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1 I. THE CONSTANT-RATE TRAVELING PROBLEMS 1.) Two cities are connected by means of a highway. A car leaves the city of Edison at 1:00 p.m. and travels at a constant rate of 40 mi/hr toward the city of Tenafly. Thirty minutes later, another car leaves B and travels toward C at a constant rate of 55 mi/hr. If the lengths of the cars are disregarded, at what time will the second car reach the first car? 2.) Two trains start from the same point and travel in opposite directions. The northbound train average 45 mi/hr and starts 2 hours before the southbound train, which averages 50 mi/hr. How long after the northbound train starts will the trains be 470 mi apart? 3.) A plane is flying at constant speed nonstop from Atlanta to Portland, a distance of about 2700 miles. After 1.5 hours in the air, the plane flies over Kansas City (a distance of 820 miles from Atlanta). Estimate the time it will take the plane to fly from Atlanta to Portland. 4.) Two trains left the same station at 3:15pm and traveled in opposite directions. The E train averaged 130 mph. The A train s speed was 110 mph. At what time were the trains 600 miles apart? 1

2 5.) Two cars traveled in opposite directions ffrom the same starting point. The rate of one car was 10 mph faster than the rate of the other car. After four hours the cars were 460 miles apart. Find each car s rate. 6.) A train left a station traveling 100 mph. A second train left two hours later and headed in the same direction at 125 mph. If the first train left at 10:30 am, at what time did the second train overtake the first train? 7.) Wilma drove at an average speed of 50 mi/hr from her home in Boston to visit her sister in Buffalo. She stayed in Buffalo 10 hours and, on the trip back, averaged 45 mi/hr. She returned home 29 hours after leaving. How many miles is Buffalo from Boston? 8.) A runner starts at the beginning of a runner s path and runs at a constant rate of 6 mi/hr. Five minutes later a second runner begins at the same point, running at a rate of 8 mi/hr and following the same course. How long will it take the second runner to reach the first? 2

3 9.) A man drove to the mountains at 60 mph. He returned at 40 mph. the return trip took 2 hours longer than the trip to the mountains. How long in time was the round trip to the mountains and back? 10.) A girl cycled 20 miles to the beach. She returned cycling one mph faster. The total time for the round trip was 9 hours. Find the girl s rate for each part of the trip. 11.) It took Dora the same time to drive 135 miles as it took Michelle to drive 180 miles. Dora s speed was 15 mph slower than Michelle s. How fast did Michelle drive? 12.) Kiran drove from Tortula to Cactus, a distance of 250 mi. She increased her speed by 10 mi/hr for the 360 mile trip from Cactus to Dry Junction. If the total trip took 11 hours, what was her speed from Tortula to Cactus? 3

4 13.) Tina hiked 15 km up a mountain trail. Her return trip along the same trail took 30 min less because she was able to increase her speed by 1 km/h. How long did it take her to climb up and down the mountain? 14.) Jacqui commutes 30 mi to her job each day. She finds that if she drives 10 mi/h faster, it takes her 6 min less to get to work. Find her new speed. 15.) Cindy and Dave left the dock to canoe downstream. Fifteen minutes later Tammy left by motorboat with the supplies. Since the motorboat traveled twice as fast as the canoe, it caught up with the canoe 3 km from the dock. What was the speed of the motorboat? 16.) Jack and Jill decide to exercise together by walking around a lake. Jack walks around the lake in 16 minutes and Jill jogs around the lake in 10 minutes. If Jack and Jill start at the same time and at the same place, and continue to exercise until they return to the starting point at the same time, how long will they be exercising? How many laps will Jill make? 4

5 17.) If you drive for 2 h at 80 km/h, how fast must you drive during the next hour in order to have an average speed of 75 km/h? 18.) (a) If you bike for 2 hours at 30 km/hr and 2 hours for 20 km/h, what is your average speed for the whole trip? (b) If you bike for 60 km at 30 km/h and return at 20 km/hr, what is your average speed for the whole trip? 19.) [AMC12A2005-6] Josh and Mike live 13 miles apart. Yesterday Josh started to ride his bicycle toward Mike s house. A little later Mike started to ride his bicycle toward Josh s house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike s rate. How many miles had Mike ridden when then met? 20.) [MMMPC ] A crew rows four miles downstream and back the same distance in one hour. If the stream flows at 3 miles per hour, the crew s rate of rowing in still water would in (in miles per hour)? 5

6 21.) A woman drove to work at an average speed of 40 miles per hour and returned along the same route at 30 miles per hour. If her total traveling time was 1 hour, what was the total number of miles in the round trip? 22.) Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages 40 miles per hour, he arrives at his workplace three minutes late. When he average 60 miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely in time? 23.) [AMC10B ] It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it? 24.) [AIME ] Rudolph bikes at a constant rate, and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the 50-mile mark at exactly the same time. How many minutes has it taken them? 6

7 Answers 1.) 2:50 pm. Let t be the time the first car traveled, then t 55( t ) t hr ) 6 hours. Let t be the time the northbound train traveled, 45t 50( t 2) 470 t 6 hr 3.) 4.94 hours. 1.5 hr t 2700 mi 4.94 hr 820 mi 4.) 5:45pm. 130t 110t 600 t 2.5 hr 5.) 52.5 mi, 62.5 mph. 4v 4( v 10) 460 v 52.5 mi / hr 6.) 8:30pm. 125( t 2) 100t t 10 hr 7.) 450 mi. Time to come back is 50(19 t) 45t t 10hours. Distance is Dist 45(10) 450mi. 5 8.) 15 min. 6t 8t t 0.25 hr 15 min. 60 d d ) 10 hours. 2 d 240 mi, t 10 hours ) 4 mi/hr and 5 mi/hr v 4 mi / hr. v v ) 60 mph. vm 60 mph. v 15 v M M 12.) 50 mph v 50 mph. v v10 13.) 5.5 hours v 5 mph, v v t hr v v 14.) 60 mi/hr v 50 mph, v mph v 10 v10 7

8 15.) 12 mi/hr. 1 3 vct 2vct 4, v 6 / c km hr, vm 2vc 12 km / hr. 16.) 8 laps. They will meet at LCM (16,10) 80 min, so Jill will run 8 laps. 17.) 65 mi/hr. 2(80) 1 x 75 x 65 km / hr 3 18.) 25 km/hr, 24 km/hr. 2(20) 2(20) 25 km / hr, km / hr 19.) 5 miles. Let, vt be Mike s speed and time. 4v vt 2t 13 vt 5 mi ) 9 mi/hr v 9 mi / hr v3 v3 21.) 34 2/7 mi. D D D mi ) 48 mi/hr. d 3 t d 3 t d v 48 mi / hr t 23.) 40 seconds. d 60vw d va 1.5 vw, d 60vw 40va ta 40 sec d 24( vw va) va vr t 49(5) ) 620 min. 3 vr t24(5) 50 4 t 620 min 8

9 II. THE SIMPLE INTEREST AND MONEY RELATED PROBLEMS 1.) Mrs. Stuart invested part of $4200 at 7% and the rest at 7.5%. If her total income from these investments was $306, how much did she invest at each rate? 2.) You invested a total of $10,000 at 4.5% and 5.5% simple interest. During one year, the two accounts earned a total of $ How much money did you invest in each account? 3.) A store has $30,000 of inventory in 13-inch and 19-inch color televisions. The profit on a 13-inch set is 22% and the profit on a 19-inch set is 40%, the profit for the entire stock is 35%. How much was invested in each type of television? 4.) Sally made two investments for a total amount of $1080. One investment is at 5% and the other at 7%. The annual simple interest from the 5% investment is $7.20 less than the interest from the 7% investment. How much money was invested at 7%. 9

10 5.) You sell computers and earn $50,000 salary plus a bonus. Your bonus is 1/20 of the amount by which your sales exceed $300,000. Find the amount of your sales must in order for you to earn a total of $60, ) You are planning to start a small business that will require an investment of $90,000. You have found some people who are willing to share equally in the venture. If you can find three more people, each person s share will decrease by $2500. How many people will be investing? 7.) A group of friends decides to buy a vacation home for $120,000, sharing the cost equally. If they could find one more person to join them, each person s contribution would drop by $6,000. How many people are in the group? 8.) A social club charters a bus at a cost of $900 to take a group of members on an excursion to Atlantic City. At the last minute, five people in the group decide not to go. This raises the transportation cost per person by $2. How many people went on the trip? 10

11 ) If the demand d and the supply s for motorcycles are d and s 200 p 1160, p find the profit p in hundreds of thousands of dollars, at the equilibrium point when demand is equal to supply. 10.) The cost of a bus trip was $180. People who signed up for the trip agreed to split the cost equally. However, six people did not show up, so that those who did go each had to pay $1.50 more. How many people actually went on the trip? 11.) The $75 cost for a party was to be shared equally by all those attending. Since five more people attended than was expected, the price per person dropped by 50 cents. How many people attended the party? 12.) An experienced plumber made $600 for working on the certain job. His apprentice, who makes $3 per hour less, also made $600. However, the apprentice worked 10 hours more than the plumber. How much does the plumber make per hour? 11

12 13.) [AMC10B2012-5] Anna enjoys at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of $27.50 for dinner. What is the cost of her dinner without tax or tip? 14.) [2010, AMC10A2010-8] Tony works 2 hours a day and is paid $0.50 per hour for each full year of his age. During a six month period Tony worked 50 days and earned $630. How old was Tony at the end of the six month period? 15.) At a party, there was one pizza for every 3 people, one salad for every 6 people, and one cake for every 8 people. If the total number of pizzas, salads, cakes was n, then, in terms of n, how many people were at the party? 16.) [AMC12A ] The state income tax where Kristin lives is levied at the rate of p % of the first $28000 of annual income plus ( p 2)% of any amount above$ Kristin noticed that the state income tax she paid amounted to ( p 0.25)% of her annual income. What was her annual income? 12

13 17.) Each day, Christina spent 20% of her money in the morning to buy breakfast. At the end of second day, $32 remained. How much money did she have originally? 18.) [AMC12A ] A charity sells 140 benefit tickets for a total of $2001. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full price tickets? 13

14 Answers 1.) $1800 in 7% and $2400 in 7.5%. 7% x 7.5%(4200 x) 306 x ) $4125 in 4.5%, $5875 in 5.5%. 4.5% x 5.5%(10000 x) x ) $8333 in 13 in and $21667 in 19 in. 22% x 40%(30000 x) 35%(30000) x ) $510. 5%(1080 x) 7% x 7.2 x , x x. 5.) $500, ) 9 people x 9. x x 3 7.) 4 people are in the group x 4. x x1 8.) 45 people went x 45. x x 5 9.) 700, p 1160 p 7 p ) 24 people n n n 6 11.) 25 people n 25 n n ) $15/hr x 15 x x 3 13.) $ 22. p 0.1p 0.15p p $22 14.) 13 years old. Tony made $1 (two hours at $0.5) times his age per day. Let a be Tony s age at the beginning of six months and x be the number of days before Tony ages 14

15 one more year. 1 ax 1 (50 x)( a 1) 630 or x 50a 580. The only solution possible is a 12 for 1x 50. So, Tony is 13 years old by the end of the six months. 15.) 8 n p p p 8 people. p 3 z, p 6 s, p 8 c, z s c n n p n p 16.) $ (28000) ( p 2)( x 28000) x( p 0.25) x ) $50. x(1 20%)(1 20%) 32 x ) $782. Let f be full price and n be the number of people. f 4002 fn (140 n) 2001 (140 n) f f. 2 (140 n) f {2, 3, 23, 29} and 0 n 140. Therefore, f or 14.2 f This implies f 23 and n fn

16 III. THE MIXTURE AND RATIO PROBLEMS 1.) A health food store sells a mixture of raisins and roasted nuts. Raisins sell for $3.50/kg and nuts sell for $4.75/kg. How many kilograms of each should be mixed to make 20 kg of this snack worth $4.00/kg? 2.) An auto mechanic has 300 ml of battery acid solution that is 60% acid. He must add water to this solution to dilute it so that it is only 45% acid. How much water should he add? 3.) The owner of the Fancy Food Shoppe wants to mix cashews selling at $8.00/kg and pecans selling at $7.00/kg. How many kilograms of each kind of nut should be mixed to get 8 kg worth $7.25/kg? 4.) If 800 ml of a juice drink has 15% grape juice, how much grape juice should be added to make a drink that is 20% grape juice? 16

17 5.) A grocer mixes 5 lb of egg noodles costing 80 cents/lb with 2 lb of spinach noodles costing $1.50/lb. What will the cost per pound of the mixture be? 6.) How many liters of water must be added to 50 L of a 30% acid solution in order to produce a 20% acid solution? 7.) A spice mixture is 25% thyme. How many grams of thyme must be added to 12 g of the mixture to increase the thyme content to 40%? 8.) Joanne makes a mixture of dried fruits by mixing dried apples costing $6.00/kg with dried apricots costing $8.00/kg. How many kilograms of each are needed to make 20 kg of a mixture worth $7.20/kg? 17

18 9.) A chemist has 10 milliliters of a solution that contains a 30%concentration of acid. How many milliliters of pure acid must be added in order to increase the concentration to 50%? 10.) Rebecca has 38 ounces of a 55% boric acid solution, which she wants to dilute to a 38% solution. How much water should she add to the boric acid? 11.) A radiator contains 8 quarts of a mixture of water and antifreeze. If 40% of the mixture is antifreeze, how much of the mixture should be drained and replaced by pure antifreeze so that the resultant mixture will contain 60% antifreeze? 12.) Peanut butter costs two cents an ounce. Jelly costs three hundred sixty eight cents per pound. Jordan wants to make 4 pounds of peanut butter and jelly. If she wants the entire mix to cost $2.96, how many pounds of jelly should she use? 18

19 13.) A grocer wants to make a mixture of three dried fruits. He decides that the ratio of pounds of banana chips to apricots to dates should be 3:1:1. Banana chips cost $1.17/lb, apricots cost $3.00/lb, and dates cost $2.30/lb. What is the cost per pound of the mixture? 14.) Tom has 1/3 more money than what Jessica has, and when Tom gives $9 to Jessica, Jessica has 4/5 more money than what Tom has. How much money does each have initially? 15.) John is 20% older than Madison, and Madison is 20% older than Adam. How much percent older than Adam is John? 16.) If a: b 21: 4and a: c 7:6, what is a: b: c? 19

20 17.) The ratio of chickens to pigs is 26:5; ratio of sheep to horses is 25:9; ratio of pigs to horses is 10:3. What is the ratio of chickens to pigs to horses to sheep? 18.) The total weight of three fruit baskets is 60 grams. If we move 3 grams from the first and second baskets respectively to the third basket, the ratio of the weights of the three baskets is 1:2:3. Find the original weight of each basket. 19.) The concentration of a solution is 30%. Then some water is added to make it 24% solution. If the same amount of water is added again, what is the concentration of the solution now? 20.) The 100 grams of salt water has 80% concentration. You pour out 40 grams and refill with the same amount of water. If you repeat this process three times, what is the final concentration of the salt water? 20

21 21.) A piece of coal contains 14.5% of water. After being exposed to air for days, the water reduces to 10%. The weight now is what percent of the original one? 22.) Cup A contains solution with 40% salt; cup B contains solution with 36% of salt; cup C contains solution with 35% salt. When mixed all three cups we get 11 grams of solution with 38.5% salt. If we know cup B has 3 grams solution than cup C does, how many grams of solution in cup A? 23.) One foot is 12 inches. How many feet are in 64 inches? 24.) There are approximately 454 grams in a pound. How many grams are in 0.35 pounds? 25.) There are approximately 3.28 feet in a meter. Determine, to the nearest square foot, how many square feet there are in 29 square meters. 21

22 26.) Dr. Kahn has ordered a special medicine from Europe. It comes with strict instructions to use 32 milliliters per kilogram that the patient weighs. However, all of Dr. Kahn s scales only tell weight in pounds. There are approximately kilograms in a pound. To the nearest 0.1 milliliter, how many milliliters per pound should Dr. Kahn use? 27.) My boss has told me that I will need one gallon of paint for every three hundred square feet of wall I must paint. Unfortunately, the store only sells cans containing 4 liters of paint, and our client has told me that she needs 370 square meters of wall painted. One liter contains approximately gallons, and there are approximately 3.28 feet in a meter. What is the smallest number of paint cans I can buy to complete the paint job? 28.) On planet Ghaap, two Gheeps are worth three Ghiips, two Ghiips are worth five Ghoops, and three Ghoops are worth two Ghuups. How many Ghuups are seven Gheeps worth? 29.) There are approximately kilograms in a pound. To the nearest whole pound, how many pounds does a steer that weighs 200 kg weigh? 30.) Janie can stuff 30 envelops in one minute. Find an expression for the number of envelopes she can stuff in n hours. 22

23 31.) I m visiting Germany, but forgot to exchange dollars to euros. My meal costs 17 euros. I give the cashier 40 dollars. If 1 euro is worth $1.32, then how much change in euros should I receive? 32.) In a far-off land three fish can be traded for two loaves of bread, and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth? 33.) Clint s Cowboy Shop buys horse feed for $10 per cubic meter. Clint s customers don t like the metric system, so they ll only buy horse feed by the cubic foot. How much should Clint charge for a cubic foot in order to double his money? (Assume 1 meter equals 3.28 feet.) 34.) The value of 10 pounds of gold is d dollars, and a pound of gold has the same value as p pounds of silver. What is the value, in dollars, of one pound of silver? 35.) Ben can type a full report in h hours. At this rate, how many reports can he type in m minutes? 23

24 36.) The price of 10 pounds of apples is d dollars. If the apples weigh an average of 1 pound for every 6 apples, what is the average price, in cents, of a dozen such apples? 37.) The price of ground coffee beans is d dollars for 8 ounces and each ounce makes c cups of brewed coffee. In terms of c and d, what is the dollar cost of the ground coffee beans required to make 1cup of brewed coffee? 38.) Eight students play a game that lasts for 80 minutes, but only allows 5 students to play at a time. If each of the eight students plays the same length of time throughout the game, how many minutes does each student play? 39.) [AMC ] One morning each member of Angela s family drank an 8-ounce mixture of coffee with milk. The amounts of coffee varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family? 24

25 40.) [AIME2008I-1] Of the students attending a school party, 60% of the students are girls, and 40% of the students like to dance. After these students are joined by 20 more boy students, all of whom like to dance, the party is now 58% girls. How many students now at the party like to dance? 41.) [AIME2011I-1] Jar A contains four liters of a solution that is 45% acid. Jar B contains five liters of a solution that is 48% acid. Jar C contains one liter of a solution that is k % acid. From jar C, m n liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end both jar A and jar B contain solutions that are 50% acid. Given that m and n are relatively prime positive integers, find k m n. 25

26 Answers 1.) 12 kg of Raisins and 8 kg of nuts. 3.5r 4.75(20 r) 4(20) r 12 kg. 2.) 100 ml. 300(60%) 45% w 100 ml 300 w. 3.) 2 kg of cashews and 6 kg of pecans. 8c 7(8 c) 7.25(8) c 2 kg 4.) 50 ml. 800(15%) g 20% g 50 ml 800 g 5.) $1/lb. 5(0.8) 2(1.5) (5 2) p, p 1 lb. 6.) 25 L. 50(30%) 20% w 25 L 50 w. 7.) 3 g. 12(25%) x 40% x 3 g. 12 x 8.) There are 8 kg of apples and 12 kg of apricots. Let e be the number of apples. Then 6e 8(20 e)7.20(20) e 8 kg 9.) 4 ml. 10(30%) a 50% a 4 ml. 10 a 10.) 17 oz. 38(55%) 38% w 17 oz 38 w 11.) 8 8(40%) 40% x x 8 quarts. 60% x quarts ) 0.5 lb. Peanut is $0.32/lb, Jelly is $3.68/lb. 3.68J 0.32(4 J ) 2.96 J 0.5 lb. 13.) $1.76/lb. 1.17(3) 3(1) 2.3(1) (3 11) x x

27 14.) J=18, A= J 9 1 J 9 J ) 44%. J (1 20%)(1 20%) A 1.44 A (1 44%) A 44% older. 16.) 21:4:18. Let b 4 a 21, c 18 a : b : c 21: 4 : ) 156:30:9:25. Let h 3. Then p 10, c 52, s c : p : h : s 52 :10 : 3 : 156 : 30 : 9 : ) 13, 23, 24. x 2x 3x 60. Then, x 10, f 3 : s 3 : t 6 1: 2 : 3 10 : 20 : 30, f 13, s 23, t ) 20%. 30% x 24% x 4w x w, 30% x 30% x 20% x 2w x x 2 20.) 17.28%. The amount left after 1 st time is (100 40)(80%) 48 g, after water is added, the concentration is 48%. Similarly, after 2 nd time, (100 40)(48%) 28.8 g, or the concentration is 28.8%. The 3 rd time is (100 40)(28.8%) g, or the concentration is 17.28%. 21.) 95%. Let x the original weight of the coal. So, the weight of pure coal is c x. w After the exposure, the water left in the coal is 10% w 0.095x w 0.855x. The weight of coal now is x' 0.855x 0.095x 0. 95x. Or, 95% of the original weight. 22.) 7 g. bc3 a b c 11 a 7 g 40% a 36% b 35% c 38.5%(11) 23.) 16/3 feet. 1 ft in 64 in ft. 12 in 3 27

28 24.) 159 g. 454 g 0.35 lb 0.35 lb 159 g. 1lb 25.) 312 square feet ft 29 m 29 m 312 ft 1 m ) 14.5 ml/lb. 32 ml 32 ml kg 14.5 ml / lb. 1 kg 1 kg 1lb 27.) 13 cans ft 1 gal 1 L 1 can m m can. 1 m 300 ft gal 4 L 28.) 35/2 Ghuups. 3i 5o 2u 35 7e 7e u. 2e 2i 3o 2 29.) 441 lb. 1lb 200 kg 200 kg 441 lb kg 30.) 1800n envelops per n hours. 60 min 30 env n hr n hr 1800n env. 1 hr 1 min 31.) euros as a change 32.) 8/3 bags 33.) $0.57/cubic feet. $ m $10 1 ft 1 ft $ ft 1 m, double the amount will be d 34.) 10 p. 1 gld d $ d 1 si 1 si $ p si 10 gld 10 p m 35.) 60 h. 1 rpt m m min m min rpt h hr 60h 28

29 1 lb d $ 100 cent 36.) 20d. 12 a 12 a 20d cent 6 a 10 lb 1$ d 37.) 8 c. 1 oz d $ d 1 cup 1 cup $ c cup 8 oz 8 c 38.) 50 min. T avg Total played time 5(80) 50 min. # of students 8 39.) 5 people. Let n be the number of people, m be the number of ounces of milk and m c 8 m c be the number of ounces of coffee. Then, 4 6 n 6. So, m is a 16 m c 8n multiple of 16 and less than 32. This implies m 16, n ) 252 students like to dance. Total number of students is 40% T 58% T 580. T 20 Before the 20 boys joined, 40% T 40%(580) 232 students like to dance. Now like to dance. 41.) 85. k m 0.45(4) 100 n 1 m k m m n 50 n n k m m 0.48(5) 1 k n m n m 2 50 n 51 n km 20n 50m k 80 kn km 60n 50m 80 m m m , k m n n n n 3 29

30 IV. THE CHEMISTRY PROBLEMS Concentration Ratios: mass of solute Mass Percent 100% mass of solution moles of solute Molarity liter of solution moles of solute Molality kg of solvent ( M ) ( m) moles of solute Mole Fraction moles of solute moles of solvent 30

31 1.) The mass of 1 mole of aluminum weighs g. How many moles is 10 g of aluminum? 2.) One mole of Isopentyl acetate weighs g and contains How many molecules does g of Isopentyl acetate have? molecules. 3.) A model Corvette has an engine with a displacement of 6.20 L. What is the displacement in units of cubic inches? One cubic feet is L. 31

32 2 4.) A solution is prepared by mixing 1.00 g (or mol ) ethanol with g (or 5.56 mol ) water to give a final volume of 101 ml. Calculate the molarity, mass percent, mole fraction, and molality of ethanol in this solution. 5.) A battery has 3.75 M sulfuric acid 2 4 H SO solution that has a density of g/ml. Calculate mass percent and molality of sulfuric acid. One mole of sulfuric acid is 98.0 g. 32

33 Answers 1 mol 1.) moles. 10 g 10 g mol g 2.) mol g 110 g g 1 mol ) in ft 12 in 6.20 L 6.20 L 378 in L 1 ft 4.) Mass Percent of ethanol = 0.990%, Molarity of ethanol = M, Mole Fraction of ethanol = , Molality of ethanol = m g ethanol Mass Percent 1 g ethanol 100 g water 100% 0.990% ethanol mol ethanol mol ethanol 1000 ml Molarity of ethanol M 101 ml solution 101 ml solution 1 L mol ethanol mol ethanol 1000 g Molality of ethanol m 100 ml water 100 g water 1 kg 5.) The density of the solution in L is g g 1000 ml ml ml 1 L g/ L mol H2SO g The mass of sulfuric acid is 368 g / L H2SO4 1L 1mol H2SO. The 4 3 mass of water is one liter is g solution 368 g H SO 862 g H O g H SO Mass Percent g Molality H SO % 29.9% H SO 3.75 mol H SO kg H2O 862 g 1000 g H O m 33

34 V. THE WORK AND RATE PROBLEMS 1.) Josh can split a cord of wood in 4 days. His father can split a cord in 2 days. How long will it take them to split a cord of wood if they work together? 2.) Robot A takes 6 min to weld a fender. Robot B takes only 5 ½ min. If they work together for 2 min, how long will it take Robot B to finish welding the fender by itself? 3.) It takes Sally 15 min to pick the apples from the tree in her backyard. Lisa can do it in 25 min. How long will it take them work together? 4.) A roofing contractor estimates that he can shingle a house in 20 hours and that his assistant can do it in 30 hours. How long will it take them to shingle the house working together? 34

35 5.) One printing machine works twice as fast as another. When both machines are used, they can print a magazine in 3 hours. How many hours would each machine require to do the jobs alone? 6.) It takes my father 3 hours to plow our cornfield with his new tractor. Using the old tractor it takes me 5 hours. If we both plow for 1 hour before I go to school, how long will it take him to finish the plowing? 7.) One pump can fill a water tank in 3 hours, and another pump takes 5 hours. When the tank was empty, both pumps were turned on for 30 min and then the faster pump was turned off. How much longer did the slower pump have to run before the tank was filled? 8.) Ramona can do a job in 12 days. After she has worked for 4 days, she is joined by Carlotta and it takes them 2 days working together to finish the job. How long would it have taken Carlotta to do the whole job herself? 35

36 9.) Nicholas and Marilyn are addressing invitations to the junior class picnic. Nicholas can address one every 30 seconds and Marilyn can do one every 40 seconds. How long will it take them to address 140 invitations? 10.) If three pipes are all opened, they can fill an empty swimming pool in 3 hours. The largest pipe alone takes one third the time that the smallest pipe takes and half the time the other pipe takes. How long would it take each pipe to fill the pool by itself? 11.) A water tank can be emptied by using one pump for 5 hours. A second smaller pump can empty the tank in 8 hours. If the larger pump is started at 1:00 pm, at what time should the smaller pump be started so that the tank will be emptied at 5:00 pm? 12.) Candy and Isha share a paper route. It takes Candy 70 minutes to deliver all the papers, whereas Isha takes 80 minutes. How long does it take them to work together? 36

37 13.) It takes Josh 14 hours to repair a car s transmission. After he had worked for 7 hours, Anish began to help him. Together they finished the job in 3 more hours. How long would it take Anish to repair the car by himself? 14.) Henry and Irene working together can wash all the windows of their house in 1 hour 48 minutes. Working alone, it takes Henry 1 hour 30 minutes longer than Irene to do the job. How long does it take Henry to wash the windows alone? 15.) Jack, Kay and Lynn deliver advertising flyers in a small town. If each person works alone, it takes Jack 4 hours to deliver all the flyers, it takes Lynn 1 hour longer than Kay to deliver all the flyers. Working together, they can deliver all of the flyers in 40% of the time it takes Kay working alone. How long does it take Lynn to deliver all the flyers working alone? 16.) [AMC12B2008B-10] Bricklayer Brenda would take 9 hours to build a chimney alone, and bricklayer Brandon would take 10 hours to build it alone. When they work together they talk a lot, and their combined output is decreased by 10 bricks per hour. Working together, they build the chimney in 5 hours. How many bricks are in the chimney? 37

38 Answers 1.) 4 3 days t 1 t days ) 5 3 min. 1 1 t t min ) t 75 8 min t 1 t min ) t 12 hours t t hours. 5.) 9 hours and 4.5 hours t 4.5 hours for the faster machine. t 2t 6.) 7 t hours t t ) 11 t hours t t ) t 4days t t 9.) 2400 seconds t 10.) 33 2 hr, 11 hr, 11 2 hr. 11.) 3:24pm. 4 t , t t t, t t, t t t1 t2 t

39 12.) min t 13.) 10.5 hours t 14.) 4.5 hours H H 1.5 H 4.5 hours. Note H 0.6 hours is not a solution. 15.) 4 hours. 16.) 900 bricks tK 1 4 tk 1 tk t 3 hours K t t 1 4hours L K x 90 hr. Total bricks is x 39

40 VI. THE COUNTING PROBLEMS 1.) How many integers are there from 25 to 79 inclusive? 2.) What is the 53 rd integer in the sequence86, 87, 88,? 3.) The largest of r consecutive integers is k. What is the smallest? 4.) What is the smallest number of coins needed to pay in exact change for any change less than one dollar? (Coins are in the denominations 1, 5, 10, 25, 50 cents) 5.) At a party, each man danced with exactly three different women and each woman danced with exactly two different men. Twelve men attended the party. How many women attended the party? 40

41 6.) A grocer puts 18 apples in n bags that each bag contains the same number of apples. If there is more than one apple in each bag and fewer than 18, what are the possible values of n? (This is a factor problem.) 7.) Lauren wants to take four courses chosen from the categories of math, sciences and humanities. Assume that each category offers at least four courses for her to take and she can take those four courses in any combination from the three categories. How many different ways can she select the courses? (This is an integer equation problem.) 8.) A worm has to climb a barrier 19 feet high. It climbs 8 feet every day and slips down 5 feet every night. Starting from the bottom of the barrier on Sunday morning, on which day of the week will the worm reach to the top of the barrier? 9.) THS has 13 clubs with numbers of members as shown below: Club # of people On the club-meeting day, the members of 12 out of 13 clubs went to the meeting held in two different rooms. The number of people at the first room was six times as many as that of the second room. Which club did not attend the meeting? (This is a remainder problem.) 41

42 10.) The integers from 1 to 15 are written in numerical order in pencil going clockwise around a circle. A student begins moving clockwise around the circle erasing every third integer that has not yet been erased until only the integer 11 remains. Which integer did the students erase first? 11.) JPS has 44 boxes of textbooks, and each box contains 113 textbooks. If JHS wants to regroup the textbooks into 12 books per box for each of the smaller boxes, how many books will be the leftover after filling the smaller boxes? (This is a modular problem.) 12.) Suppose that 80% of students at JPS own iphones and 70% of the students own Apple laptops. What is range of possible percentages of students that own both? 13.) [AMC10B ] On a 50-question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and -1 point for an incorrect answer. Jesse s total score on the contest was a 99. What is the maximum number of questions that Jesse could have answered correctly? 42

43 14.) A bag contains red, blue and yellow socks. It has 10 socks of each of the three colors. How many socks do you need to pick to guarantee to have obtained two identical pairs that each pair has two-different-color socks? For instance, two pairs of socks that each has a red sock and a yellow sock. (This is a pigeonhole problem.) 15.) A group of children try to share some candies. If each child takes k candies, 14 candies will be left; if each child before him takes 9 candies, the last child will only have 6 candies. How many children are there in the group? (This is an integer problem.) 16.) [AMC10A ] A majority of 30 students in Ms. Demeanor s class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all pencils was $ What was the cost of a pencil in cents? (This is a factor problem.) 17.) [AMC12A2010-5] Halfway through a100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye, she scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea s next n shots are bullseyes, she will be guaranteed victory. What is the minimum value of n? 43

44 18.) Three chickens can lay three eggs in a day and a half. How long will it take for two chickens to lay 200 eggs? 19.) You have an unlimited supply of unbreakable sticks of length 2, 4, and 6 inches. Using these sticks, how many non-congruent triangles can you make? Two sticks can be joined only at a vertex of triangle (A triangle with sides of length 4, 6, 6 is an example of one such triangle to be included.) 20.) How many factors does 180 have? 21.) You climb a flight of 10 stairs to go to the 2 nd floor of the THS building. With each step, you can climb either one or two stairs. In how many different ways can you reach the 2 nd floor by climbing the flight of stairs. (This is a recursion problem.) 22.) [AMC10A ] In 1991, the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. What is the percent growth of the town s population during this twenty-year period? (This is a factor problem.) 44

45 23.) How many 3-digit numbers formed by using 0, 3, 5 and 7 are divisible by 2, 3 and 5? (This is a divisibility problem.) 24.) What are the three digits abcsuch,, that the number 865abc is the smallest number that is divisible by 3, 4 and 5? 25.) [AMC10B2012-7] Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 corns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide? 45

46 Answers 1.) 55. Subtract 24 from all 25, 26,, 79, the result is 1, 2,, 55. So, there is 55 integers. 2.) 138. Subtract 85 from all 86, 87, 88,, the result is 1, 2, 3,. So, the 53 rd term should be ) x k r 1. Let the smallest integer be x. Then x, x 1,, x r 1, and hence x k r 1. 4.) Nine coins: one half dollar, one quarter, two dimes, one nickel, 4 pennies. 1, 11 2, 111 3, , , , , , 5.) 18. The number of dances was 12(3) 36, and each woman danced twice, so, the number of women was 36 / ) The factors of 18 are 1, 2, 3, 6, 9,18. The answer could be any of those numbers except 1 and 18. So, there are 4 possible choices for n. 7.) 15. It is equivalent to solve m s h 4for 0 m, s, h 4. A table can be used to tabulate the cases: m s h # of choices 0 sh sh sh sh sh 0 1 So, there are total of 15 selections. 8.) Thursday. Sun Mon Tue Wed Thu Morning high Evening low

47 The answer is on Thursday the worm will reach to the top. 9.) 13 people. There are total of 160 people in all clubs. If the number in the first room was x, the number of people in the second room was 6x, and the total number of people attended the meeting was 7x. That is, 160 7x y, where y is the number of people in a club y y who did not attend the meeting. x 22. Or the remainder of is 6, which implies that the club was Club #9 with 13 people. 10.) 9. If it starts from 1, 2, 3,, 15, the first number erased is 3, then 6,.. and the last is 5. Re-label 5 to 11, 4 to 10, 3 to 9, etc, so the first number that is erased is ) 4. Since (312 8)(9 12 5) , the leftover should be the remainder of 40 /12, which is 4. The problem can also be solved by using the modular arithmetic a b(mod n), where a b is an integer. In this problem, 44 8(mod 12), 113 5(mod 12), and n (mod 12) 12.) 50%~70%. The maximum overlapping is the 70% of the students who own laptops also own iphones, and minimum overlapping is 30% of students who do not own the laptops own iphones, so 50% own both. So, the range is between 50% and 70%. 13.) 29. Let x be the number of questions that Jesse could answer correctly and y be the questions Jesse leaves blank. Then, maximum of x is achieved at leaving the fewest questions blank. So, 149 y 4 x (50 y x) 99 x For x to be maximum and an integer, y is 4 and x ) 13. The worst case is to pick the same color 10 times for the first 10 socks, for instance, the first 10 socks are red. Since there is no more red socks, the 11 th, 12 th picks could be either yellow or blue. If they are the same colors, then you are done; if they are different colors, then you need to pick one more sock so that the 11 th, 12 th, and 13 th picks have to have at two same-color socks. So, the answer is that you need to pick at least 13 socks to guarantee to have two pairs with the same different-color socks. 5 47

48 15.) 17. Let n be the number of children. Then, kn 14 9( n 1) 6 or (9 kn ) 17. Since n is a non-negative integer, the only number for k is 8, which implies n ) 7. Let c be the cost for a pencil, p be the number of pencils, and n be the number of students. Then, c p n Since p 1, c p and 15 n 30, this implies p 7, n 23 and c ) 42. Let n be the minimum number of bullseyes. The maximum points for the remaining 50 shots are 500 when every shot hits bullseyes. Therefore, the minimum number is n 50 4(50 n) 500 n egg 2 18.) 150 days. The rate is r egg / ( chicken day), Let x be # of 3 chicken 1.5 day 3 days 2 so x ( egg / chicken day)( chicken day), x 150 days The question can be solved by cases: a.). All sides are the same: 2,2,2, 4,4,4, and 6,6,6, total three equilateral triangles. b.) Two sides are the same: 2,2,4, 4,4,2, 4,4, 6, and 6,6,4, total four isosceles triangles. c.) All three sides are different: none. The answer is ) 18 factors (3)(2) 18 factors th 21.) 89 ways. Let f ( n) be the number of different ways to reach the n stair. Obviously, f (1) 1, there is only one way to climb the 1 st stair, and f (2) 2, two ways to reach the 48

49 2 nd stair by either two one-stair steps or one two-stair step. The rest of the problem can be solved from counting the ways down from the last stair. To reach 10 th stair, there are two possible ways to go: either one stair from the 9 th stair or two stairs from the 8 th stair. That is, f (10) f (9) f (8). By the same token, f (9) f (8) f (7),, f (3) f (2) f (1). Because f (1) 1and f (2) 2, f (3) f (2) f (1) 1 2 3, f (4) f (3) f (2) 3 2 5, f (5) f (4) f (3) f (10) f (9) f (8) There are 89 ways to climb the 10 stairs ) 62%. Assume at 1991, the population was n, and at 2001, the population was m 9, in 2011, the population was q. Then m 9 n 150 and q 9 m 150, and 2 2 mn47 m n m 25, n 22 mn q So the percentage change was / /121, or about 62%. 23.) 570, 750. Since the number is divisible by 2 and 5, so the number will end with zero. Since the number is also divisible by three, the sum of its digits should be a multiple of three. That is, the rest of two digits are 5 and 7. The two possible numbers are 570, ) Divisible by 4 and 5 implies c 0, or 865ab 0. To make 865ab0divisible by 3, the sum of digits is a multiple of 3. Since and 19 10, then ab 2, hence, a 0, b2 will make 865abc the smallest as ) 48 acorns. 49

50 VII. THE NUMBER PROBLEMS 1.) The sum of a number and its reciprocal is Find the number. 2.) The sum of a number and its reciprocal is Find the number. 3.) The sum of the reciprocals of two consecutive odd integers is 8. Find the integers ) The sum of the reciprocals of two consecutive even integers is 11. Find the integers ) The numerator of a fraction is 1 more than the denominator. If the numerator and denominator are both increased by 2, the new fraction will be 1 less than the original 4 fraction. Find the original fraction. 50

51 6.) The numerator of a fraction is 1 less than the denominator. If the numerator and the denominator are both increased by 4, the new fraction will be 1 more than the original 8 fraction. Find the original fraction. 7.) The sum of two numbers is 10 and the sum of their reciprocals is 5. Find the 12 numbers. 8.) The two numbers differ by 11. When the larger number is divided by the smaller, the quotient is 2 and the remainder is 4. Find the numbers. 51

52 Answers 1.) 1 5, 5 2.) 2, ) 3, 5 4.) 10, 12 5.) 3, 3 2 4, x1 x3 1 x x ) 9, 3 8 4, 1 x1 x3 8 x x 4 7.) 4, 6, 8.) 7, 18, n 10 n 12 n n n 52