The correct answer is (D). In the weighted-average formula, 315 annual gallons receives a weight of 4, while the average annual number of gallons for the next six years (x) receives a weight of 6: 378 5 1260 1 6x 10 3780 5 1260 1 6x 3780 2 1260 5 6x 2520 5 6x 420 5 x This solution (420) is the average number of gallons needed per year, on average, during the next 6 years. To guard against calculation errors, check your answer by sizing up the question. Generally, how great a number are you looking for? Notice that the stated goal is a bit greater than the annual average production over the first four years. So you’re looking for an answer that is greater than the goal—a number somewhat greater than 378 gallons per year. You can eliminate choice (A) and (B) out of hand. The number 420 fits the bill. CURRENCY PROBLEMS Currency problems are similar to weighted-average problems in that each item (bill or coin) is weighted according to its monetary value. Unlike weighted average problems, however, the “average” value of all the bills or coins is not at issue. In solving currency problems, remember the following: • You must formulate algebraic expressions involving both number of items (bills or coins) and value of items. • You should convert the value of all moneys to a common currency unit before formulating an equation. If converting to cents, for example, you must multiply the number of nickels by 5, dimes by 10, and so forth. 29. Jim has $2.05 in dimes and quarters. If he has four fewer dimes than quarters, how much money does he have in dimes? (A) 20 cents (B) 30 cents (C) 40 cents (D) 50 cents (E) 60 cents The correct answer is (B). Letting x equal the number of dimes, x 1 4 represents the number of quarters. The total value of the dimes (in cents) is 10x, and the total value of the Chapter 10: Math Review: Number Theory and Algebra 263 www.petersons.com quarters (in cents) is 25(x 1 4) or 25x 1 100. Given that Jim has $2.05, the following equation emerges: 10x 1 25x 1 100 5 205 35x 5 105 x 5 3 Jim has three dimes, so he has 30 cents in dimes. You could also solve this problem without formal algebra, by plugging in each answer choice in turn. Let’s try this strategy for choices (A) and (B): A. 20 cents is 2 dimes, so Jim has 6 quarters. 20 cents plus $1.50 adds up to $1.70. Wrong answer! B. 30 cents is 3 dimes, so Jim has 7 quarters. 30 cents plus $1.75 adds up to $2.05. Correct answer! MIXTURE PROBLEMS In GMAT mixture problems, you combine substances with different characteristics, resulting in a particular mixture or proportion, usually expressed as percentages. Substances are measured and mixed by either volume or weight—rather than by number (quantity). 30. How many quarts of pure alcohol must you add to 15 quarts of a solution that is 40% alcohol to strengthen it to a solution that is 50% alcohol? (A) 4.0 (B) 3.5 (C) 3.25 (D) 3.0 (E) 2.5 The correct answer is (D). You can solve this problem by working backward from the answer choices—trying out each one in turn. Or, you can solve the problem algebraically. The original amount of alcohol is 40% of 15. Letting x equal the number of quarts of alcohol that you must add to achieve a 50% alcohol solution, 0.4(15) 1 x equals the amount of alcohol in the solution after adding more alcohol. You can express this amount as 50% of (15 1 x). Thus, you can express the mixture algebraically as follows: ~0.4!~15!1x 5~0.5!~15 1 x! 6 1 x 5 7.5 1 0.5x 0.5x 5 1.5 x 5 3 You must add 3 quarts of alcohol to obtain a 50% alcohol solution. 264 PART IV: GMAT Quantitative Section TIP You can solve most GMAT currency problems by working backward from the answer choices. www.petersons.com INVESTMENT PROBLEMS GMAT investment problems involve interest earned (at a certain percentage rate) on money over a certain time period (usually a year). To calculate interest earned, multiply the original amount of money by the interest rate: amount of money 3 interest rate 5 amount of interest on money For example, if you deposit $1000 in a savings account that earns 5% interest annually, the total amount in the account after one year will be $1000 1 0.05($1000) 5 $1000 1 $50 5 $1050. GMAT investment questions usually involve more than simply calculating interest earned on a given principal amount at a given rate. They usually call for you to set up and solve an algebraic equation. When handling these problems, it’s best to eliminate percent signs. 31. Dr. Kramer plans to invest $20,000 in an account paying 6% interest annually. How much more must she invest at the same time at 3% so that her total annual income during the first year is 4% of her entire investment? (A) $32,000 (B) $36,000 (C) $40,000 (D) $47,000 (E) $49,000 The correct answer is (C). Letting x equal the amount invested at 3%, you can express Dr. Kramer’s total investment as 20,000 1 x. The interest on $20,000 plus the interest on the additional investment equals the total interest from both investments. You can state this algebraically as follows: 0.06(20,000) 1 0.03x 5 0.04(20,000 1 x) Multiply all terms by 100 to eliminate decimals, then solve for x: 6~20,000!13x 5 4~20,000 1 x! 120,000 1 3x 5 80,000 1 4x 40,000 5 x She must invest $40,000 at 3% for her total annual income to be 4% of her total investment ($60,000). Beware: In solving GMAT investment problems, by all means size up the question to make sure your calculated answer is in the ballpark. But don’t rely on your intuition to derive a precise solution. Interest problems can be misleading. For instance, you might have guessed that Dr. Kramer would need to invest more than twice as much at 3% than at 6% to lower the overall interest rate to 4%, which is not true. Chapter 10: Math Review: Number Theory and Algebra 265 www.petersons.com PROBLEMS OF RATE OF PRODUCTION OR WORK A rate is a fraction that expresses a quantity per unit of time. For example, the rate at which a machine produces a certain product is expressed this way: rate of production 5 number of units produced time A simple GMAT rate question might provide two of the three terms and then ask you for the value of the third term. To complicate matters, the question might also require you to convert a number from one unit of measurement to another. 32. If a printer can print pages at a rate of 15 pages per minute, how many pages can it print in 2 1 2 hours? (A) 1375 (B) 1500 (C) 1750 (D) 2250 (E) 2500 The correct answer is (D). Apply the following formula: rate 5 no. of pages time The rate is given as 15 minutes, so convert the time (2 1 2 hours) to 150 minutes. Determine the number of pages by applying the formula to these numbers: 15 5 no. of pages 150 ~15!~150!5no. of pages 2250 5 no. of pages A more challenging type of rate-of-production (work) problem involves two or more workers (people or machines) working together to accomplish a task or job. In these scenarios, there’s an inverse relationship between the number of workers and the time that it takes to complete the job; in other words, the more workers, the quicker the job gets done. A GMAT work problem might specify the rates at which certain workers work alone and ask you to determine the rate at which they work together, or vice versa. Here’s the basic formula for solving a work problem: A x 1 A y 5 1 In this formula: • x and y represent the time needed for each of two workers, x and y, to complete the job alone. • A represents the time it takes for both x and y to complete the job working in the aggregate (together). 266 PART IV: GMAT Quantitative Section www.petersons.com So each fraction represents the portion of the job completed by a worker. The sum of the two fractions must be 1 if the job is completed. 33. One printing press can print a daily newspaper in 12 hours, while another press can print it in 18 hours. How long will the job take if both presses work simultaneously? (A) 7 hours, 12 minutes (B) 9 hours, 30 minutes (C) 10 hours, 45 minutes (D) 15 hours (E) 30 hours The correct answer is (A). Just plug the two numbers 12 and 18 into our work formula, then solve for A: A 12 1 A 18 5 1 3A 36 1 2A 36 5 1 5A 36 5 1 5A 5 36 A 5 36 5 ,or7 1 5 . Both presses working simultaneously can do the job in 7 1 5 hours—or 7 hours, 12 minutes. PROBLEMS OF RATE OF TRAVEL (SPEED) GMAT rate problems often involve rate of travel (speed). You can express a rate of travel this way: rate of travel 5 distance time An easier speed problem will involve a single distance, rate, and time. A tougher speed problem might involve different rates, such as: • Two different times over the same distance • Two different distances covered in the same time In either type, apply the basic rate-of-travel formula to each of the two events. Then solve for the missing information by algebraic substitution. Use essentially the same approach for any of the following scenarios: • One object making two separate “legs” of a trip—either in the same direction or as a round trip • Two objects moving in the same direction • Two objects moving in opposite directions NOTE In the real world, a team may be more efficient than the individuals working alone. But for GMAT questions, assume that no additional efficiency is gained this way. TIP In work problems, use your common sense to narrow down answer choices. Chapter 10: Math Review: Number Theory and Algebra 267 www.petersons.com 34. Janice left her home at 11 a.m., traveling along Route 1 at 30 mph. At 1 p.m., her brother Richard left home and started after her on the same road at 45 mph. At what time did Richard catch up to Janice? (A) 2:45 p.m. (B) 3:00 p.m. (C) 3:30 p.m. (D) 4:15 p.m. (E) 5:00 p.m. The correct answer is (E). Notice that the distance Janice covered is equal to that of Richard—that is, distance is constant. Letting x equal Janice’s time, you can express Richard’s time as x 2 2. Substitute these values for time and the values for rate given in the problem into the speed formula for Richard and Janice: Formula: rate 3 time 5 distance Janice: (30)(x) 5 30x Richard: (45)(x 2 2) 5 45x 2 90 Because the distance is constant, you can equate Janice’s distance to Richard’s, then solve for x: 30x 5 45x 2 90 15x 5 90 x 5 6 Janice had traveled 6 hours when Richard caught up with her. Because Janice left at 11:00 a.m., Richard caught up with her at 5:00 p.m. 35. How far in kilometers can Scott drive into the country if he drives out at 40 kilome- ters per hour (kph), returns over the same road at 30 kph, and spends 8 hours away from home, including a 1-hour stop for lunch? (A) 105 (B) 120 (C) 145 (D) 180 (E) 210 The correct answer is (B). Scott’s actual driving time is 7 hours, which you must divide into two parts: his time spent driving into the country and his time spent returning. Letting the first part equal x, the return time is what remains of the 7 hours, or 7 2 x. Substitute these expressions into the motion formula for each of the two parts of Scott’s journey: Formula: rate 3 time 5 distance Going: (40)(x) 5 40x Returning: (30)(7 2 x) 5 210 2 30x 268 PART IV: GMAT Quantitative Section www.petersons.com Because the journey is round trip, the distance going equals the distance returning. Simply equate the two algebraic expressions, then solve for x: 40x 5 210 2 30x 70x 5 210 x 5 3 Scott traveled 40 kph for 3 hours, so he traveled 120 kilometers. PROBLEMS INVOLVING OVERLAPPING SETS Overlapping set problems involve distinct sets that share some number of members. GMAT overlapping set problems come in one of two varieties: Single overlap (easier) Double overlap (tougher) 36. Each of the 24 people auditioning for a community-theater production is an actor, a musician, or both. If 10 of the people auditioning are actors and 19 of the people auditioning are musicians, how many of the people auditioning are musicians but not actors? (A) 10 (B) 14 (C) 19 (D) 21 (E) 24 The correct answer is (B). You can approach this relatively simple problem without formal algebra: The number of actors plus the number of musicians equals 29 (10 1 19 5 29). However, only 24 people are auditioning. Thus, 5 of the 24 are actor-musicians, so 14 of the 19 musicians must not be actors. You can also solve this problem algebraically. The question describes three mutually exclusive sets: (1) actors who are not musicians, (2) musicians who are not actors, and (3) actors who are also musicians. The total number of people among these three sets is 24. You can represent this scenario with the following algebraic equation (n 5 number of actors/ musicians), solving for 19 2 n to answer the question: ~10 2 n!1n 1~19 2 n!524 29 2 n 5 24 n 5 5 19 2 5 5 14 TIP Regardless of the type of speed problem, start by setting up two distinct equations patterned after the simple rate-of-travel formula (r 3 t 5 d). Chapter 10: Math Review: Number Theory and Algebra 269 www.petersons.com 37. Adrian owns 60 neckties, each of which is either 100% silk or 100% polyester. Forty percent of each type of tie is striped, and 25 of the ties are silk. How many of the ties are polyester but not striped? (A) 18 (B) 21 (C) 24 (D) 35 (E) 40 The correct answer is (B). This double-overlap problem involves four distinct sets: striped silk ties, striped polyester ties, non-striped silk ties, and non-striped polyester ties. Set up a table representing the four sets, filling in the information given in the problem, as shown in the next figure: striped non-striped silk polyester 25 35 40% 60%? Given that 25 ties are silk (see the left column), 35 ties must be polyester (see the right column). Also, given that 40% of the ties are striped (see the top row), 60% must be non-striped (see the bottom row). Thus, 60% of 35 ties, or 21 ties, are polyester and non-striped. 270 PART IV: GMAT Quantitative Section www.petersons.com SUMMING IT UP • Make sure you’re up to speed on the definitions of absolute numbers, integers, factors, and prime numbers to better prepare yourself for the number theory and algebra questions on the GMAT Quantitative section. • Use prime factorization to factor composite integers. • GMAT questions involving exponents usually require that you combine two or more terms that contain exponents, so review the basic rules for adding, subtracting, multiplying, and dividing them. • On the GMAT, always look for a way to simplify radicals by moving what’s under the radical sign to the outside of the sign. • Most algebraic equations you’ll see on the GMAT exam are linear. Remember the operations for isolating the unknown on one side of the equation. Solving algebraic inequalities is similar to solving equations: Isolate the variable on one side of the inequality symbol first. • Weighted average problems and currency problems can be solved in a similar manner by using the arithmetic mean formula. • Mixture and investment problems on the GMAT can be solved using what you’ve learned about solving proportion and percentage questions. Rate of production and travel questions can be solved using the strategies you’ve learned about fraction problems. Chapter 10: Math Review: Number Theory and Algebra 271 www.petersons.com . solution. 264 PART IV: GMAT Quantitative Section TIP You can solve most GMAT currency problems by working backward from the answer choices. www.petersons.com INVESTMENT PROBLEMS GMAT investment. motion formula for each of the two parts of Scott’s journey: Formula: rate 3 time 5 distance Going: (40)(x) 5 40x Returning: (30)(7 2 x) 5 210 2 30x 268 PART IV: GMAT Quantitative Section www.petersons.com Because. time it takes for both x and y to complete the job working in the aggregate (together). 266 PART IV: GMAT Quantitative Section www.petersons.com So each fraction represents the portion of