GMAT Word Problems: Fundamentals (Excerpt) Je¤ Sackmann / GMAT HACKS June 2009 Contents 1 Introduction 2 2 Di¢ culty Levels 3 3 Problem Solving 4 4 Data Su¢ ciency 6 5 Answer Key 8 6 Explanations 9 11. INTRODUCTION Je¤ Sackmann GMAT Tutor www.gmathacks.com 1 Introduction This document contains nothing but GMAT Word Problems questions100 of them, to be exact. Theres nothing inherently di¢ cult about a word problem, but very few GMAT word problems are very easy. The biggest challenge most students face is translating the word problem to an algebra problem (or whatever content area it is). The math may not be hard, but getting to the point where you can do the math takes practice. As in all of my GMAT preparation resources, youll nd these questions indexed by di¢ culty. For further, more specic practice, I have produced several other resources that may help you. There are many word problems in nearly all of my practice sets, but youll nd the most in "Rates, Ratios, and Percents," and "Problem Solving: Fundamentals." There are also many tricky ones in "Algebra: Funda- mentals," in particular several that have variables in the answer choices. Also, The GMAT Math Bible has several chapters (along with focused prac- tice) on just about every type of word problem, including individual chapters on ratios, rates, averages, weighted averages, overlapping sets, probability, permu- tations, combinations, and much more. If you nd you are struggling with the mechanics of these problems, your time is probably better spent with the GMAT Math Bible than in doing dozens and dozens of practice problems, hoping to pick up those skills along the way. As far as strategy is concerned, there are dozens of articles at GMAT HACKS to help you with your strategic approach to Arithmetic questions. Most impor- tantly, you should make sure you understand every practice problem you do. It doesnt matter if you get it right the rst timewhat matters is whether youll get it right the next time you see it, because the next time you see it could be on the GMAT. With that in mind, carefully analyze the explanations. Redo questions that took you too long the rst time around. Review questions over multiple sessions, rather than cramming for eight hours straight each Saturday. These basic study skills may not feel like the key to GMAT preparation, but they are the di¤erence between those people who reach their score goals and those who never do. Enough talking; there are 100 Word Problems questions waiting inside. Get to work! 2 Copyright 2008-09 Je¤ Sackmann www.gmathacks.com2. DIFFICULTY LEVELS Je¤ Sackmann GMAT Tutor www.gmathacks.com 2 Di¢ culty Levels In general, the level 5 questions in this guide are 500- to 560-level questions. The level 4 questions represent a broad range of di¢ culty from about 380 to 500, while the level 3 questions are lower. Easy (3) PS 9, 12, 30 DS 47, 50, 55, 59, 63, 74, 76, 82, 84, 96 Moderately Easy (4) PS 3, 4, 7, 8, 11, 18, 22, 23, 27, 28, 32, 33, 34, 36, 38, 40, 42, 43 DS 46, 48, 49, 53, 54, 56, 57, 58, 60, 61, 62, 64, 65, 66, 75, 77, 78, 79, 83, 85, 91, 92, 94, 95, 97, 98, 99, 100 Moderate (5) PS 1, 2, 5, 6, 10, 13, 14, 15, 16, 17, 19, 20, 21, 24, 25, 26, 29, 31, 35, 37, 39, 41, 44, 45 DS 51, 52, 67, 68, 69, 70, 71, 72, 73, 80, 81, 86, 87, 88, 89, 90, 93 3 Copyright 2008-09 Je¤ Sackmann www.gmathacks.com3. PROBLEM SOLVING Je¤ Sackmann GMAT Tutor www.gmathacks.com 3 Problem Solving Note: this guide contains both an answer key (so you can quickly check your answers) and full explanations. 41. Country X taxes each of its citizens an amount equal to 12 percent of the rst $40,000 of income, plus 20 percent of all income in excess of $40,000. If a citizen of Country X is taxed a total of $8,000, what is her income? (A) $40,000 (B) $56,000 (C) $64,000 (D) $66,667 (E) $80,000 42. The manager of a concert venue noted that for every 100 admission tickets sold, the venue sells 12 compact discs at $15.00 each, 8 t-shirts at $20.00 each, and 5 concert posters at $12.00 each. What is the average (arithmetic mean) of these merchandise sales per ticket sold? (A) $3.00 (B) $3.76 (C) $3.81 (D) $4.00 (E) $4.40 $10, $11, $11, $12, $12, $14, $16, $17, $21, $22 43. The various prices that a certain product were sold for at retailers in City X is shown above. How many prices were greater than the median price but less than the mean price? (A) None (B) One (C) Two (D) Three (E) Four 4 Copyright 2008-09 Je¤ Sackmann www.gmathacks.com3. PROBLEM SOLVING Je¤ Sackmann GMAT Tutor www.gmathacks.com 44. Tara drove at an average speed of 50 miles per hour for the rst 50 miles of her trip and then at an average speed of 75 miles per hour for the remaining 50 miles of her trip. If she made no stops during the trip, what was Taras average speed, in miles per hour, for the entire trip? (A) 50 (B) 55 (C) 60 (D) 65 (E) 70 45. Running at the same constant rate, a identical machines can produce a total of 120 bolts per minute. At this rate, how many bolts could 4 such machines produce in 5 minutes? (A) a6 (B) a 600 (C) 150 a (D) 480 a (E) 2;400 a 5 Copyright 2008-09 Je¤ Sackmann www.gmathacks.com4. DATA SUFFICIENCY Je¤ Sackmann GMAT Tutor www.gmathacks.com 4 Data Su¢ ciency For all Data Su¢ ciency questions, the answer choices are as follows: (A) Statement (1) ALONE is su¢ cient, but statement (2) alone is not su¢ cient. (B) Statement (2) ALONE is su¢ cient, but statement (1) alone is not su¢ cient. (C) BOTH statements TOGETHER are su¢ cient, but NEITHER statement ALONE is su¢ cient. (D) EACH statement ALONE is su¢ cient. (E) Statements (1) and (2) TOGETHER are NOT su¢ cient. 46. If x equally priced shares of a certain stock were valued at $25,000 on Monday, what was the value per share of the stock on Friday? (1) The value of each share increased by 3% between Monday and Friday. (2) The value of each share increased by $1.50 between Monday and Friday. 47. Steve and Carl each received an increase in salary. Which one received the larger percent increase? (1) Steves salary increased by $1,000 per month. (2) Carls salary increased by $1,200 per month 48. A sporting goods store that sells tennis balls sells each of its packages of Brand X balls for a certain price and each of its packages of Brand Y balls for a certain price. If Oscar, Rennie, and Thom bought tennis balls in this store, how much did Rennie pay for 2 packages of Brand X balls and 2 packages of Brand Y balls? (1) Oscar bought 2 packages of Brand X balls and 1 package of Brand Y balls for $9.49. (2) Thom bought 1 package of Brand X balls and 1 package of Brand Y balls for $6.49. 49. If an item is purchased for x percent less than its retail price, and then sold for y percent less than its retail price, what was the retail price of the item? (1) x y = 10 (2) y = 5 6 Copyright 2008-09 Je¤ Sackmann www.gmathacks.com4. DATA SUFFICIENCY Je¤ Sackmann GMAT Tutor www.gmathacks.com 50. Board member A wants to schedule a 30-minute meeting on Tuesday with himself and three other board members, B, C, and D. Is there a 30-minute period on Tuesday that is open for all four members? (1) On Tuesday A and C have an open period from 9:30 a.m to 11:00 a.m. (2) On Tuesday D has an open period from 10:30 a.m. to 1:00 p.m. and B has an open period from 8:00 a.m. to 12:00 p.m. 7 Copyright 2008-09 Je¤ Sackmann www.gmathacks.com5. ANSWER KEY Je¤ Sackmann GMAT Tutor www.gmathacks.com 5 Answer Key For full explanations, see the next section. 41. B 42. D 43. B 44. C 45. E 46. C 47. E 48. B 49. E 50. C 8 Copyright 2008-09 Je¤ Sackmann www.gmathacks.com6. EXPLANATIONS Je¤ Sackmann GMAT Tutor www.gmathacks.com 6 Explanations For a quick-reference answer key, see the previous section. 41. B Explanation: 12 percent of the rst $40,000 in income is: 0:12(40; 000) = 12(400) = 4800 Thats less than $8,000, so the citizens income must be more than $40,000. If the citizen was taxed $4,800 on the rst $40,000 in income, that leaves $8; 000 $4; 800 = $3; 200 in tax on the remaining income. The tax rate on additional income is 20%. If we call the additional income a, we can set 20% of a equal to $3,200: 3200 = 0:2a a = 3200(5) = 16; 000 If the citizen earns $16,000 after the rst $40,000, thats a total of $56,000, choice (B). 42. D Explanation: First, nd the total merchandise sales per 100 admission tickets: 12 compact discs at $15 each is: 12(15) = $180 8 t-shirts at $20 each is: 8(20) = $160 5 posters at $12 each is 5(12) = $60 The total is $180 + $160 + $60 = $400 $400 per 100 is $4 per ticket, choice (D). 43. B Explanation: To nd the median, rst arrange the terms in ascending order. (Thats already done for you.) Since there are an even number of terms in the set, the median is the average of the two middle terms, $12 and $14, for a median of $13. To nd the mean price, divide the sum of the prices by the number of prices (10): 10+11+11+12+12+14+16+17+21+22 10 = 146 10 = 14:6 To answer the question: how many terms are greater than the median price of $13 and less than the mean price of $14.60? The only term that ts that description is $14, so the answer is one, choice (B). 44. C Explanation: To nd average speed, nd the total distance and the total time. Total distance is straightforward: there are two 50-mile segments, for a total of 100 miles. Total time requires more e¤ort. If she drove at a speed of 50 miles per hour for the rst 50 miles, thats a time of 1 hour. 9 Copyright 2008-09 Je¤ Sackmann www.gmathacks.com6. EXPLANATIONS Je¤ Sackmann GMAT Tutor www.gmathacks.com The next segment is 75 miles per hour for 50 miles: that requires some algebra: t = dr = 50 75 = 23 Total time for the entire trip is 1 hour plus 23 of an hour. Average speed is total distance divided by total time: s = 100 1 23 = 100 53 = 100( 35 ) = 60, choice (C). 45. E Explanation: This is a bit challenging to get a handle on, so try choosing a value for a. If a = 10 machines, then 10 identical machines can produce a total of 120 bolts per minute. That 12 bolts per minute per machine. In other words, we can nd the rate per minute per machine by dividing 120 by a: algebraically, thats 120 a bolts per minute per machine. If you have four machines, you can multiply that rate per minute per machine by 4: 120 a (4) bolts per minute for four machines. If the machines will work for 5 minutes, multiply that rate per minute by 5: 120 a (4)(5) bolts for four machines in ve minutes. Simplied, that is: 120(4)(5) a = 2;400 a , choice (E). 46. C Explanation: Statement (1) is insu¢ cient: we can determine from this the total value of the shares on Friday, but that doesnt help us nd the value of each individual share. Statement (2) is also insu¢ cient: this gives us some information about indi- vidual shares, but since we dont know the value per share on Monday, we cant determine the value per share on Friday. Taken together, the statements are su¢ cient. If the total value increased by 3% from Monday to Friday, the value per share did as well. If that increase is equal to $1.50, we can nd the original price: 0:03x = 1:50 where x is the Monday price per share. Find that, add $1.50, and you have the Friday price per share. Choice (C) is correct. 47. E Explanation: Statements (1) and (2) are insu¢ cient for the same reason: without knowing each persons original salary, we dont know what percent increase those dollar amounts represent. Taken together, the statements are still insu¢ cient. While Carl received the larger dollar amount increase, its possible that Steve received the larger percent increase. If Carls starting salary was much larger, a $1,200 per month increase may not be as large of a percent increase as a $1,000 per month increase on a smaller starting salary. Choice (E) is correct. 48. B 10 Copyright 2008-09 Je¤ Sackmann www.gmathacks.com6. EXPLANATIONS Je¤ Sackmann GMAT Tutor www.gmathacks.com Explanation: Statement (1) is insu¢ cient: it allows us to set up an equa- tion with two variables, but thats not enough to solve for each variable: 2x + y = 9:49 Statement (2) is su¢ cient. Again, we get an equation with two variables, but this time we can translate it into the answer to the question: x + y = 6:49 Multiply both sides by 2 to get the price of 2 packages of Brand X balls and 2 packages of Brand Y balls: 2x + 2y = 2(6:49) Choice (B) is correct. 49. E Explanation: Many data su¢ ciency questions require that you have an actual number, whether it be a price, a population size, or the length of a geometric gure. This is one of them: x and y represent percents, but in order to know the retail price, youll need to know some numbersuch as $30 or $100. It only takes a quick glance at the statements to realize that (E) must be the correct choice, as the statements are concerned with the percents, not the price. Without a price, you cant answer the question. 50. C Explanation: Statement (1) is insu¢ cient: it gives us no information about B or D. Statement (2) is also insu¢ cient: there is an overlap between Bs free time and Ds free time, but we dont know whether that time period overlaps with availability from A and C. Taken together, the statements are su¢ cient. (2) tells us that B and D are both available from 10:30 a.m. to 12:00 p.m., and (1) tells us that A and C are available from 10:30 a.m. to 11:00 a.m. They can meet for that 30-minute period. Choice (C) is correct. 11 Copyright 2008-09 Je¤ Sackmann www.gmathacks.com