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.
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.
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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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12. problem-solving-s02-q02
A printer produces 12 pages per minute. How long will it take to print 300 pages?
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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?
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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?
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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?
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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?
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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?
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18. problem-solving-s02-q08
A recipe for 4 people needs 300 g of rice. How much rice is needed for 10 people?
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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?
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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?
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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?
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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?
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23. problem-solving-s03-q03
A number is divided by 3, then 5 is subtracted, giving 7. What was the original number?
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24. problem-solving-s03-q04
A shop reduces a coat by 20% to a sale price of £48. What was the original price?
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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?
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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?
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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?
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28. problem-solving-s03-q08
After a 10% pay rise, Omar earns £22,000. What did he earn before the rise?
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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?
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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?
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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?
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32. problem-solving-s04-q02
Four runners finish a race. Dev finishes before Eli but after Fatima. Gus finishes last. Who wins the race?
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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?
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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?
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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?
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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?
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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?
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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?
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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)?
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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?
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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, …?
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42. problem-solving-s05-q02
One number breaks the doubling pattern in this list: 3, 6, 12, 24, 45, 96. Which one?
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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?
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44. problem-solving-s05-q04
In a simple code, APPLE is written as BQQMF. Using the same rule, how would LEMON be written?
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45. problem-solving-s05-q05
What is the missing number: 1, 4, 9, 16, __, 36?
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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?
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47. problem-solving-s05-q07
Shapes repeat in the fixed cycle: star, moon, moon, sun, star, moon, moon, sun, … What is the 11th shape?
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48. problem-solving-s05-q08
A rule maps 6 → 15, 8 → 19, and 10 → 23. Using the same rule, what does 12 map to?
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49. problem-solving-s05-q09
What is the next term in the sequence 3, 4, 6, 9, 13, 18, …?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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82. problem-solving-s09-q02
How many different ways can the three letters of the word TAP be arranged?
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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?
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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?
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85. problem-solving-s09-q05
Four people each shake hands with every other person exactly once. How many handshakes happen?
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86. problem-solving-s09-q06
A committee of 2 people is chosen from 5 volunteers. How many different committees are possible?
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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?
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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?
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89. problem-solving-s09-q09
Two fair six-sided dice are rolled. What is the probability that the total is exactly 4?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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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?
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