Problem solving: multi-step reasoning in depth: open practice pack ################################################################## 100 untimed questions across 10 authored sections. Pack: cognitive-skills-problem-solving Source: cognitive-skills Pack version: v1 Worksheet items: 100 Authored hints: 0 Worked explanations: 100 ! This is untimed educational practice for one cognitive skill. It is not a clinical, diagnostic, employment, or professionally recognized assessment, and it does not produce an IQ score or credential. ! The answer key and worked explanations in this file belong only to this public, untimed educational practice source. No active aptitude suite or Novus Reasoning Challenge answer key is ever included in any download. Learner worksheet ================= Problem solving is the skill of combining several pieces of information and reaching a practical, well-reasoned answer. Every item is self-contained. All the data you need is in the question. Work through each section at a comfortable pace; the sections build from single careful steps to integrated challenges. Choose one answer for each item. For a delayed-recall item, read its study text, cover it, and then answer without looking back. Multi-step foundations ---------------------- Warm up by combining two or three simple facts into a single answer. 1. problem-solving-s01-q01 -------------------------- Three coworkers restock shared printer paper on different cycles: every 2 days, every 3 days, and every 4 days. All three restocked together today. In how many days will all three restock on the same day again? - A. 6 - B. 8 - C. 12 - D. 24 Your answer: ______ 2. problem-solving-s01-q02 -------------------------- A delivery van can carry either 40 small boxes or 25 large boxes (space scales proportionally by count). It currently carries 16 small boxes. How many large boxes can it still fit? - A. 9 - B. 12 - C. 15 - D. 18 Your answer: ______ 3. problem-solving-s01-q03 -------------------------- A 4-character password uses only the digits 1, 2, 3, and 4, with no digit repeated. How many different passwords are possible? - A. 4 - B. 12 - C. 24 - D. 64 Your answer: ______ 4. problem-solving-s01-q04 -------------------------- Item A sells 8 units per day from a 40-unit stock; Item B sells 5 units per day from a 40-unit stock. Assuming no restocking, which item runs out first, and by how many days sooner? - A. A, by 2 days - B. A, by 3 days - C. B, by 3 days - D. B, by 5 days Your answer: ______ 5. problem-solving-s01-q05 -------------------------- A task force needs exactly one representative from each of three departments. Sales has 4 possible reps, Engineering has 3, and Support has 2. How many different three-person task forces are possible? - A. 9 - B. 18 - C. 24 - D. 32 Your answer: ______ 6. problem-solving-s01-q06 -------------------------- Four friends (Sam, Priya, Marcus, and Yuki) sit in a row of four seats numbered 1 to 4, left to right. Priya is not at either end. Sam is immediately to the left of Priya. Marcus is in seat 4. Yuki is not in seat 1. Which seat is Yuki in? - A. Seat 1 - B. Seat 2 - C. Seat 3 - D. Seat 4 Your answer: ______ 7. problem-solving-s01-q07 -------------------------- A cinema ticket costs £9 and a snack combo costs £6. Ana buys 2 tickets and 1 combo, paying with £30. How much change does she receive? - A. £6 - B. £4 - C. £5 - D. £7 Your answer: ______ 8. problem-solving-s01-q08 -------------------------- A lift can carry at most 600 kg. Four crates weigh 150 kg, 180 kg, 120 kg, and 200 kg. Which single crate should be left behind so the other three weigh as much as possible without exceeding the limit? - A. The 150 kg crate - B. The 180 kg crate - C. The 200 kg crate - D. The 120 kg crate Your answer: ______ 9. problem-solving-s01-q09 -------------------------- Tea bags come in boxes of 40. The office uses 25 tea bags per week. How many full weeks will 2 boxes last? - A. 3 - B. 4 - C. 2 - D. 5 Your answer: ______ 10. problem-solving-s01-q10 --------------------------- A bus departs every 15 minutes starting at 09:00. Leo arrives at the stop at 09:50. How long must he wait for the next bus? - A. 5 minutes - B. 15 minutes - C. 25 minutes - D. 10 minutes Your answer: ______ Rates, ratios and fair shares ----------------------------- Scale quantities up and down, and split amounts in a stated proportion. 11. problem-solving-s02-q01 --------------------------- Three friends share a £45 restaurant bill in the ratio 2 : 3 : 4. How much does the friend with the middle share pay? - A. £15 - B. £10 - C. £20 - D. £12 Your answer: ______ 12. problem-solving-s02-q02 --------------------------- A printer produces 12 pages per minute. How long will it take to print 300 pages? - A. 20 minutes - B. 25 minutes - C. 30 minutes - D. 24 minutes Your answer: ______ 13. problem-solving-s02-q03 --------------------------- Orange squash is mixed using 1 part concentrate to 4 parts water. How much concentrate is needed to make 2,000 ml of mixed squash? - A. 500 ml - B. 250 ml - C. 200 ml - D. 400 ml Your answer: ______ 14. problem-solving-s02-q04 --------------------------- Two decorators working together at the same rate paint a room in 6 hours. How long would one decorator take alone? - A. 12 hours - B. 9 hours - C. 3 hours - D. 8 hours Your answer: ______ 15. problem-solving-s02-q05 --------------------------- A car uses 6 litres of fuel per 100 km. How much fuel is needed for a 250 km journey? - A. 12 litres - B. 15 litres - C. 18 litres - D. 14 litres Your answer: ______ 16. problem-solving-s02-q06 --------------------------- Nia earns £14 per hour, and time-and-a-half for overtime. One week she works 35 standard hours and 4 overtime hours. What is her total pay? - A. £546 - B. £560 - C. £588 - D. £574 Your answer: ______ 17. problem-solving-s02-q07 --------------------------- A map is drawn at a scale of 1 : 50,000. Two villages are 6 cm apart on the map. What is the real distance between them? - A. 3 km - B. 30 km - C. 0.3 km - D. 5 km Your answer: ______ 18. problem-solving-s02-q08 --------------------------- A recipe for 4 people needs 300 g of rice. How much rice is needed for 10 people? - A. 650 g - B. 700 g - C. 750 g - D. 800 g Your answer: ______ 19. problem-solving-s02-q09 --------------------------- £120 is shared between two grandchildren so that the elder receives twice as much as the younger. How much does the younger receive? - A. £60 - B. £40 - C. £30 - D. £80 Your answer: ______ 20. problem-solving-s02-q10 --------------------------- A tap fills a 90-litre water butt in 45 minutes at a steady rate. After 30 minutes the tap is turned off. How many more litres are needed to fill the butt? - A. 20 litres - B. 45 litres - C. 25 litres - D. 30 litres Your answer: ______ Working backwards ----------------- Start from the final result and reverse each step to recover the starting value. 21. problem-solving-s03-q01 --------------------------- Mia thinks of a number, doubles it, then adds 7 to get 31. What number did she start with? - A. 10 - B. 12 - C. 14 - D. 19 Your answer: ______ 22. problem-solving-s03-q02 --------------------------- After spending half his money on lunch and then £4 on a coffee, Ben has £6 left. How much did he start with? - A. £20 - B. £16 - C. £24 - D. £18 Your answer: ______ 23. problem-solving-s03-q03 --------------------------- A number is divided by 3, then 5 is subtracted, giving 7. What was the original number? - A. 26 - B. 21 - C. 30 - D. 36 Your answer: ______ 24. problem-solving-s03-q04 --------------------------- A shop reduces a coat by 20% to a sale price of £48. What was the original price? - A. £57.60 - B. £56 - C. £60 - D. £64 Your answer: ______ 25. problem-solving-s03-q05 --------------------------- A patch of mould on a slice of bread doubles in area every day. It covers the whole slice on day 12. On which day did it cover exactly a quarter of the slice? - A. Day 3 - B. Day 10 - C. Day 9 - D. Day 11 Your answer: ______ 26. problem-solving-s03-q06 --------------------------- Three siblings take sweets from a jar in turn. Aisha takes half of the sweets, Ben takes half of what remains, and Cara takes the final 6, leaving the jar empty. How many sweets were in the jar at the start? - A. 18 - B. 30 - C. 20 - D. 24 Your answer: ______ 27. problem-solving-s03-q07 --------------------------- A train arrives at 14:20 after a journey of 2 hours 45 minutes. It departed 10 minutes late. What was its scheduled departure time? - A. 11:25 - B. 11:35 - C. 11:45 - D. 11:15 Your answer: ______ 28. problem-solving-s03-q08 --------------------------- After a 10% pay rise, Omar earns £22,000. What did he earn before the rise? - A. £19,800 - B. £21,000 - C. £20,000 - D. £19,000 Your answer: ______ 29. problem-solving-s03-q09 --------------------------- In a knockout quiz tournament, half the teams are eliminated in each round. After three rounds, 4 teams remain. How many teams started? - A. 24 - B. 32 - C. 16 - D. 48 Your answer: ______ 30. problem-solving-s03-q10 --------------------------- A photocopier's counter reads 5,230 at the end of the day. During the day it printed two jobs of 340 pages each and one job of 195 pages. What did the counter read at the start of the day? - A. 4,455 - B. 4,365 - C. 4,255 - D. 4,355 Your answer: ______ Constraints and elimination --------------------------- Use each clue to rule options out until only one possibility survives. 31. problem-solving-s04-q01 --------------------------- Three lockers (numbered 1, 2, and 3) belong to Finn, Grace, and Hana, one each. Grace's locker is number 2 and Hana's is number 3. Which locker is Finn's? - A. Locker 1 - B. Locker 2 - C. Locker 3 - D. It cannot be determined Your answer: ______ 32. problem-solving-s04-q02 --------------------------- Four runners finish a race. Dev finishes before Eli but after Fatima. Gus finishes last. Who wins the race? - A. Dev - B. Eli - C. Fatima - D. Gus Your answer: ______ 33. problem-solving-s04-q03 --------------------------- A cafe offers sandwiches on white, brown, or rye bread. Jo will not eat rye, and the cafe has run out of white. Which bread will Jo's sandwich be on? - A. White - B. Brown - C. Rye - D. Jo cannot order a sandwich Your answer: ______ 34. problem-solving-s04-q04 --------------------------- Kim, Raj, and Sofia each play exactly one instrument: piano, violin, or drums. Raj plays the violin, and Kim does not play the drums. Which instrument does Kim play? - A. Piano - B. Violin - C. Drums - D. It cannot be determined Your answer: ______ 35. problem-solving-s04-q05 --------------------------- A 3-digit code uses digits from 1 to 9. The first digit is twice the second, the third digit is the sum of the first two, and the second digit is 3. What is the code? - A. 369 - B. 636 - C. 963 - D. 639 Your answer: ______ 36. problem-solving-s04-q06 --------------------------- Five houses are numbered 1 to 5 along a lane. The blue house is at number 5, the green house is at number 1, and the red house is next to the blue house. What number is the red house? - A. 2 - B. 3 - C. 4 - D. 5 Your answer: ______ 37. problem-solving-s04-q07 --------------------------- Tara is older than Uma. Uma is older than Vik. Wren is younger than Vik. Who is the second oldest? - A. Uma - B. Vik - C. Tara - D. Wren Your answer: ______ 38. problem-solving-s04-q08 --------------------------- A password is one of: 4412, 4421, 2144, or 1442. It starts with 4 and its last digit is odd. Which is it? - A. 4412 - B. 4421 - C. 2144 - D. 1442 Your answer: ______ 39. problem-solving-s04-q09 --------------------------- Four books (an atlas, a biography, a cookbook, and a dictionary) stand in a row of four positions, left to right. The cookbook is at the far right, the atlas is immediately to the left of the dictionary, and the biography is not in the first position. Which book is first (far left)? - A. The biography - B. The dictionary - C. The cookbook - D. The atlas Your answer: ______ 40. problem-solving-s04-q10 --------------------------- Four suspects each make one statement. Ana: 'Ben did it.' Ben: 'I did not do it.' Carl: 'I did not do it.' Dee: 'Ana is telling the truth.' Exactly one statement is true, and exactly one person is responsible. Who is responsible? - A. Ana - B. Ben - C. Carl - D. Dee Your answer: ______ Finding the rule ---------------- Work out the rule that generates each pattern, then apply it. 41. problem-solving-s05-q01 --------------------------- What is the next number in the sequence 2, 5, 11, 23, 47, …? - A. 94 - B. 95 - C. 93 - D. 71 Your answer: ______ 42. problem-solving-s05-q02 --------------------------- One number breaks the doubling pattern in this list: 3, 6, 12, 24, 45, 96. Which one? - A. 6 - B. 24 - C. 96 - D. 45 Your answer: ______ 43. problem-solving-s05-q03 --------------------------- A sequence alternates between two operations: add 4, then subtract 1. Starting at 5 it runs 5, 9, 8, 12, 11, … What is the next number? - A. 15 - B. 10 - C. 14 - D. 16 Your answer: ______ 44. problem-solving-s05-q04 --------------------------- In a simple code, APPLE is written as BQQMF. Using the same rule, how would LEMON be written? - A. MFOPN - B. KDLNM - C. MFNPO - D. MGNQO Your answer: ______ 45. problem-solving-s05-q05 --------------------------- What is the missing number: 1, 4, 9, 16, __, 36? - A. 24 - B. 25 - C. 20 - D. 30 Your answer: ______ 46. problem-solving-s05-q06 --------------------------- In this sequence each term is the sum of the two before it: 4, 7, 11, 18, 29, … What comes next? - A. 40 - B. 36 - C. 44 - D. 47 Your answer: ______ 47. problem-solving-s05-q07 --------------------------- Shapes repeat in the fixed cycle: star, moon, moon, sun, star, moon, moon, sun, … What is the 11th shape? - A. Star - B. Sun - C. Moon - D. It cannot be determined Your answer: ______ 48. problem-solving-s05-q08 --------------------------- A rule maps 6 → 15, 8 → 19, and 10 → 23. Using the same rule, what does 12 map to? - A. 27 - B. 25 - C. 26 - D. 29 Your answer: ______ 49. problem-solving-s05-q09 --------------------------- What is the next term in the sequence 3, 4, 6, 9, 13, 18, …? - A. 22 - B. 23 - C. 25 - D. 24 Your answer: ______ 50. problem-solving-s05-q10 --------------------------- A sequence follows the rule: double the number, then subtract the digit sum of the result. Starting from 8 it runs 8 → (16 − 7) = 9 → (18 − 9) = 9 → … What happens to the sequence from this point on? - A. It grows without limit - B. It stays at 9 forever - C. It alternates between 9 and 18 - D. It eventually reaches 0 Your answer: ______ Planning and scheduling ----------------------- Fit tasks and journeys into time, watching for gaps, breaks, and overlaps. 51. problem-solving-s06-q01 --------------------------- Rosa needs 3 hours to paint a fence. She starts at 15:30 and takes one 20-minute break. At what time will she finish? - A. 18:50 - B. 18:30 - C. 18:20 - D. 19:00 Your answer: ______ 52. problem-solving-s06-q02 --------------------------- A load of laundry takes 50 minutes to wash and 70 minutes to dry, and drying can only start once washing ends. If the load goes in at 10:15, when is it fully dry? - A. 11:55 - B. 12:05 - C. 12:15 - D. 12:25 Your answer: ______ 53. problem-solving-s06-q03 --------------------------- A meal needs three tasks: chop vegetables (10 minutes, hands-on), simmer the sauce (30 minutes, unattended once started), and boil the pasta (12 minutes, can run alongside the simmering sauce). Chopping must finish before the sauce starts. What is the shortest total time until everything is ready? - A. 52 minutes - B. 30 minutes - C. 42 minutes - D. 40 minutes Your answer: ______ 54. problem-solving-s06-q04 --------------------------- Meetings A and B each last 45 minutes, and B cannot start until 15 minutes after A ends (to allow the room to be reset). A starts at 09:00. What is B's earliest finish time? - A. 10:45 - B. 10:30 - C. 11:00 - D. 10:15 Your answer: ______ 55. problem-solving-s06-q05 --------------------------- A ferry crosses to an island every 40 minutes, with the first crossing at 08:00 and the last at 18:00. How many crossings run in the day? - A. 15 - B. 16 - C. 17 - D. 14 Your answer: ______ 56. problem-solving-s06-q06 --------------------------- An article needs three full-day stages in strict order: drafting (2 days), review (1 day), and publishing (1 day). Reviews are only held on Tuesdays. Drafting starts on a Monday and fills that Monday and Tuesday. What is the earliest day the article can publish? - A. Thursday of the same week - B. Tuesday of the following week - C. Friday of the same week - D. Wednesday of the following week Your answer: ______ 57. problem-solving-s06-q07 --------------------------- Amir must catch a 17:12 train. The walk to the station takes 25 minutes, and he wants to arrive 10 minutes early. What is the latest time he should leave? - A. 16:47 - B. 16:42 - C. 16:37 - D. 16:27 Your answer: ______ 58. problem-solving-s06-q08 --------------------------- Two machines process orders continuously from 09:00. Machine 1 finishes an order every 8 minutes; Machine 2 finishes one every 12 minutes. At what time do they next finish an order at the same moment? - A. 09:24 - B. 09:48 - C. 09:20 - D. 09:36 Your answer: ______ 59. problem-solving-s06-q09 --------------------------- A project has three tasks: P takes 3 days, Q takes 2 days, and R takes 4 days. Q and R can run at the same time, but both may only start after P finishes. What is the shortest overall project duration? - A. 9 days - B. 7 days - C. 5 days - D. 6 days Your answer: ______ 60. problem-solving-s06-q10 --------------------------- A clinic runs 20-minute appointments back to back from 09:00, pauses completely from 12:00 to 12:30, and closes at 14:30. Appointments cannot be shortened and cannot overlap the break. How many appointments fit in the day? - A. 13 - B. 16 - C. 14 - D. 15 Your answer: ______ Trade-offs and best choices --------------------------- Compare the options fairly. Put them on the same footing before deciding which wins. 61. problem-solving-s07-q01 --------------------------- Shop A sells 6 eggs for £1.50. Shop B sells 10 eggs for £2.30. Which shop is cheaper per egg, and at what price? - A. Shop A, at 25p per egg - B. Shop B, at 23p per egg - C. Shop A, at 23p per egg - D. They cost the same per egg Your answer: ______ 62. problem-solving-s07-q02 --------------------------- One phone plan costs £12 per month plus £2 per gigabyte of data used. Another costs a flat £20 per month with unlimited data. At what monthly data use do the two plans cost the same? - A. 2 GB - B. 3 GB - C. 4 GB - D. 5 GB Your answer: ______ 63. problem-solving-s07-q03 --------------------------- A car park charges £2.50 per hour or £9 for a full-day ticket. From how many whole hours of parking does the day ticket become the cheaper option? - A. 4 hours - B. 3 hours - C. 5 hours - D. 6 hours Your answer: ______ 64. problem-solving-s07-q04 --------------------------- A courier's bike carries at most 16 kg. The parcels available weigh 9 kg, 7 kg, 6 kg, and 4 kg. Which combination carries the most weight without exceeding the limit? - A. 9 kg + 6 kg - B. 7 kg + 6 kg + 4 kg - C. 9 kg + 4 kg - D. 9 kg + 7 kg Your answer: ______ 65. problem-solving-s07-q05 --------------------------- An online store offers a choice of discounts: 15% off everything, or £10 off orders over £60. Which saves more on a £70 order? - A. £10 off, saving £10 - B. 15% off, saving £10.50 - C. Both save the same - D. 15% off, saving £9.50 Your answer: ______ 66. problem-solving-s07-q06 --------------------------- A gardener has 24 metres of fencing for a rectangular bed with whole-metre sides. Which dimensions give the largest area? - A. 8 m × 4 m - B. 10 m × 2 m - C. 6 m × 6 m - D. 7 m × 5 m Your answer: ______ 67. problem-solving-s07-q07 --------------------------- A commuter can drive (35 minutes, £4.20 in fuel), take the bus (50 minutes, £2.60), or cycle (55 minutes, free). Her firm rule is to arrive within 45 minutes; among options that meet the rule she picks the cheapest. Which should she choose? - A. The bus - B. Cycling - C. The bus and cycling tie - D. Driving Your answer: ______ 68. problem-solving-s07-q08 --------------------------- Printer cartridges are sold singly for £18 (400 pages each) or as a twin pack for £30 (800 pages). How much does the twin pack save per 800 pages compared with buying singles? - A. £4 - B. £6 - C. £8 - D. £12 Your answer: ______ 69. problem-solving-s07-q09 --------------------------- A bake sale prices cupcakes at 80p each, or 3 for £2. What is the cheapest cost of exactly 7 cupcakes? - A. £4.80 - B. £5.60 - C. £4.66 - D. £5.20 Your answer: ______ 70. problem-solving-s07-q10 --------------------------- A 90-minute workshop must fit 4 talks of 20 minutes each, with breaks of at least 5 minutes between consecutive talks. Does the schedule fit? - A. Yes, with 10 minutes spare - B. Yes, with exactly no spare time - C. No: it overruns by 5 minutes - D. No: it overruns by 15 minutes Your answer: ______ Multi-stage processes --------------------- Follow a quantity through several stages in turn, updating it carefully at each step. 71. problem-solving-s08-q01 --------------------------- A workshop makes items in three stages: cutting (2 minutes per unit), assembly (3 minutes per unit), and packing (1 minute per unit). One worker completes all three stages for each unit before starting the next unit. How long does it take to finish 5 units? - A. 30 minutes - B. 25 minutes - C. 36 minutes - D. 28 minutes Your answer: ______ 72. problem-solving-s08-q02 --------------------------- Water flows into a 100-litre tank at 8 litres per minute while a leak drains 3 litres per minute. The tank starts empty. How long until it overflows? - A. 12.5 minutes - B. 20 minutes - C. 25 minutes - D. 33 minutes Your answer: ______ 73. problem-solving-s08-q03 --------------------------- A document passes through three approvers in sequence. Each approver takes 2 working days, and there is a 1-day handover between consecutive approvers. How many working days pass from submission to final approval? - A. 6 - B. 9 - C. 7 - D. 8 Your answer: ______ 74. problem-solving-s08-q04 --------------------------- In a game, each level doubles your coins, but you pay a 10-coin entry fee before each level starts. Starting with 30 coins, how many do you have after two levels? - A. 60 - B. 80 - C. 100 - D. 50 Your answer: ______ 75. problem-solving-s08-q05 --------------------------- A batch of 200 parts goes through two inspections: 10% fail the first check, and of those that pass, 5% fail the second. How many parts pass both checks? - A. 180 - B. 170 - C. 171 - D. 190 Your answer: ______ 76. problem-solving-s08-q06 --------------------------- A savings account pays 5% interest at each year end, and £100 is deposited at the start of each year. Starting from £0, what is the balance at the end of year two? - A. £210 - B. £215.25 - C. £205 - D. £220.50 Your answer: ______ 77. problem-solving-s08-q07 --------------------------- A recycling plant receives 400 kg of mixed waste. 60% of the input is paper, but 25% of that paper is too contaminated to recycle. How much paper is actually recycled? - A. 240 kg - B. 100 kg - C. 60 kg - D. 180 kg Your answer: ______ 78. problem-solving-s08-q08 --------------------------- A relay team of three swimmers must finish in under 6 minutes in total. The first swims in 1 minute 55 seconds and the second in 2 minutes 5 seconds. What is the slowest the third swimmer can be while keeping the team under 6 minutes? - A. Just under 2 minutes - B. Just under 1 minute 50 seconds - C. Just under 2 minutes 10 seconds - D. Just under 1 minute 30 seconds Your answer: ______ 79. problem-solving-s08-q09 --------------------------- An email campaign sends 1,000 emails. 40% are opened, 15% of opened emails receive a click, and half of those who click go on to sign up. How many people sign up? - A. 60 - B. 30 - C. 45 - D. 20 Your answer: ______ 80. problem-solving-s08-q10 --------------------------- Machine X makes 30 widgets per hour but needs a 15-minute clean after every 2 hours of running. It starts at 08:00. How many complete widgets are made by 12:00? - A. 120 - B. 110 - C. 112 - D. 115 Your answer: ______ Arrangements, counting and chance --------------------------------- Count possibilities systematically, and reason about likelihood from equally likely outcomes. 81. problem-solving-s09-q01 --------------------------- A cafe offers 3 soups and 4 sandwiches. How many different soup-and-sandwich lunches are possible? - A. 7 - B. 12 - C. 9 - D. 16 Your answer: ______ 82. problem-solving-s09-q02 --------------------------- How many different ways can the three letters of the word TAP be arranged? - A. 3 - B. 9 - C. 6 - D. 12 Your answer: ______ 83. problem-solving-s09-q03 --------------------------- A drawer holds 6 black socks and 4 white socks. Picking in complete darkness, how many socks must you take to be certain of having a matching pair? - A. 3 - B. 2 - C. 5 - D. 7 Your answer: ______ 84. problem-solving-s09-q04 --------------------------- A spinner has 8 equal sectors numbered 1 to 8. What is the probability of spinning an even number greater than 3? - A. 1/2 - B. 1/4 - C. 5/8 - D. 3/8 Your answer: ______ 85. problem-solving-s09-q05 --------------------------- Four people each shake hands with every other person exactly once. How many handshakes happen? - A. 12 - B. 4 - C. 6 - D. 8 Your answer: ______ 86. problem-solving-s09-q06 --------------------------- A committee of 2 people is chosen from 5 volunteers. How many different committees are possible? - A. 20 - B. 10 - C. 25 - D. 15 Your answer: ______ 87. problem-solving-s09-q07 --------------------------- A bag holds 5 red, 3 green, and 2 yellow marbles. One marble is drawn at random. What is the probability it is NOT green? - A. 3/10 - B. 1/2 - C. 3/7 - D. 7/10 Your answer: ______ 88. problem-solving-s09-q08 --------------------------- How many 3-digit numbers can be formed from the digits 2, 5, 7, and 8 if no digit may repeat? - A. 24 - B. 12 - C. 64 - D. 36 Your answer: ______ 89. problem-solving-s09-q09 --------------------------- Two fair six-sided dice are rolled. What is the probability that the total is exactly 4? - A. 1/9 - B. 1/12 - C. 1/6 - D. 1/18 Your answer: ______ 90. problem-solving-s09-q10 --------------------------- A lift starts empty and stops at 4 floors. At each floor exactly one of the two people waiting there gets in. How many different groups of passengers could be in the lift at the end? - A. 8 - B. 12 - C. 16 - D. 24 Your answer: ______ Integrated challenge problems ----------------------------- The hardest set: each item combines several techniques from earlier sections. 91. problem-solving-s10-q01 --------------------------- A bookshop deal gives a sixth paperback free with every five bought. Paperbacks cost £8 each. Leah wants to leave with 20 paperbacks. How much does she pay? - A. £160 - B. £128 - C. £144 - D. £136 Your answer: ______ 92. problem-solving-s10-q02 --------------------------- Two cyclists start 60 km apart and ride directly towards each other, one at 14 km/h and the other at 16 km/h. How long until they meet? - A. 2 hours - B. 1 hour 45 minutes - C. 2 hours 15 minutes - D. 1 hour 30 minutes Your answer: ______ 93. problem-solving-s10-q03 --------------------------- A tank contains 240 litres. Pump P drains it at 4 litres per minute. After 20 minutes, pump Q joins in, draining a further 6 litres per minute. How long does emptying the tank take in total? - A. 40 minutes - B. 32 minutes - C. 36 minutes - D. 24 minutes Your answer: ______ 94. problem-solving-s10-q04 --------------------------- In an office of 40 people, 22 drink coffee, 17 drink tea, and 9 drink both. How many drink neither? - A. 8 - B. 10 - C. 12 - D. 6 Your answer: ______ 95. problem-solving-s10-q05 --------------------------- A train leaves a station at 10:00 travelling at 90 km/h. A second train leaves the same station on the same route at 10:30, travelling at 120 km/h. At what time does the second train catch the first? - A. 11:30 - B. 12:30 - C. 11:45 - D. 12:00 Your answer: ______ 96. problem-solving-s10-q06 --------------------------- Anya is now three times as old as her son. In 12 years she will be exactly twice as old as him. How old is her son now? - A. 12 - B. 10 - C. 8 - D. 15 Your answer: ______ 97. problem-solving-s10-q07 --------------------------- A two-digit number has digits that sum to 9. Reversing its digits produces a number 27 larger. What is the original number? - A. 63 - B. 45 - C. 36 - D. 27 Your answer: ______ 98. problem-solving-s10-q08 --------------------------- Three flatmates split rent of £900 so that Jo pays £60 more than Kai, and Kai pays twice as much as Lee. How much does Kai pay? - A. £300 - B. £336 - C. £360 - D. £276 Your answer: ______ 99. problem-solving-s10-q09 --------------------------- A ferry can carry at most 8 tonnes per crossing. Waiting to cross: a lorry (5 tonnes), a van (2.5 tonnes), and three cars (1.5 tonnes each). The lorry must travel on the first crossing. What is the minimum number of crossings needed to move all the vehicles? - A. 4 - B. 3 - C. It cannot be done - D. 2 Your answer: ______ 100. problem-solving-s10-q10 ---------------------------- In a quiz, a correct answer scores 4 points and a wrong answer loses 1 point; there are no skips. Zara answered all 20 questions and scored 55. How many did she answer correctly? - A. 15 - B. 14 - C. 16 - D. 13 Your answer: ______ Authored hints ============== This cognitive practice source does not publish authored hints. Every item is listed as missing a hint, and no hint text is generated or inferred. 100 worksheet items do not have an authored hint. No hint text was generated for them. Answer key ========== ! The answer key and worked explanations in this file belong only to this public, untimed educational practice source. No active aptitude suite or Novus Reasoning Challenge answer key is ever included in any download. Problem solving: multi-step reasoning in depth: open practice pack answer key Item Choice Answer ----------------------- ------ ------------------------------- problem-solving-s01-q01 C 12 problem-solving-s01-q02 C 15 problem-solving-s01-q03 C 24 problem-solving-s01-q04 B A, by 3 days problem-solving-s01-q05 C 24 problem-solving-s01-q06 C Seat 3 problem-solving-s01-q07 A £6 problem-solving-s01-q08 D The 120 kg crate problem-solving-s01-q09 A 3 problem-solving-s01-q10 D 10 minutes problem-solving-s02-q01 A £15 problem-solving-s02-q02 B 25 minutes problem-solving-s02-q03 D 400 ml problem-solving-s02-q04 A 12 hours problem-solving-s02-q05 B 15 litres problem-solving-s02-q06 D £574 problem-solving-s02-q07 A 3 km problem-solving-s02-q08 C 750 g problem-solving-s02-q09 B £40 problem-solving-s02-q10 D 30 litres problem-solving-s03-q01 B 12 problem-solving-s03-q02 A £20 problem-solving-s03-q03 D 36 problem-solving-s03-q04 C £60 problem-solving-s03-q05 B Day 10 problem-solving-s03-q06 D 24 problem-solving-s03-q07 A 11:25 problem-solving-s03-q08 C £20,000 problem-solving-s03-q09 B 32 problem-solving-s03-q10 D 4,355 problem-solving-s04-q01 A Locker 1 problem-solving-s04-q02 C Fatima problem-solving-s04-q03 B Brown problem-solving-s04-q04 A Piano problem-solving-s04-q05 D 639 problem-solving-s04-q06 C 4 problem-solving-s04-q07 A Uma problem-solving-s04-q08 B 4421 problem-solving-s04-q09 D The atlas problem-solving-s04-q10 C Carl problem-solving-s05-q01 B 95 problem-solving-s05-q02 D 45 problem-solving-s05-q03 A 15 problem-solving-s05-q04 C MFNPO problem-solving-s05-q05 B 25 problem-solving-s05-q06 D 47 problem-solving-s05-q07 C Moon problem-solving-s05-q08 A 27 problem-solving-s05-q09 D 24 problem-solving-s05-q10 B It stays at 9 forever problem-solving-s06-q01 A 18:50 problem-solving-s06-q02 C 12:15 problem-solving-s06-q03 D 40 minutes problem-solving-s06-q04 A 10:45 problem-solving-s06-q05 B 16 problem-solving-s06-q06 D Wednesday of the following week problem-solving-s06-q07 C 16:37 problem-solving-s06-q08 A 09:24 problem-solving-s06-q09 B 7 days problem-solving-s06-q10 D 15 problem-solving-s07-q01 B Shop B, at 23p per egg problem-solving-s07-q02 C 4 GB problem-solving-s07-q03 A 4 hours problem-solving-s07-q04 D 9 kg + 7 kg problem-solving-s07-q05 B 15% off, saving £10.50 problem-solving-s07-q06 C 6 m × 6 m problem-solving-s07-q07 D Driving problem-solving-s07-q08 B £6 problem-solving-s07-q09 A £4.80 problem-solving-s07-q10 C No: it overruns by 5 minutes problem-solving-s08-q01 A 30 minutes problem-solving-s08-q02 B 20 minutes problem-solving-s08-q03 D 8 problem-solving-s08-q04 A 60 problem-solving-s08-q05 C 171 problem-solving-s08-q06 B £215.25 problem-solving-s08-q07 D 180 kg problem-solving-s08-q08 A Just under 2 minutes problem-solving-s08-q09 B 30 problem-solving-s08-q10 C 112 problem-solving-s09-q01 B 12 problem-solving-s09-q02 C 6 problem-solving-s09-q03 A 3 problem-solving-s09-q04 D 3/8 problem-solving-s09-q05 C 6 problem-solving-s09-q06 B 10 problem-solving-s09-q07 D 7/10 problem-solving-s09-q08 A 24 problem-solving-s09-q09 B 1/12 problem-solving-s09-q10 C 16 problem-solving-s10-q01 D £136 problem-solving-s10-q02 A 2 hours problem-solving-s10-q03 C 36 minutes problem-solving-s10-q04 B 10 problem-solving-s10-q05 D 12:00 problem-solving-s10-q06 A 12 problem-solving-s10-q07 C 36 problem-solving-s10-q08 B £336 problem-solving-s10-q09 D 2 problem-solving-s10-q10 A 15 Worked explanations =================== problem-solving-s01-q01 The next joint restock day is the least common multiple of 2, 3, and 4, which is 12. problem-solving-s01-q02 16 of 40 small boxes uses 40% of the space, leaving 60% free: 60% of 25 large boxes = 15. problem-solving-s01-q03 With 4 distinct digits and no repeats, this is 4! = 24 arrangements. problem-solving-s01-q04 Item A lasts 40 ÷ 8 = 5 days; Item B lasts 40 ÷ 5 = 8 days. A runs out 3 days sooner. problem-solving-s01-q05 Multiply the independent choices: 4 × 3 × 2 = 24. problem-solving-s01-q06 Priya must be in seat 2 or 3. If Priya were in seat 3, Sam would be in seat 2, forcing Yuki into seat 1, which is excluded. So Priya is in seat 2, Sam in seat 1, Marcus in seat 4, and Yuki in seat 3. problem-solving-s01-q07 Work out the total first: 2 × £9 + £6 = £24. Then subtract from the payment: £30 − £24 = £6. problem-solving-s01-q08 All four crates total 650 kg: 50 kg over the limit. Removing the lightest crate that clears the excess (120 kg) leaves 530 kg, the heaviest legal load. problem-solving-s01-q09 Two boxes hold 80 tea bags, and 80 ÷ 25 = 3.2, so only 3 complete weeks are covered before the supply runs short. problem-solving-s01-q10 List the departures: 09:00, 09:15, 09:30, 09:45, 10:00. Arriving at 09:50, the next bus is at 10:00. A 10-minute wait. problem-solving-s02-q01 The ratio has 2 + 3 + 4 = 9 parts, so one part is £45 ÷ 9 = £5. The middle share is 3 parts: £15. problem-solving-s02-q02 Divide the total by the rate: 300 ÷ 12 = 25 minutes. problem-solving-s02-q03 The mix has 1 + 4 = 5 equal parts, so each part is 2,000 ÷ 5 = 400 ml, and concentrate is exactly one part. problem-solving-s02-q04 Halving the workforce doubles the time when rates are equal: 6 × 2 = 12 hours. problem-solving-s02-q05 250 km is 2.5 blocks of 100 km, so fuel scales the same way: 6 × 2.5 = 15 litres. problem-solving-s02-q06 Standard pay is 35 × £14 = £490. The overtime rate is £21 per hour, adding 4 × £21 = £84, for £574 in total. problem-solving-s02-q07 6 cm × 50,000 = 300,000 cm. Converting units step by step: 300,000 cm = 3,000 m = 3 km. problem-solving-s02-q08 Find the per-person amount first: 300 ÷ 4 = 75 g. Then scale up: 75 × 10 = 750 g. problem-solving-s02-q09 A 2 : 1 split makes 3 equal parts of £40 each; the younger grandchild gets one part. problem-solving-s02-q10 The rate is 90 ÷ 45 = 2 litres per minute, so 30 minutes delivers 60 litres, leaving 90 − 60 = 30 litres to go. problem-solving-s03-q01 Reverse the steps in reverse order: subtract 7 (31 − 7 = 24), then halve (24 ÷ 2 = 12). problem-solving-s03-q02 Undo the last step first: before the coffee he had £6 + £4 = £10, which was half his money, so he started with £20. problem-solving-s03-q03 Reverse each operation: add 5 back (7 + 5 = 12), then multiply by 3 (12 × 3 = 36). problem-solving-s03-q04 The sale price is 80% of the original, so divide rather than add: £48 ÷ 0.8 = £60. Adding 20% of £48 back is the classic trap. Percentages of different bases are not interchangeable. problem-solving-s03-q05 Work backwards by halving: half covered on day 11, a quarter covered on day 10. problem-solving-s03-q06 Cara's 6 sweets were what remained after two halvings, i.e. a quarter of the jar, so the jar held 6 × 4 = 24. problem-solving-s03-q07 Subtract the journey from the arrival: 14:20 − 2:45 = 11:35 actual departure. It left 10 minutes late, so the schedule said 11:25. problem-solving-s03-q08 The new salary is 110% of the old one, so divide: £22,000 ÷ 1.1 = £20,000. Subtracting 10% of the new figure gives the wrong base. problem-solving-s03-q09 Double once per round going backwards: 4 → 8 → 16 → 32 teams at the start. problem-solving-s03-q10 Total pages printed = 340 + 340 + 195 = 875. Undo the day's printing: 5,230 − 875 = 4,355. problem-solving-s04-q01 With lockers 2 and 3 assigned, elimination leaves only locker 1 for Finn. problem-solving-s04-q02 Chain the clues into one order: Fatima before Dev before Eli, with Gus last, so the full order is Fatima, Dev, Eli, Gus. problem-solving-s04-q03 One constraint removes rye and another removes white, leaving brown as the only workable option. problem-solving-s04-q04 The violin is taken by Raj and the drums are ruled out for Kim, so the piano is the only instrument left for her. problem-solving-s04-q05 Apply the clues in order: the second digit is 3, so the first is 6, and the third is 6 + 3 = 9, giving 639. problem-solving-s04-q06 Number 5 is at the end of the lane, so its only neighbour is number 4. The red house must be there. problem-solving-s04-q07 Linking the comparisons gives one chain: Tara, Uma, Vik, Wren from oldest to youngest. Uma sits second. problem-solving-s04-q08 Starting with 4 eliminates 2144 and 1442; needing an odd final digit then eliminates 4412, leaving 4421. problem-solving-s04-q09 With the cookbook fixed in position 4, the atlas–dictionary pair fits at positions 1–2 or 2–3. The second option forces the biography into position 1, which is forbidden, so the atlas is first. problem-solving-s04-q10 Test each possibility against the 'exactly one true statement' condition. Only if Carl is responsible is a single statement true (Ben's denial); every other candidate makes two or more statements true. problem-solving-s05-q01 Each term is double the previous term plus 1, so the next is 47 × 2 + 1 = 95. problem-solving-s05-q02 Doubling 24 should give 48, not 45, and doubling 48 would correctly give 96, confirming 45 is the misfit. problem-solving-s05-q03 Track whose turn it is: the last step was a subtraction, so the next is an addition, 11 + 4 = 15. problem-solving-s05-q04 Each letter shifts forward by one place in the alphabet, so L, E, M, O, N become M, F, N, P, O. problem-solving-s05-q05 These are the square numbers 1², 2², 3², …, the missing term is 5² = 25. problem-solving-s05-q06 Apply the stated rule to the last two terms: 18 + 29 = 47. problem-solving-s05-q07 The cycle has length 4, and 11 = 4 × 2 + 3, so the 11th shape matches the 3rd in the cycle: moon. problem-solving-s05-q08 Each output is double the input plus 3 (check: 2 × 6 + 3 = 15), so 12 maps to 2 × 12 + 3 = 27. problem-solving-s05-q09 The gaps grow by one each time: +1, +2, +3, +4, +5. The next gap is +6, giving 18 + 6 = 24. problem-solving-s05-q10 Applying the rule to 9 gives 18 − (1 + 8) = 9 again, so 9 is a fixed point. Once reached, the sequence never changes. problem-solving-s06-q01 Add the work and the break together: 3 hours 20 minutes after 15:30 is 18:50. problem-solving-s06-q02 The stages are strictly sequential: 50 + 70 = 120 minutes, and two hours after 10:15 is 12:15. problem-solving-s06-q03 Chop for 10 minutes, then start the 30-minute simmer and boil the pasta during it. Only the chopping and simmering lie on the critical path: 10 + 30 = 40 minutes. problem-solving-s06-q04 A ends at 09:45; adding the 15-minute changeover lets B start at 10:00 and finish 45 minutes later, at 10:45. problem-solving-s06-q05 From 08:00 to 18:00 is 600 minutes, which holds 600 ÷ 40 = 15 gaps between crossings, but counting both endpoints gives 15 + 1 = 16 crossings. problem-solving-s06-q06 Drafting occupies this week's Tuesday, so the earliest available review slot is the following Tuesday, and publishing takes the day after, Wednesday of the following week. problem-solving-s06-q07 Combine the walk and the margin: 35 minutes before 17:12 is 16:37. problem-solving-s06-q08 They coincide at the least common multiple of 8 and 12 minutes, which is 24, so at 09:24. problem-solving-s06-q09 After P's 3 days, Q and R run in parallel, so only the longer of the two matters: 3 + max(2, 4) = 7 days. problem-solving-s06-q10 The morning block (09:00–12:00) is 180 minutes, fitting 9 appointments; the afternoon block (12:30–14:30) is 120 minutes, fitting 6, a total of 15. problem-solving-s07-q01 Convert both to a unit price: £1.50 ÷ 6 = 25p per egg against £2.30 ÷ 10 = 23p per egg. Shop B wins. problem-solving-s07-q02 Set the costs equal: £12 + £2 per GB matches £20 when the data charge is £8, i.e. at 4 GB. problem-solving-s07-q03 Hourly parking costs £7.50 for 3 hours but £10 for 4 hours, so from 4 hours onward the £9 day ticket is cheaper. problem-solving-s07-q04 Check each total against the limit: 9 + 7 = 16 kg uses the capacity exactly, while 7 + 6 + 4 = 17 kg is over and the rest carry less. problem-solving-s07-q05 Work out each saving on the actual order value: 15% of £70 is £10.50, which beats the flat £10. problem-solving-s07-q06 All the options use the same perimeter, so compare areas: the closer the sides are to equal, the larger the area, and 6 × 6 = 36 m² beats 35, 32, and 20. problem-solving-s07-q07 Apply the hard constraint before comparing costs: only driving arrives within 45 minutes, so cost never comes into play. problem-solving-s07-q08 Match the quantities first: 800 pages needs two singles at £36, so the £30 twin pack saves £6. problem-solving-s07-q09 Use the deal as many times as possible: two 3-for-£2 bundles cover 6 cupcakes for £4, plus one single at 80p makes £4.80. problem-solving-s07-q10 Count the gaps as well as the talks: 4 × 20 = 80 minutes of talks plus 3 × 5 = 15 minutes of breaks totals 95 minutes, 5 more than the slot allows. problem-solving-s08-q01 Each unit takes 2 + 3 + 1 = 6 minutes from start to finish, so 5 units take 5 × 6 = 30 minutes. problem-solving-s08-q02 Work with the net rate: 8 − 3 = 5 litres per minute, so 100 ÷ 5 = 20 minutes. problem-solving-s08-q03 Three approvals take 3 × 2 = 6 days, and there are only 2 handovers between 3 approvers, adding 2 more days: 8 in total. problem-solving-s08-q04 Apply fee then doubling, twice: (30 − 10) × 2 = 40, then (40 − 10) × 2 = 60. The order of operations within each level matters. problem-solving-s08-q05 Take the percentages of the surviving quantity each time: 200 → 180 after the first check, then 5% of 180 (9 parts) fail, leaving 171. problem-solving-s08-q06 Year one: £100 × 1.05 = £105. Year two starts with £105 + £100 = £205, which grows to £205 × 1.05 = £215.25: interest compounds on the running balance, not just the deposits. problem-solving-s08-q07 Chain the percentages: 60% of 400 kg is 240 kg of paper, and 75% of that survives contamination checks, 0.75 × 240 = 180 kg. problem-solving-s08-q08 The first two legs total exactly 4 minutes (1:55 + 2:05), leaving anything under 2 minutes for the final leg. problem-solving-s08-q09 Follow the funnel stage by stage: 1,000 → 400 opens → 60 clicks → 30 sign-ups. Each percentage applies to the previous stage's result, not the original total. problem-solving-s08-q10 Running 08:00–10:00 gives 60 widgets, then cleaning lasts until 10:15. From 10:15 to 12:00 is 105 minutes at 2 minutes per widget, 52 complete widgets, for 112 in total. problem-solving-s09-q01 Independent choices multiply rather than add: 3 × 4 = 12 different lunches. problem-solving-s09-q02 Three positions filled by three distinct letters gives 3 × 2 × 1 = 6 arrangements. problem-solving-s09-q03 Plan for the worst case: the first two socks could be one of each colour, but a third sock must match one of them. There are only two colours. problem-solving-s09-q04 List the outcomes that satisfy both conditions: 4, 6, and 8: three of the eight equally likely sectors, so 3/8. problem-solving-s09-q05 Each of the 4 people shakes 3 hands, but that counts every handshake twice: 4 × 3 ÷ 2 = 6. problem-solving-s09-q06 There are 5 × 4 = 20 ordered picks, but a committee ignores order, so halve it: 10 distinct pairs. problem-solving-s09-q07 Use the complement: the chance of green is 3/10, so the chance of not-green is 1 − 3/10 = 7/10. problem-solving-s09-q08 Fill the positions one at a time: 4 choices, then 3, then 2, giving 4 × 3 × 2 = 24 numbers. problem-solving-s09-q09 Count ordered outcomes over the 36 possible rolls: 1+3, 2+2, and 3+1 give a total of 4, so the probability is 3/36 = 1/12. problem-solving-s09-q10 Each floor contributes an independent two-way choice, so the possibilities multiply: 2 × 2 × 2 × 2 = 16 distinct groups. problem-solving-s10-q01 Group the books in sixes: three groups cover 18 books while paying for only 15, and the last 2 books are paid in full, 17 paid × £8 = £136. problem-solving-s10-q02 When two movers approach each other their speeds add: the gap closes at 30 km/h, so 60 km takes 2 hours. problem-solving-s10-q03 Split the job into phases: the first 20 minutes drain 80 litres, leaving 160; the combined rate of 10 litres per minute then takes 16 more minutes, for 36 in total. problem-solving-s10-q04 Avoid double-counting the overlap: coffee or tea drinkers number 22 + 17 − 9 = 30, so 40 − 30 = 10 drink neither. problem-solving-s10-q05 By 10:30 the first train is 45 km ahead. The second closes that gap at 120 − 90 = 30 km/h, taking 1.5 hours, so it catches up at 12:00. problem-solving-s10-q06 Let the son's age be s, so Anya is 3s. In 12 years: 3s + 12 = 2(s + 12), which solves to s = 12. problem-solving-s10-q07 Reversing changes the value by 9 times the digit difference, so the digits differ by 27 ÷ 9 = 3. Together with a digit sum of 9, the digits are 3 and 6, and the reversal must grow, so the number is 36. problem-solving-s10-q08 Express everyone in terms of Lee's share x: Kai pays 2x and Jo pays 2x + 60, so 5x + 60 = 900 gives x = £168, and Kai pays 2 × £168 = £336. problem-solving-s10-q09 Pack to the limit: the lorry plus two cars is exactly 5 + 1.5 + 1.5 = 8 tonnes for crossing one, leaving the van and the last car (4 tonnes) for crossing two. problem-solving-s10-q10 With r correct answers there are 20 − r wrong ones, so 4r − (20 − r) = 55. That simplifies to 5r = 75, giving r = 15. 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