{ "format": "novus-learn-open-practice", "version": 1, "packId": "cognitive-skills-problem-solving", "source": "cognitive-skills", "slug": "problem-solving", "title": "Problem solving: multi-step reasoning in depth: open practice pack", "subtitle": "100 untimed questions across 10 authored sections.", "sourceHref": "/cognitive-skills/problem-solving", "disclaimer": "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.", "worksheet": { "title": "Learner worksheet", "instructions": [ "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." ], "groups": [ { "id": "problem-solving-s01", "title": "Multi-step foundations", "instructions": "Warm up by combining two or three simple facts into a single answer.", "items": [ { "id": "problem-solving-s01-q01", "prompt": "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?", "choices": [ { "id": "a", "label": "6" }, { "id": "b", "label": "8" }, { "id": "c", "label": "12" }, { "id": "d", "label": "24" } ] }, { "id": "problem-solving-s01-q02", "prompt": "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?", "choices": [ { "id": "a", "label": "9" }, { "id": "b", "label": "12" }, { "id": "c", "label": "15" }, { "id": "d", "label": "18" } ] }, { "id": "problem-solving-s01-q03", "prompt": "A 4-character password uses only the digits 1, 2, 3, and 4, with no digit repeated. How many different passwords are possible?", "choices": [ { "id": "a", "label": "4" }, { "id": "b", "label": "12" }, { "id": "c", "label": "24" }, { "id": "d", "label": "64" } ] }, { "id": "problem-solving-s01-q04", "prompt": "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?", "choices": [ { "id": "a", "label": "A, by 2 days" }, { "id": "b", "label": "A, by 3 days" }, { "id": "c", "label": "B, by 3 days" }, { "id": "d", "label": "B, by 5 days" } ] }, { "id": "problem-solving-s01-q05", "prompt": "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?", "choices": [ { "id": "a", "label": "9" }, { "id": "b", "label": "18" }, { "id": "c", "label": "24" }, { "id": "d", "label": "32" } ] }, { "id": "problem-solving-s01-q06", "prompt": "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?", "choices": [ { "id": "a", "label": "Seat 1" }, { "id": "b", "label": "Seat 2" }, { "id": "c", "label": "Seat 3" }, { "id": "d", "label": "Seat 4" } ] }, { "id": "problem-solving-s01-q07", "prompt": "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?", "choices": [ { "id": "a", "label": "£6" }, { "id": "b", "label": "£4" }, { "id": "c", "label": "£5" }, { "id": "d", "label": "£7" } ] }, { "id": "problem-solving-s01-q08", "prompt": "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?", "choices": [ { "id": "a", "label": "The 150 kg crate" }, { "id": "b", "label": "The 180 kg crate" }, { "id": "c", "label": "The 200 kg crate" }, { "id": "d", "label": "The 120 kg crate" } ] }, { "id": "problem-solving-s01-q09", "prompt": "Tea bags come in boxes of 40. The office uses 25 tea bags per week. How many full weeks will 2 boxes last?", "choices": [ { "id": "a", "label": "3" }, { "id": "b", "label": "4" }, { "id": "c", "label": "2" }, { "id": "d", "label": "5" } ] }, { "id": "problem-solving-s01-q10", "prompt": "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?", "choices": [ { "id": "a", "label": "5 minutes" }, { "id": "b", "label": "15 minutes" }, { "id": "c", "label": "25 minutes" }, { "id": "d", "label": "10 minutes" } ] } ] }, { "id": "problem-solving-s02", "title": "Rates, ratios and fair shares", "instructions": "Scale quantities up and down, and split amounts in a stated proportion.", "items": [ { "id": "problem-solving-s02-q01", "prompt": "Three friends share a £45 restaurant bill in the ratio 2 : 3 : 4. How much does the friend with the middle share pay?", "choices": [ { "id": "a", "label": "£15" }, { "id": "b", "label": "£10" }, { "id": "c", "label": "£20" }, { "id": "d", "label": "£12" } ] }, { "id": "problem-solving-s02-q02", "prompt": "A printer produces 12 pages per minute. How long will it take to print 300 pages?", "choices": [ { "id": "a", "label": "20 minutes" }, { "id": "b", "label": "25 minutes" }, { "id": "c", "label": "30 minutes" }, { "id": "d", "label": "24 minutes" } ] }, { "id": "problem-solving-s02-q03", "prompt": "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?", "choices": [ { "id": "a", "label": "500 ml" }, { "id": "b", "label": "250 ml" }, { "id": "c", "label": "200 ml" }, { "id": "d", "label": "400 ml" } ] }, { "id": "problem-solving-s02-q04", "prompt": "Two decorators working together at the same rate paint a room in 6 hours. How long would one decorator take alone?", "choices": [ { "id": "a", "label": "12 hours" }, { "id": "b", "label": "9 hours" }, { "id": "c", "label": "3 hours" }, { "id": "d", "label": "8 hours" } ] }, { "id": "problem-solving-s02-q05", "prompt": "A car uses 6 litres of fuel per 100 km. How much fuel is needed for a 250 km journey?", "choices": [ { "id": "a", "label": "12 litres" }, { "id": "b", "label": "15 litres" }, { "id": "c", "label": "18 litres" }, { "id": "d", "label": "14 litres" } ] }, { "id": "problem-solving-s02-q06", "prompt": "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?", "choices": [ { "id": "a", "label": "£546" }, { "id": "b", "label": "£560" }, { "id": "c", "label": "£588" }, { "id": "d", "label": "£574" } ] }, { "id": "problem-solving-s02-q07", "prompt": "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?", "choices": [ { "id": "a", "label": "3 km" }, { "id": "b", "label": "30 km" }, { "id": "c", "label": "0.3 km" }, { "id": "d", "label": "5 km" } ] }, { "id": "problem-solving-s02-q08", "prompt": "A recipe for 4 people needs 300 g of rice. How much rice is needed for 10 people?", "choices": [ { "id": "a", "label": "650 g" }, { "id": "b", "label": "700 g" }, { "id": "c", "label": "750 g" }, { "id": "d", "label": "800 g" } ] }, { "id": "problem-solving-s02-q09", "prompt": "£120 is shared between two grandchildren so that the elder receives twice as much as the younger. How much does the younger receive?", "choices": [ { "id": "a", "label": "£60" }, { "id": "b", "label": "£40" }, { "id": "c", "label": "£30" }, { "id": "d", "label": "£80" } ] }, { "id": "problem-solving-s02-q10", "prompt": "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?", "choices": [ { "id": "a", "label": "20 litres" }, { "id": "b", "label": "45 litres" }, { "id": "c", "label": "25 litres" }, { "id": "d", "label": "30 litres" } ] } ] }, { "id": "problem-solving-s03", "title": "Working backwards", "instructions": "Start from the final result and reverse each step to recover the starting value.", "items": [ { "id": "problem-solving-s03-q01", "prompt": "Mia thinks of a number, doubles it, then adds 7 to get 31. What number did she start with?", "choices": [ { "id": "a", "label": "10" }, { "id": "b", "label": "12" }, { "id": "c", "label": "14" }, { "id": "d", "label": "19" } ] }, { "id": "problem-solving-s03-q02", "prompt": "After spending half his money on lunch and then £4 on a coffee, Ben has £6 left. How much did he start with?", "choices": [ { "id": "a", "label": "£20" }, { "id": "b", "label": "£16" }, { "id": "c", "label": "£24" }, { "id": "d", "label": "£18" } ] }, { "id": "problem-solving-s03-q03", "prompt": "A number is divided by 3, then 5 is subtracted, giving 7. What was the original number?", "choices": [ { "id": "a", "label": "26" }, { "id": "b", "label": "21" }, { "id": "c", "label": "30" }, { "id": "d", "label": "36" } ] }, { "id": "problem-solving-s03-q04", "prompt": "A shop reduces a coat by 20% to a sale price of £48. What was the original price?", "choices": [ { "id": "a", "label": "£57.60" }, { "id": "b", "label": "£56" }, { "id": "c", "label": "£60" }, { "id": "d", "label": "£64" } ] }, { "id": "problem-solving-s03-q05", "prompt": "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?", "choices": [ { "id": "a", "label": "Day 3" }, { "id": "b", "label": "Day 10" }, { "id": "c", "label": "Day 9" }, { "id": "d", "label": "Day 11" } ] }, { "id": "problem-solving-s03-q06", "prompt": "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?", "choices": [ { "id": "a", "label": "18" }, { "id": "b", "label": "30" }, { "id": "c", "label": "20" }, { "id": "d", "label": "24" } ] }, { "id": "problem-solving-s03-q07", "prompt": "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?", "choices": [ { "id": "a", "label": "11:25" }, { "id": "b", "label": "11:35" }, { "id": "c", "label": "11:45" }, { "id": "d", "label": "11:15" } ] }, { "id": "problem-solving-s03-q08", "prompt": "After a 10% pay rise, Omar earns £22,000. What did he earn before the rise?", "choices": [ { "id": "a", "label": "£19,800" }, { "id": "b", "label": "£21,000" }, { "id": "c", "label": "£20,000" }, { "id": "d", "label": "£19,000" } ] }, { "id": "problem-solving-s03-q09", "prompt": "In a knockout quiz tournament, half the teams are eliminated in each round. After three rounds, 4 teams remain. How many teams started?", "choices": [ { "id": "a", "label": "24" }, { "id": "b", "label": "32" }, { "id": "c", "label": "16" }, { "id": "d", "label": "48" } ] }, { "id": "problem-solving-s03-q10", "prompt": "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?", "choices": [ { "id": "a", "label": "4,455" }, { "id": "b", "label": "4,365" }, { "id": "c", "label": "4,255" }, { "id": "d", "label": "4,355" } ] } ] }, { "id": "problem-solving-s04", "title": "Constraints and elimination", "instructions": "Use each clue to rule options out until only one possibility survives.", "items": [ { "id": "problem-solving-s04-q01", "prompt": "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?", "choices": [ { "id": "a", "label": "Locker 1" }, { "id": "b", "label": "Locker 2" }, { "id": "c", "label": "Locker 3" }, { "id": "d", "label": "It cannot be determined" } ] }, { "id": "problem-solving-s04-q02", "prompt": "Four runners finish a race. Dev finishes before Eli but after Fatima. Gus finishes last. Who wins the race?", "choices": [ { "id": "a", "label": "Dev" }, { "id": "b", "label": "Eli" }, { "id": "c", "label": "Fatima" }, { "id": "d", "label": "Gus" } ] }, { "id": "problem-solving-s04-q03", "prompt": "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?", "choices": [ { "id": "a", "label": "White" }, { "id": "b", "label": "Brown" }, { "id": "c", "label": "Rye" }, { "id": "d", "label": "Jo cannot order a sandwich" } ] }, { "id": "problem-solving-s04-q04", "prompt": "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?", "choices": [ { "id": "a", "label": "Piano" }, { "id": "b", "label": "Violin" }, { "id": "c", "label": "Drums" }, { "id": "d", "label": "It cannot be determined" } ] }, { "id": "problem-solving-s04-q05", "prompt": "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?", "choices": [ { "id": "a", "label": "369" }, { "id": "b", "label": "636" }, { "id": "c", "label": "963" }, { "id": "d", "label": "639" } ] }, { "id": "problem-solving-s04-q06", "prompt": "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?", "choices": [ { "id": "a", "label": "2" }, { "id": "b", "label": "3" }, { "id": "c", "label": "4" }, { "id": "d", "label": "5" } ] }, { "id": "problem-solving-s04-q07", "prompt": "Tara is older than Uma. Uma is older than Vik. Wren is younger than Vik. Who is the second oldest?", "choices": [ { "id": "a", "label": "Uma" }, { "id": "b", "label": "Vik" }, { "id": "c", "label": "Tara" }, { "id": "d", "label": "Wren" } ] }, { "id": "problem-solving-s04-q08", "prompt": "A password is one of: 4412, 4421, 2144, or 1442. It starts with 4 and its last digit is odd. Which is it?", "choices": [ { "id": "a", "label": "4412" }, { "id": "b", "label": "4421" }, { "id": "c", "label": "2144" }, { "id": "d", "label": "1442" } ] }, { "id": "problem-solving-s04-q09", "prompt": "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)?", "choices": [ { "id": "a", "label": "The biography" }, { "id": "b", "label": "The dictionary" }, { "id": "c", "label": "The cookbook" }, { "id": "d", "label": "The atlas" } ] }, { "id": "problem-solving-s04-q10", "prompt": "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?", "choices": [ { "id": "a", "label": "Ana" }, { "id": "b", "label": "Ben" }, { "id": "c", "label": "Carl" }, { "id": "d", "label": "Dee" } ] } ] }, { "id": "problem-solving-s05", "title": "Finding the rule", "instructions": "Work out the rule that generates each pattern, then apply it.", "items": [ { "id": "problem-solving-s05-q01", "prompt": "What is the next number in the sequence 2, 5, 11, 23, 47, …?", "choices": [ { "id": "a", "label": "94" }, { "id": "b", "label": "95" }, { "id": "c", "label": "93" }, { "id": "d", "label": "71" } ] }, { "id": "problem-solving-s05-q02", "prompt": "One number breaks the doubling pattern in this list: 3, 6, 12, 24, 45, 96. Which one?", "choices": [ { "id": "a", "label": "6" }, { "id": "b", "label": "24" }, { "id": "c", "label": "96" }, { "id": "d", "label": "45" } ] }, { "id": "problem-solving-s05-q03", "prompt": "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?", "choices": [ { "id": "a", "label": "15" }, { "id": "b", "label": "10" }, { "id": "c", "label": "14" }, { "id": "d", "label": "16" } ] }, { "id": "problem-solving-s05-q04", "prompt": "In a simple code, APPLE is written as BQQMF. Using the same rule, how would LEMON be written?", "choices": [ { "id": "a", "label": "MFOPN" }, { "id": "b", "label": "KDLNM" }, { "id": "c", "label": "MFNPO" }, { "id": "d", "label": "MGNQO" } ] }, { "id": "problem-solving-s05-q05", "prompt": "What is the missing number: 1, 4, 9, 16, __, 36?", "choices": [ { "id": "a", "label": "24" }, { "id": "b", "label": "25" }, { "id": "c", "label": "20" }, { "id": "d", "label": "30" } ] }, { "id": "problem-solving-s05-q06", "prompt": "In this sequence each term is the sum of the two before it: 4, 7, 11, 18, 29, … What comes next?", "choices": [ { "id": "a", "label": "40" }, { "id": "b", "label": "36" }, { "id": "c", "label": "44" }, { "id": "d", "label": "47" } ] }, { "id": "problem-solving-s05-q07", "prompt": "Shapes repeat in the fixed cycle: star, moon, moon, sun, star, moon, moon, sun, … What is the 11th shape?", "choices": [ { "id": "a", "label": "Star" }, { "id": "b", "label": "Sun" }, { "id": "c", "label": "Moon" }, { "id": "d", "label": "It cannot be determined" } ] }, { "id": "problem-solving-s05-q08", "prompt": "A rule maps 6 → 15, 8 → 19, and 10 → 23. Using the same rule, what does 12 map to?", "choices": [ { "id": "a", "label": "27" }, { "id": "b", "label": "25" }, { "id": "c", "label": "26" }, { "id": "d", "label": "29" } ] }, { "id": "problem-solving-s05-q09", "prompt": "What is the next term in the sequence 3, 4, 6, 9, 13, 18, …?", "choices": [ { "id": "a", "label": "22" }, { "id": "b", "label": "23" }, { "id": "c", "label": "25" }, { "id": "d", "label": "24" } ] }, { "id": "problem-solving-s05-q10", "prompt": "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?", "choices": [ { "id": "a", "label": "It grows without limit" }, { "id": "b", "label": "It stays at 9 forever" }, { "id": "c", "label": "It alternates between 9 and 18" }, { "id": "d", "label": "It eventually reaches 0" } ] } ] }, { "id": "problem-solving-s06", "title": "Planning and scheduling", "instructions": "Fit tasks and journeys into time, watching for gaps, breaks, and overlaps.", "items": [ { "id": "problem-solving-s06-q01", "prompt": "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?", "choices": [ { "id": "a", "label": "18:50" }, { "id": "b", "label": "18:30" }, { "id": "c", "label": "18:20" }, { "id": "d", "label": "19:00" } ] }, { "id": "problem-solving-s06-q02", "prompt": "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?", "choices": [ { "id": "a", "label": "11:55" }, { "id": "b", "label": "12:05" }, { "id": "c", "label": "12:15" }, { "id": "d", "label": "12:25" } ] }, { "id": "problem-solving-s06-q03", "prompt": "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?", "choices": [ { "id": "a", "label": "52 minutes" }, { "id": "b", "label": "30 minutes" }, { "id": "c", "label": "42 minutes" }, { "id": "d", "label": "40 minutes" } ] }, { "id": "problem-solving-s06-q04", "prompt": "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?", "choices": [ { "id": "a", "label": "10:45" }, { "id": "b", "label": "10:30" }, { "id": "c", "label": "11:00" }, { "id": "d", "label": "10:15" } ] }, { "id": "problem-solving-s06-q05", "prompt": "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?", "choices": [ { "id": "a", "label": "15" }, { "id": "b", "label": "16" }, { "id": "c", "label": "17" }, { "id": "d", "label": "14" } ] }, { "id": "problem-solving-s06-q06", "prompt": "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?", "choices": [ { "id": "a", "label": "Thursday of the same week" }, { "id": "b", "label": "Tuesday of the following week" }, { "id": "c", "label": "Friday of the same week" }, { "id": "d", "label": "Wednesday of the following week" } ] }, { "id": "problem-solving-s06-q07", "prompt": "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?", "choices": [ { "id": "a", "label": "16:47" }, { "id": "b", "label": "16:42" }, { "id": "c", "label": "16:37" }, { "id": "d", "label": "16:27" } ] }, { "id": "problem-solving-s06-q08", "prompt": "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?", "choices": [ { "id": "a", "label": "09:24" }, { "id": "b", "label": "09:48" }, { "id": "c", "label": "09:20" }, { "id": "d", "label": "09:36" } ] }, { "id": "problem-solving-s06-q09", "prompt": "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?", "choices": [ { "id": "a", "label": "9 days" }, { "id": "b", "label": "7 days" }, { "id": "c", "label": "5 days" }, { "id": "d", "label": "6 days" } ] }, { "id": "problem-solving-s06-q10", "prompt": "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?", "choices": [ { "id": "a", "label": "13" }, { "id": "b", "label": "16" }, { "id": "c", "label": "14" }, { "id": "d", "label": "15" } ] } ] }, { "id": "problem-solving-s07", "title": "Trade-offs and best choices", "instructions": "Compare the options fairly. Put them on the same footing before deciding which wins.", "items": [ { "id": "problem-solving-s07-q01", "prompt": "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?", "choices": [ { "id": "a", "label": "Shop A, at 25p per egg" }, { "id": "b", "label": "Shop B, at 23p per egg" }, { "id": "c", "label": "Shop A, at 23p per egg" }, { "id": "d", "label": "They cost the same per egg" } ] }, { "id": "problem-solving-s07-q02", "prompt": "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?", "choices": [ { "id": "a", "label": "2 GB" }, { "id": "b", "label": "3 GB" }, { "id": "c", "label": "4 GB" }, { "id": "d", "label": "5 GB" } ] }, { "id": "problem-solving-s07-q03", "prompt": "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?", "choices": [ { "id": "a", "label": "4 hours" }, { "id": "b", "label": "3 hours" }, { "id": "c", "label": "5 hours" }, { "id": "d", "label": "6 hours" } ] }, { "id": "problem-solving-s07-q04", "prompt": "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?", "choices": [ { "id": "a", "label": "9 kg + 6 kg" }, { "id": "b", "label": "7 kg + 6 kg + 4 kg" }, { "id": "c", "label": "9 kg + 4 kg" }, { "id": "d", "label": "9 kg + 7 kg" } ] }, { "id": "problem-solving-s07-q05", "prompt": "An online store offers a choice of discounts: 15% off everything, or £10 off orders over £60. Which saves more on a £70 order?", "choices": [ { "id": "a", "label": "£10 off, saving £10" }, { "id": "b", "label": "15% off, saving £10.50" }, { "id": "c", "label": "Both save the same" }, { "id": "d", "label": "15% off, saving £9.50" } ] }, { "id": "problem-solving-s07-q06", "prompt": "A gardener has 24 metres of fencing for a rectangular bed with whole-metre sides. Which dimensions give the largest area?", "choices": [ { "id": "a", "label": "8 m × 4 m" }, { "id": "b", "label": "10 m × 2 m" }, { "id": "c", "label": "6 m × 6 m" }, { "id": "d", "label": "7 m × 5 m" } ] }, { "id": "problem-solving-s07-q07", "prompt": "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?", "choices": [ { "id": "a", "label": "The bus" }, { "id": "b", "label": "Cycling" }, { "id": "c", "label": "The bus and cycling tie" }, { "id": "d", "label": "Driving" } ] }, { "id": "problem-solving-s07-q08", "prompt": "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?", "choices": [ { "id": "a", "label": "£4" }, { "id": "b", "label": "£6" }, { "id": "c", "label": "£8" }, { "id": "d", "label": "£12" } ] }, { "id": "problem-solving-s07-q09", "prompt": "A bake sale prices cupcakes at 80p each, or 3 for £2. What is the cheapest cost of exactly 7 cupcakes?", "choices": [ { "id": "a", "label": "£4.80" }, { "id": "b", "label": "£5.60" }, { "id": "c", "label": "£4.66" }, { "id": "d", "label": "£5.20" } ] }, { "id": "problem-solving-s07-q10", "prompt": "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?", "choices": [ { "id": "a", "label": "Yes, with 10 minutes spare" }, { "id": "b", "label": "Yes, with exactly no spare time" }, { "id": "c", "label": "No: it overruns by 5 minutes" }, { "id": "d", "label": "No: it overruns by 15 minutes" } ] } ] }, { "id": "problem-solving-s08", "title": "Multi-stage processes", "instructions": "Follow a quantity through several stages in turn, updating it carefully at each step.", "items": [ { "id": "problem-solving-s08-q01", "prompt": "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?", "choices": [ { "id": "a", "label": "30 minutes" }, { "id": "b", "label": "25 minutes" }, { "id": "c", "label": "36 minutes" }, { "id": "d", "label": "28 minutes" } ] }, { "id": "problem-solving-s08-q02", "prompt": "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?", "choices": [ { "id": "a", "label": "12.5 minutes" }, { "id": "b", "label": "20 minutes" }, { "id": "c", "label": "25 minutes" }, { "id": "d", "label": "33 minutes" } ] }, { "id": "problem-solving-s08-q03", "prompt": "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?", "choices": [ { "id": "a", "label": "6" }, { "id": "b", "label": "9" }, { "id": "c", "label": "7" }, { "id": "d", "label": "8" } ] }, { "id": "problem-solving-s08-q04", "prompt": "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?", "choices": [ { "id": "a", "label": "60" }, { "id": "b", "label": "80" }, { "id": "c", "label": "100" }, { "id": "d", "label": "50" } ] }, { "id": "problem-solving-s08-q05", "prompt": "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?", "choices": [ { "id": "a", "label": "180" }, { "id": "b", "label": "170" }, { "id": "c", "label": "171" }, { "id": "d", "label": "190" } ] }, { "id": "problem-solving-s08-q06", "prompt": "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?", "choices": [ { "id": "a", "label": "£210" }, { "id": "b", "label": "£215.25" }, { "id": "c", "label": "£205" }, { "id": "d", "label": "£220.50" } ] }, { "id": "problem-solving-s08-q07", "prompt": "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?", "choices": [ { "id": "a", "label": "240 kg" }, { "id": "b", "label": "100 kg" }, { "id": "c", "label": "60 kg" }, { "id": "d", "label": "180 kg" } ] }, { "id": "problem-solving-s08-q08", "prompt": "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?", "choices": [ { "id": "a", "label": "Just under 2 minutes" }, { "id": "b", "label": "Just under 1 minute 50 seconds" }, { "id": "c", "label": "Just under 2 minutes 10 seconds" }, { "id": "d", "label": "Just under 1 minute 30 seconds" } ] }, { "id": "problem-solving-s08-q09", "prompt": "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?", "choices": [ { "id": "a", "label": "60" }, { "id": "b", "label": "30" }, { "id": "c", "label": "45" }, { "id": "d", "label": "20" } ] }, { "id": "problem-solving-s08-q10", "prompt": "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?", "choices": [ { "id": "a", "label": "120" }, { "id": "b", "label": "110" }, { "id": "c", "label": "112" }, { "id": "d", "label": "115" } ] } ] }, { "id": "problem-solving-s09", "title": "Arrangements, counting and chance", "instructions": "Count possibilities systematically, and reason about likelihood from equally likely outcomes.", "items": [ { "id": "problem-solving-s09-q01", "prompt": "A cafe offers 3 soups and 4 sandwiches. How many different soup-and-sandwich lunches are possible?", "choices": [ { "id": "a", "label": "7" }, { "id": "b", "label": "12" }, { "id": "c", "label": "9" }, { "id": "d", "label": "16" } ] }, { "id": "problem-solving-s09-q02", "prompt": "How many different ways can the three letters of the word TAP be arranged?", "choices": [ { "id": "a", "label": "3" }, { "id": "b", "label": "9" }, { "id": "c", "label": "6" }, { "id": "d", "label": "12" } ] }, { "id": "problem-solving-s09-q03", "prompt": "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?", "choices": [ { "id": "a", "label": "3" }, { "id": "b", "label": "2" }, { "id": "c", "label": "5" }, { "id": "d", "label": "7" } ] }, { "id": "problem-solving-s09-q04", "prompt": "A spinner has 8 equal sectors numbered 1 to 8. What is the probability of spinning an even number greater than 3?", "choices": [ { "id": "a", "label": "1/2" }, { "id": "b", "label": "1/4" }, { "id": "c", "label": "5/8" }, { "id": "d", "label": "3/8" } ] }, { "id": "problem-solving-s09-q05", "prompt": "Four people each shake hands with every other person exactly once. How many handshakes happen?", "choices": [ { "id": "a", "label": "12" }, { "id": "b", "label": "4" }, { "id": "c", "label": "6" }, { "id": "d", "label": "8" } ] }, { "id": "problem-solving-s09-q06", "prompt": "A committee of 2 people is chosen from 5 volunteers. How many different committees are possible?", "choices": [ { "id": "a", "label": "20" }, { "id": "b", "label": "10" }, { "id": "c", "label": "25" }, { "id": "d", "label": "15" } ] }, { "id": "problem-solving-s09-q07", "prompt": "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?", "choices": [ { "id": "a", "label": "3/10" }, { "id": "b", "label": "1/2" }, { "id": "c", "label": "3/7" }, { "id": "d", "label": "7/10" } ] }, { "id": "problem-solving-s09-q08", "prompt": "How many 3-digit numbers can be formed from the digits 2, 5, 7, and 8 if no digit may repeat?", "choices": [ { "id": "a", "label": "24" }, { "id": "b", "label": "12" }, { "id": "c", "label": "64" }, { "id": "d", "label": "36" } ] }, { "id": "problem-solving-s09-q09", "prompt": "Two fair six-sided dice are rolled. What is the probability that the total is exactly 4?", "choices": [ { "id": "a", "label": "1/9" }, { "id": "b", "label": "1/12" }, { "id": "c", "label": "1/6" }, { "id": "d", "label": "1/18" } ] }, { "id": "problem-solving-s09-q10", "prompt": "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?", "choices": [ { "id": "a", "label": "8" }, { "id": "b", "label": "12" }, { "id": "c", "label": "16" }, { "id": "d", "label": "24" } ] } ] }, { "id": "problem-solving-s10", "title": "Integrated challenge problems", "instructions": "The hardest set: each item combines several techniques from earlier sections.", "items": [ { "id": "problem-solving-s10-q01", "prompt": "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?", "choices": [ { "id": "a", "label": "£160" }, { "id": "b", "label": "£128" }, { "id": "c", "label": "£144" }, { "id": "d", "label": "£136" } ] }, { "id": "problem-solving-s10-q02", "prompt": "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?", "choices": [ { "id": "a", "label": "2 hours" }, { "id": "b", "label": "1 hour 45 minutes" }, { "id": "c", "label": "2 hours 15 minutes" }, { "id": "d", "label": "1 hour 30 minutes" } ] }, { "id": "problem-solving-s10-q03", "prompt": "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?", "choices": [ { "id": "a", "label": "40 minutes" }, { "id": "b", "label": "32 minutes" }, { "id": "c", "label": "36 minutes" }, { "id": "d", "label": "24 minutes" } ] }, { "id": "problem-solving-s10-q04", "prompt": "In an office of 40 people, 22 drink coffee, 17 drink tea, and 9 drink both. How many drink neither?", "choices": [ { "id": "a", "label": "8" }, { "id": "b", "label": "10" }, { "id": "c", "label": "12" }, { "id": "d", "label": "6" } ] }, { "id": "problem-solving-s10-q05", "prompt": "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?", "choices": [ { "id": "a", "label": "11:30" }, { "id": "b", "label": "12:30" }, { "id": "c", "label": "11:45" }, { "id": "d", "label": "12:00" } ] }, { "id": "problem-solving-s10-q06", "prompt": "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?", "choices": [ { "id": "a", "label": "12" }, { "id": "b", "label": "10" }, { "id": "c", "label": "8" }, { "id": "d", "label": "15" } ] }, { "id": "problem-solving-s10-q07", "prompt": "A two-digit number has digits that sum to 9. Reversing its digits produces a number 27 larger. What is the original number?", "choices": [ { "id": "a", "label": "63" }, { "id": "b", "label": "45" }, { "id": "c", "label": "36" }, { "id": "d", "label": "27" } ] }, { "id": "problem-solving-s10-q08", "prompt": "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?", "choices": [ { "id": "a", "label": "£300" }, { "id": "b", "label": "£336" }, { "id": "c", "label": "£360" }, { "id": "d", "label": "£276" } ] }, { "id": "problem-solving-s10-q09", "prompt": "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?", "choices": [ { "id": "a", "label": "4" }, { "id": "b", "label": "3" }, { "id": "c", "label": "It cannot be done" }, { "id": "d", "label": "2" } ] }, { "id": "problem-solving-s10-q10", "prompt": "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?", "choices": [ { "id": "a", "label": "15" }, { "id": "b", "label": "14" }, { "id": "c", "label": "16" }, { "id": "d", "label": "13" } ] } ] } ] }, "authoredHints": { "title": "Authored hints", "policy": "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.", "items": [], "missingItemIds": [ "problem-solving-s01-q01", "problem-solving-s01-q02", "problem-solving-s01-q03", "problem-solving-s01-q04", "problem-solving-s01-q05", "problem-solving-s01-q06", "problem-solving-s01-q07", "problem-solving-s01-q08", "problem-solving-s01-q09", "problem-solving-s01-q10", "problem-solving-s02-q01", "problem-solving-s02-q02", "problem-solving-s02-q03", "problem-solving-s02-q04", "problem-solving-s02-q05", "problem-solving-s02-q06", "problem-solving-s02-q07", "problem-solving-s02-q08", "problem-solving-s02-q09", "problem-solving-s02-q10", "problem-solving-s03-q01", "problem-solving-s03-q02", "problem-solving-s03-q03", "problem-solving-s03-q04", "problem-solving-s03-q05", "problem-solving-s03-q06", "problem-solving-s03-q07", "problem-solving-s03-q08", "problem-solving-s03-q09", "problem-solving-s03-q10", "problem-solving-s04-q01", "problem-solving-s04-q02", 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"problem-solving-s01-q02", "correctChoiceId": "c", "correctChoiceLabel": "15" }, { "itemId": "problem-solving-s01-q03", "correctChoiceId": "c", "correctChoiceLabel": "24" }, { "itemId": "problem-solving-s01-q04", "correctChoiceId": "b", "correctChoiceLabel": "A, by 3 days" }, { "itemId": "problem-solving-s01-q05", "correctChoiceId": "c", "correctChoiceLabel": "24" }, { "itemId": "problem-solving-s01-q06", "correctChoiceId": "c", "correctChoiceLabel": "Seat 3" }, { "itemId": "problem-solving-s01-q07", "correctChoiceId": "a", "correctChoiceLabel": "£6" }, { "itemId": "problem-solving-s01-q08", "correctChoiceId": "d", "correctChoiceLabel": "The 120 kg crate" }, { "itemId": "problem-solving-s01-q09", "correctChoiceId": "a", "correctChoiceLabel": "3" }, { "itemId": "problem-solving-s01-q10", "correctChoiceId": "d", "correctChoiceLabel": "10 minutes" }, { "itemId": "problem-solving-s02-q01", "correctChoiceId": "a", "correctChoiceLabel": "£15" }, { "itemId": "problem-solving-s02-q02", "correctChoiceId": "b", "correctChoiceLabel": "25 minutes" }, { "itemId": "problem-solving-s02-q03", "correctChoiceId": "d", "correctChoiceLabel": "400 ml" }, { "itemId": "problem-solving-s02-q04", "correctChoiceId": "a", "correctChoiceLabel": "12 hours" }, { "itemId": "problem-solving-s02-q05", "correctChoiceId": "b", "correctChoiceLabel": "15 litres" }, { "itemId": "problem-solving-s02-q06", "correctChoiceId": "d", "correctChoiceLabel": "£574" }, { "itemId": "problem-solving-s02-q07", "correctChoiceId": "a", "correctChoiceLabel": "3 km" }, { "itemId": "problem-solving-s02-q08", "correctChoiceId": "c", "correctChoiceLabel": "750 g" }, { "itemId": "problem-solving-s02-q09", "correctChoiceId": "b", "correctChoiceLabel": "£40" }, { "itemId": "problem-solving-s02-q10", "correctChoiceId": "d", "correctChoiceLabel": "30 litres" }, { "itemId": "problem-solving-s03-q01", "correctChoiceId": "b", "correctChoiceLabel": "12" }, { "itemId": "problem-solving-s03-q02", "correctChoiceId": "a", "correctChoiceLabel": "£20" }, { "itemId": "problem-solving-s03-q03", "correctChoiceId": "d", "correctChoiceLabel": "36" }, { "itemId": "problem-solving-s03-q04", "correctChoiceId": "c", "correctChoiceLabel": "£60" }, { "itemId": "problem-solving-s03-q05", "correctChoiceId": "b", "correctChoiceLabel": "Day 10" }, { "itemId": "problem-solving-s03-q06", "correctChoiceId": "d", "correctChoiceLabel": "24" }, { "itemId": "problem-solving-s03-q07", "correctChoiceId": "a", "correctChoiceLabel": "11:25" }, { "itemId": "problem-solving-s03-q08", "correctChoiceId": "c", "correctChoiceLabel": "£20,000" }, { "itemId": "problem-solving-s03-q09", "correctChoiceId": "b", "correctChoiceLabel": "32" }, { "itemId": "problem-solving-s03-q10", "correctChoiceId": "d", "correctChoiceLabel": "4,355" }, { "itemId": "problem-solving-s04-q01", "correctChoiceId": "a", "correctChoiceLabel": "Locker 1" }, { "itemId": "problem-solving-s04-q02", "correctChoiceId": "c", "correctChoiceLabel": "Fatima" }, { "itemId": "problem-solving-s04-q03", "correctChoiceId": "b", "correctChoiceLabel": "Brown" }, { "itemId": "problem-solving-s04-q04", "correctChoiceId": "a", "correctChoiceLabel": "Piano" }, { "itemId": "problem-solving-s04-q05", "correctChoiceId": "d", "correctChoiceLabel": "639" }, { "itemId": "problem-solving-s04-q06", "correctChoiceId": "c", "correctChoiceLabel": "4" }, { "itemId": "problem-solving-s04-q07", "correctChoiceId": "a", "correctChoiceLabel": "Uma" }, { "itemId": "problem-solving-s04-q08", "correctChoiceId": "b", "correctChoiceLabel": "4421" }, { "itemId": "problem-solving-s04-q09", "correctChoiceId": "d", "correctChoiceLabel": "The atlas" }, { "itemId": "problem-solving-s04-q10", "correctChoiceId": "c", "correctChoiceLabel": "Carl" }, { "itemId": "problem-solving-s05-q01", "correctChoiceId": "b", "correctChoiceLabel": "95" }, { "itemId": "problem-solving-s05-q02", "correctChoiceId": "d", "correctChoiceLabel": "45" }, { "itemId": "problem-solving-s05-q03", "correctChoiceId": "a", "correctChoiceLabel": "15" }, { "itemId": "problem-solving-s05-q04", "correctChoiceId": "c", "correctChoiceLabel": "MFNPO" }, { "itemId": "problem-solving-s05-q05", "correctChoiceId": "b", "correctChoiceLabel": "25" }, { "itemId": "problem-solving-s05-q06", "correctChoiceId": "d", "correctChoiceLabel": "47" }, { "itemId": "problem-solving-s05-q07", "correctChoiceId": "c", "correctChoiceLabel": "Moon" }, { "itemId": "problem-solving-s05-q08", "correctChoiceId": "a", "correctChoiceLabel": "27" }, { "itemId": "problem-solving-s05-q09", "correctChoiceId": "d", "correctChoiceLabel": "24" }, { "itemId": "problem-solving-s05-q10", "correctChoiceId": "b", "correctChoiceLabel": "It stays at 9 forever" }, { "itemId": "problem-solving-s06-q01", "correctChoiceId": "a", "correctChoiceLabel": "18:50" }, { "itemId": "problem-solving-s06-q02", "correctChoiceId": "c", "correctChoiceLabel": "12:15" }, { "itemId": "problem-solving-s06-q03", "correctChoiceId": "d", "correctChoiceLabel": "40 minutes" }, { "itemId": "problem-solving-s06-q04", "correctChoiceId": "a", "correctChoiceLabel": "10:45" }, { "itemId": "problem-solving-s06-q05", "correctChoiceId": "b", "correctChoiceLabel": "16" }, { "itemId": "problem-solving-s06-q06", "correctChoiceId": "d", "correctChoiceLabel": "Wednesday of the following week" }, { "itemId": "problem-solving-s06-q07", "correctChoiceId": "c", "correctChoiceLabel": "16:37" }, { "itemId": "problem-solving-s06-q08", "correctChoiceId": "a", "correctChoiceLabel": "09:24" }, { "itemId": "problem-solving-s06-q09", "correctChoiceId": "b", "correctChoiceLabel": "7 days" }, { "itemId": "problem-solving-s06-q10", "correctChoiceId": "d", "correctChoiceLabel": "15" }, { "itemId": "problem-solving-s07-q01", "correctChoiceId": "b", "correctChoiceLabel": "Shop B, at 23p per egg" }, { "itemId": "problem-solving-s07-q02", "correctChoiceId": "c", "correctChoiceLabel": "4 GB" }, { "itemId": "problem-solving-s07-q03", "correctChoiceId": "a", "correctChoiceLabel": "4 hours" }, { "itemId": "problem-solving-s07-q04", "correctChoiceId": "d", "correctChoiceLabel": "9 kg + 7 kg" }, { "itemId": "problem-solving-s07-q05", "correctChoiceId": "b", "correctChoiceLabel": "15% off, saving £10.50" }, { "itemId": "problem-solving-s07-q06", "correctChoiceId": "c", "correctChoiceLabel": "6 m × 6 m" }, { "itemId": "problem-solving-s07-q07", "correctChoiceId": "d", "correctChoiceLabel": "Driving" }, { "itemId": "problem-solving-s07-q08", "correctChoiceId": "b", "correctChoiceLabel": "£6" }, { "itemId": "problem-solving-s07-q09", "correctChoiceId": "a", "correctChoiceLabel": "£4.80" }, { "itemId": "problem-solving-s07-q10", "correctChoiceId": "c", "correctChoiceLabel": "No: it overruns by 5 minutes" }, { "itemId": "problem-solving-s08-q01", "correctChoiceId": "a", "correctChoiceLabel": "30 minutes" }, { "itemId": "problem-solving-s08-q02", "correctChoiceId": "b", "correctChoiceLabel": "20 minutes" }, { "itemId": "problem-solving-s08-q03", "correctChoiceId": "d", "correctChoiceLabel": "8" }, { "itemId": "problem-solving-s08-q04", "correctChoiceId": "a", "correctChoiceLabel": "60" }, { "itemId": "problem-solving-s08-q05", "correctChoiceId": "c", "correctChoiceLabel": "171" }, { "itemId": "problem-solving-s08-q06", "correctChoiceId": "b", "correctChoiceLabel": "£215.25" }, { "itemId": "problem-solving-s08-q07", "correctChoiceId": "d", "correctChoiceLabel": "180 kg" }, { "itemId": "problem-solving-s08-q08", "correctChoiceId": "a", "correctChoiceLabel": "Just under 2 minutes" }, { "itemId": "problem-solving-s08-q09", "correctChoiceId": "b", "correctChoiceLabel": "30" }, { "itemId": "problem-solving-s08-q10", "correctChoiceId": "c", "correctChoiceLabel": "112" }, { "itemId": "problem-solving-s09-q01", "correctChoiceId": "b", "correctChoiceLabel": "12" }, { "itemId": "problem-solving-s09-q02", "correctChoiceId": "c", "correctChoiceLabel": "6" }, { "itemId": "problem-solving-s09-q03", "correctChoiceId": "a", "correctChoiceLabel": "3" }, { "itemId": "problem-solving-s09-q04", "correctChoiceId": "d", "correctChoiceLabel": "3/8" }, { "itemId": "problem-solving-s09-q05", "correctChoiceId": "c", "correctChoiceLabel": "6" }, { "itemId": "problem-solving-s09-q06", "correctChoiceId": "b", "correctChoiceLabel": "10" }, { "itemId": "problem-solving-s09-q07", "correctChoiceId": "d", "correctChoiceLabel": "7/10" }, { "itemId": "problem-solving-s09-q08", "correctChoiceId": "a", "correctChoiceLabel": "24" }, { "itemId": "problem-solving-s09-q09", "correctChoiceId": "b", "correctChoiceLabel": "1/12" }, { "itemId": "problem-solving-s09-q10", "correctChoiceId": "c", "correctChoiceLabel": "16" }, { "itemId": "problem-solving-s10-q01", "correctChoiceId": "d", "correctChoiceLabel": "£136" }, { "itemId": "problem-solving-s10-q02", "correctChoiceId": "a", "correctChoiceLabel": "2 hours" }, { "itemId": "problem-solving-s10-q03", "correctChoiceId": "c", "correctChoiceLabel": "36 minutes" }, { "itemId": "problem-solving-s10-q04", "correctChoiceId": "b", "correctChoiceLabel": "10" }, { "itemId": "problem-solving-s10-q05", "correctChoiceId": "d", "correctChoiceLabel": "12:00" }, { "itemId": "problem-solving-s10-q06", "correctChoiceId": "a", "correctChoiceLabel": "12" }, { "itemId": "problem-solving-s10-q07", "correctChoiceId": "c", "correctChoiceLabel": "36" }, { "itemId": "problem-solving-s10-q08", "correctChoiceId": "b", "correctChoiceLabel": "£336" }, { "itemId": "problem-solving-s10-q09", "correctChoiceId": "d", "correctChoiceLabel": "2" }, { "itemId": "problem-solving-s10-q10", "correctChoiceId": "a", "correctChoiceLabel": "15" } ] }, "workedExplanations": { "title": "Worked explanations", "items": [ { "itemId": "problem-solving-s01-q01", "explanation": "The next joint restock day is the least common multiple of 2, 3, and 4, which is 12." }, { "itemId": "problem-solving-s01-q02", "explanation": "16 of 40 small boxes uses 40% of the space, leaving 60% free: 60% of 25 large boxes = 15." }, { "itemId": "problem-solving-s01-q03", "explanation": "With 4 distinct digits and no repeats, this is 4! = 24 arrangements." }, { "itemId": "problem-solving-s01-q04", "explanation": "Item A lasts 40 ÷ 8 = 5 days; Item B lasts 40 ÷ 5 = 8 days. A runs out 3 days sooner." }, { "itemId": "problem-solving-s01-q05", "explanation": "Multiply the independent choices: 4 × 3 × 2 = 24." }, { "itemId": "problem-solving-s01-q06", "explanation": "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." }, { "itemId": "problem-solving-s01-q07", "explanation": "Work out the total first: 2 × £9 + £6 = £24. Then subtract from the payment: £30 − £24 = £6." }, { "itemId": "problem-solving-s01-q08", "explanation": "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." }, { "itemId": "problem-solving-s01-q09", "explanation": "Two boxes hold 80 tea bags, and 80 ÷ 25 = 3.2, so only 3 complete weeks are covered before the supply runs short." }, { "itemId": "problem-solving-s01-q10", "explanation": "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." }, { "itemId": "problem-solving-s02-q01", "explanation": "The ratio has 2 + 3 + 4 = 9 parts, so one part is £45 ÷ 9 = £5. The middle share is 3 parts: £15." }, { "itemId": "problem-solving-s02-q02", "explanation": "Divide the total by the rate: 300 ÷ 12 = 25 minutes." }, { "itemId": "problem-solving-s02-q03", "explanation": "The mix has 1 + 4 = 5 equal parts, so each part is 2,000 ÷ 5 = 400 ml, and concentrate is exactly one part." }, { "itemId": "problem-solving-s02-q04", "explanation": "Halving the workforce doubles the time when rates are equal: 6 × 2 = 12 hours." }, { "itemId": "problem-solving-s02-q05", "explanation": "250 km is 2.5 blocks of 100 km, so fuel scales the same way: 6 × 2.5 = 15 litres." }, { "itemId": "problem-solving-s02-q06", "explanation": "Standard pay is 35 × £14 = £490. The overtime rate is £21 per hour, adding 4 × £21 = £84, for £574 in total." }, { "itemId": "problem-solving-s02-q07", "explanation": "6 cm × 50,000 = 300,000 cm. Converting units step by step: 300,000 cm = 3,000 m = 3 km." }, { "itemId": "problem-solving-s02-q08", "explanation": "Find the per-person amount first: 300 ÷ 4 = 75 g. Then scale up: 75 × 10 = 750 g." }, { "itemId": "problem-solving-s02-q09", "explanation": "A 2 : 1 split makes 3 equal parts of £40 each; the younger grandchild gets one part." }, { "itemId": "problem-solving-s02-q10", "explanation": "The rate is 90 ÷ 45 = 2 litres per minute, so 30 minutes delivers 60 litres, leaving 90 − 60 = 30 litres to go." }, { "itemId": "problem-solving-s03-q01", "explanation": "Reverse the steps in reverse order: subtract 7 (31 − 7 = 24), then halve (24 ÷ 2 = 12)." }, { "itemId": "problem-solving-s03-q02", "explanation": "Undo the last step first: before the coffee he had £6 + £4 = £10, which was half his money, so he started with £20." }, { "itemId": "problem-solving-s03-q03", "explanation": "Reverse each operation: add 5 back (7 + 5 = 12), then multiply by 3 (12 × 3 = 36)." }, { "itemId": "problem-solving-s03-q04", "explanation": "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." }, { "itemId": "problem-solving-s03-q05", "explanation": "Work backwards by halving: half covered on day 11, a quarter covered on day 10." }, { "itemId": "problem-solving-s03-q06", "explanation": "Cara's 6 sweets were what remained after two halvings, i.e. a quarter of the jar, so the jar held 6 × 4 = 24." }, { "itemId": "problem-solving-s03-q07", "explanation": "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." }, { "itemId": "problem-solving-s03-q08", "explanation": "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." }, { "itemId": "problem-solving-s03-q09", "explanation": "Double once per round going backwards: 4 → 8 → 16 → 32 teams at the start." }, { "itemId": "problem-solving-s03-q10", "explanation": "Total pages printed = 340 + 340 + 195 = 875. Undo the day's printing: 5,230 − 875 = 4,355." }, { "itemId": "problem-solving-s04-q01", "explanation": "With lockers 2 and 3 assigned, elimination leaves only locker 1 for Finn." }, { "itemId": "problem-solving-s04-q02", "explanation": "Chain the clues into one order: Fatima before Dev before Eli, with Gus last, so the full order is Fatima, Dev, Eli, Gus." }, { "itemId": "problem-solving-s04-q03", "explanation": "One constraint removes rye and another removes white, leaving brown as the only workable option." }, { "itemId": "problem-solving-s04-q04", "explanation": "The violin is taken by Raj and the drums are ruled out for Kim, so the piano is the only instrument left for her." }, { "itemId": "problem-solving-s04-q05", "explanation": "Apply the clues in order: the second digit is 3, so the first is 6, and the third is 6 + 3 = 9, giving 639." }, { "itemId": "problem-solving-s04-q06", "explanation": "Number 5 is at the end of the lane, so its only neighbour is number 4. The red house must be there." }, { "itemId": "problem-solving-s04-q07", "explanation": "Linking the comparisons gives one chain: Tara, Uma, Vik, Wren from oldest to youngest. Uma sits second." }, { "itemId": "problem-solving-s04-q08", "explanation": "Starting with 4 eliminates 2144 and 1442; needing an odd final digit then eliminates 4412, leaving 4421." }, { "itemId": "problem-solving-s04-q09", "explanation": "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." }, { "itemId": "problem-solving-s04-q10", "explanation": "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." }, { "itemId": "problem-solving-s05-q01", "explanation": "Each term is double the previous term plus 1, so the next is 47 × 2 + 1 = 95." }, { "itemId": "problem-solving-s05-q02", "explanation": "Doubling 24 should give 48, not 45, and doubling 48 would correctly give 96, confirming 45 is the misfit." }, { "itemId": "problem-solving-s05-q03", "explanation": "Track whose turn it is: the last step was a subtraction, so the next is an addition, 11 + 4 = 15." }, { "itemId": "problem-solving-s05-q04", "explanation": "Each letter shifts forward by one place in the alphabet, so L, E, M, O, N become M, F, N, P, O." }, { "itemId": "problem-solving-s05-q05", "explanation": "These are the square numbers 1², 2², 3², …, the missing term is 5² = 25." }, { "itemId": "problem-solving-s05-q06", "explanation": "Apply the stated rule to the last two terms: 18 + 29 = 47." }, { "itemId": "problem-solving-s05-q07", "explanation": "The cycle has length 4, and 11 = 4 × 2 + 3, so the 11th shape matches the 3rd in the cycle: moon." }, { "itemId": "problem-solving-s05-q08", "explanation": "Each output is double the input plus 3 (check: 2 × 6 + 3 = 15), so 12 maps to 2 × 12 + 3 = 27." }, { "itemId": "problem-solving-s05-q09", "explanation": "The gaps grow by one each time: +1, +2, +3, +4, +5. The next gap is +6, giving 18 + 6 = 24." }, { "itemId": "problem-solving-s05-q10", "explanation": "Applying the rule to 9 gives 18 − (1 + 8) = 9 again, so 9 is a fixed point. Once reached, the sequence never changes." }, { "itemId": "problem-solving-s06-q01", "explanation": "Add the work and the break together: 3 hours 20 minutes after 15:30 is 18:50." }, { "itemId": "problem-solving-s06-q02", "explanation": "The stages are strictly sequential: 50 + 70 = 120 minutes, and two hours after 10:15 is 12:15." }, { "itemId": "problem-solving-s06-q03", "explanation": "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." }, { "itemId": "problem-solving-s06-q04", "explanation": "A ends at 09:45; adding the 15-minute changeover lets B start at 10:00 and finish 45 minutes later, at 10:45." }, { "itemId": "problem-solving-s06-q05", "explanation": "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." }, { "itemId": "problem-solving-s06-q06", "explanation": "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." }, { "itemId": "problem-solving-s06-q07", "explanation": "Combine the walk and the margin: 35 minutes before 17:12 is 16:37." }, { "itemId": "problem-solving-s06-q08", "explanation": "They coincide at the least common multiple of 8 and 12 minutes, which is 24, so at 09:24." }, { "itemId": "problem-solving-s06-q09", "explanation": "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." }, { "itemId": "problem-solving-s06-q10", "explanation": "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." }, { "itemId": "problem-solving-s07-q01", "explanation": "Convert both to a unit price: £1.50 ÷ 6 = 25p per egg against £2.30 ÷ 10 = 23p per egg. Shop B wins." }, { "itemId": "problem-solving-s07-q02", "explanation": "Set the costs equal: £12 + £2 per GB matches £20 when the data charge is £8, i.e. at 4 GB." }, { "itemId": "problem-solving-s07-q03", "explanation": "Hourly parking costs £7.50 for 3 hours but £10 for 4 hours, so from 4 hours onward the £9 day ticket is cheaper." }, { "itemId": "problem-solving-s07-q04", "explanation": "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." }, { "itemId": "problem-solving-s07-q05", "explanation": "Work out each saving on the actual order value: 15% of £70 is £10.50, which beats the flat £10." }, { "itemId": "problem-solving-s07-q06", "explanation": "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." }, { "itemId": "problem-solving-s07-q07", "explanation": "Apply the hard constraint before comparing costs: only driving arrives within 45 minutes, so cost never comes into play." }, { "itemId": "problem-solving-s07-q08", "explanation": "Match the quantities first: 800 pages needs two singles at £36, so the £30 twin pack saves £6." }, { "itemId": "problem-solving-s07-q09", "explanation": "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." }, { "itemId": "problem-solving-s07-q10", "explanation": "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." }, { "itemId": "problem-solving-s08-q01", "explanation": "Each unit takes 2 + 3 + 1 = 6 minutes from start to finish, so 5 units take 5 × 6 = 30 minutes." }, { "itemId": "problem-solving-s08-q02", "explanation": "Work with the net rate: 8 − 3 = 5 litres per minute, so 100 ÷ 5 = 20 minutes." }, { "itemId": "problem-solving-s08-q03", "explanation": "Three approvals take 3 × 2 = 6 days, and there are only 2 handovers between 3 approvers, adding 2 more days: 8 in total." }, { "itemId": "problem-solving-s08-q04", "explanation": "Apply fee then doubling, twice: (30 − 10) × 2 = 40, then (40 − 10) × 2 = 60. The order of operations within each level matters." }, { "itemId": "problem-solving-s08-q05", "explanation": "Take the percentages of the surviving quantity each time: 200 → 180 after the first check, then 5% of 180 (9 parts) fail, leaving 171." }, { "itemId": "problem-solving-s08-q06", "explanation": "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." }, { "itemId": "problem-solving-s08-q07", "explanation": "Chain the percentages: 60% of 400 kg is 240 kg of paper, and 75% of that survives contamination checks, 0.75 × 240 = 180 kg." }, { "itemId": "problem-solving-s08-q08", "explanation": "The first two legs total exactly 4 minutes (1:55 + 2:05), leaving anything under 2 minutes for the final leg." }, { "itemId": "problem-solving-s08-q09", "explanation": "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." }, { "itemId": "problem-solving-s08-q10", "explanation": "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." }, { "itemId": "problem-solving-s09-q01", "explanation": "Independent choices multiply rather than add: 3 × 4 = 12 different lunches." }, { "itemId": "problem-solving-s09-q02", "explanation": "Three positions filled by three distinct letters gives 3 × 2 × 1 = 6 arrangements." }, { "itemId": "problem-solving-s09-q03", "explanation": "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." }, { "itemId": "problem-solving-s09-q04", "explanation": "List the outcomes that satisfy both conditions: 4, 6, and 8: three of the eight equally likely sectors, so 3/8." }, { "itemId": "problem-solving-s09-q05", "explanation": "Each of the 4 people shakes 3 hands, but that counts every handshake twice: 4 × 3 ÷ 2 = 6." }, { "itemId": "problem-solving-s09-q06", "explanation": "There are 5 × 4 = 20 ordered picks, but a committee ignores order, so halve it: 10 distinct pairs." }, { "itemId": "problem-solving-s09-q07", "explanation": "Use the complement: the chance of green is 3/10, so the chance of not-green is 1 − 3/10 = 7/10." }, { "itemId": "problem-solving-s09-q08", "explanation": "Fill the positions one at a time: 4 choices, then 3, then 2, giving 4 × 3 × 2 = 24 numbers." }, { "itemId": "problem-solving-s09-q09", "explanation": "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." }, { "itemId": "problem-solving-s09-q10", "explanation": "Each floor contributes an independent two-way choice, so the possibilities multiply: 2 × 2 × 2 × 2 = 16 distinct groups." }, { "itemId": "problem-solving-s10-q01", "explanation": "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." }, { "itemId": "problem-solving-s10-q02", "explanation": "When two movers approach each other their speeds add: the gap closes at 30 km/h, so 60 km takes 2 hours." }, { "itemId": "problem-solving-s10-q03", "explanation": "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." }, { "itemId": "problem-solving-s10-q04", "explanation": "Avoid double-counting the overlap: coffee or tea drinkers number 22 + 17 − 9 = 30, so 40 − 30 = 10 drink neither." }, { "itemId": "problem-solving-s10-q05", "explanation": "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." }, { "itemId": "problem-solving-s10-q06", "explanation": "Let the son's age be s, so Anya is 3s. In 12 years: 3s + 12 = 2(s + 12), which solves to s = 12." }, { "itemId": "problem-solving-s10-q07", "explanation": "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." }, { "itemId": "problem-solving-s10-q08", "explanation": "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." }, { "itemId": "problem-solving-s10-q09", "explanation": "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." }, { "itemId": "problem-solving-s10-q10", "explanation": "With r correct answers there are 20 − r wrong ones, so 4r − (20 − r) = 55. That simplifies to 5r = 75, giving r = 15." } ] } }