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The first executive gets 120 minutes of lunch after working for 60 months. That's 2 minutes for each month that he's been working. The second executive gets 220 minutes so he must have been working 110 months which is a bit over 9 years (C).

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9. If w = (x + y)/k and z=(x – 2y)/(2k) and k≠0 and x=k2+k then w+z =

(A) k + 1
(B) 3 · (k + 1) / 2
(C) 3 · (k + 1)2 / 2
(D) 2k · (k + 1)
(E) 2k · (k + 1)2

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To get a common denominator of 2k, multiply the top and bottom of the first fraction by 2. You a +2y from the first fraction and a –2y from the second and they cancel. You'll have 2x + x = 3x on top. You're given x = k(k+1), so you substitute for x cancel the k on top with the k on the bottom and get (3/2)(k+1) or (B).

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10. The quantity (1 – x2) / (1 – 1/x2) is defined for x≠0,±1 and may be expressed as:

(A) 1 / (x2 – 1)
(B) –x2
(C) 1 / x2
(D) –1 / x 2
(E) –1

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Whenever you see a 1 minus some fraction, get a common denominator and do the subtraction. So 1 - 1/x2 is (x2-1)/x2. Since that's on the bottom, you flip it and multiply. Remember (a-b)/(b-a) is –1 so you get –x2 or (B).

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