{ "format": "novus-learn-open-practice", "version": 1, "packId": "cognitive-skills-numerical-reasoning", "source": "cognitive-skills", "slug": "numerical-reasoning", "title": "Numerical reasoning: full practice bank: open practice pack", "subtitle": "100 untimed questions across 10 authored sections.", "sourceHref": "/cognitive-skills/numerical-reasoning", "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": [ "One hundred short problems in ten themed sections, starting gently and building up. Work at your own pace from the numbers given in each question. No outside data or calculator tricks needed.", "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": "numerical-reasoning-s01", "title": "Everyday arithmetic", "instructions": "Warm up with whole-number addition, subtraction, multiplication, and division.", "items": [ { "id": "numerical-reasoning-s01-q01", "prompt": "One shelf holds 24 books and a second shelf holds 38 books. How many books are there altogether?", "choices": [ { "id": "a", "label": "62" }, { "id": "b", "label": "64" }, { "id": "c", "label": "68" }, { "id": "d", "label": "72" } ] }, { "id": "numerical-reasoning-s01-q02", "prompt": "A box of 96 pencils is shared equally among 8 pupils. How many pencils does each pupil receive?", "choices": [ { "id": "a", "label": "6" }, { "id": "b", "label": "8" }, { "id": "c", "label": "10" }, { "id": "d", "label": "12" } ] }, { "id": "numerical-reasoning-s01-q03", "prompt": "Cinema tickets cost £9 each. What is the total cost of 7 tickets?", "choices": [ { "id": "a", "label": "£56" }, { "id": "b", "label": "£63" }, { "id": "c", "label": "£66" }, { "id": "d", "label": "£72" } ] }, { "id": "numerical-reasoning-s01-q04", "prompt": "A tank holds 500 litres of water. If 137 litres are drained out, how many litres remain?", "choices": [ { "id": "a", "label": "337 litres" }, { "id": "b", "label": "353 litres" }, { "id": "c", "label": "363 litres" }, { "id": "d", "label": "373 litres" } ] }, { "id": "numerical-reasoning-s01-q05", "prompt": "What is the value of 18 − 12 ÷ 3?", "choices": [ { "id": "a", "label": "2" }, { "id": "b", "label": "6" }, { "id": "c", "label": "10" }, { "id": "d", "label": "14" } ] }, { "id": "numerical-reasoning-s01-q06", "prompt": "A train has 6 carriages and each carriage seats 72 passengers. How many seats does the train have in total?", "choices": [ { "id": "a", "label": "432" }, { "id": "b", "label": "440" }, { "id": "c", "label": "452" }, { "id": "d", "label": "468" } ] }, { "id": "numerical-reasoning-s01-q07", "prompt": "A crate contains 12 boxes, and each box holds 15 apples. How many apples are in the crate?", "choices": [ { "id": "a", "label": "175" }, { "id": "b", "label": "180" }, { "id": "c", "label": "185" }, { "id": "d", "label": "190" } ] }, { "id": "numerical-reasoning-s01-q08", "prompt": "75 chairs are arranged into complete rows of 9, with any spare chairs left at the side. How many chairs are left over?", "choices": [ { "id": "a", "label": "0" }, { "id": "b", "label": "1" }, { "id": "c", "label": "2" }, { "id": "d", "label": "3" } ] }, { "id": "numerical-reasoning-s01-q09", "prompt": "What is the value of 7 × 8 − 6?", "choices": [ { "id": "a", "label": "50" }, { "id": "b", "label": "52" }, { "id": "c", "label": "56" }, { "id": "d", "label": "62" } ] }, { "id": "numerical-reasoning-s01-q10", "prompt": "A shop opens at 9:00 am and closes at 5:30 pm. For how many hours is it open?", "choices": [ { "id": "a", "label": "7.5 hours" }, { "id": "b", "label": "8 hours" }, { "id": "c", "label": "8.5 hours" }, { "id": "d", "label": "9 hours" } ] } ] }, { "id": "numerical-reasoning-s02", "title": "Fractions and decimals", "instructions": "Add, compare, and convert simple fractions and decimals.", "items": [ { "id": "numerical-reasoning-s02-q01", "prompt": "What is 1/2 + 1/4?", "choices": [ { "id": "a", "label": "2/6" }, { "id": "b", "label": "3/4" }, { "id": "c", "label": "5/6" }, { "id": "d", "label": "1 1/4" } ] }, { "id": "numerical-reasoning-s02-q02", "prompt": "What is 3/5 of 40?", "choices": [ { "id": "a", "label": "24" }, { "id": "b", "label": "26" }, { "id": "c", "label": "28" }, { "id": "d", "label": "32" } ] }, { "id": "numerical-reasoning-s02-q03", "prompt": "What is 0.25 × 80?", "choices": [ { "id": "a", "label": "8" }, { "id": "b", "label": "12" }, { "id": "c", "label": "16" }, { "id": "d", "label": "20" } ] }, { "id": "numerical-reasoning-s02-q04", "prompt": "Which of these fractions is the largest?", "choices": [ { "id": "a", "label": "3/5" }, { "id": "b", "label": "7/12" }, { "id": "c", "label": "2/3" }, { "id": "d", "label": "5/8" } ] }, { "id": "numerical-reasoning-s02-q05", "prompt": "What is 7.2 − 3.85?", "choices": [ { "id": "a", "label": "3.35" }, { "id": "b", "label": "3.45" }, { "id": "c", "label": "3.55" }, { "id": "d", "label": "4.35" } ] }, { "id": "numerical-reasoning-s02-q06", "prompt": "A ribbon 3.6 metres long is cut into pieces each 0.4 metres long. How many pieces are made?", "choices": [ { "id": "a", "label": "8" }, { "id": "b", "label": "9" }, { "id": "c", "label": "10" }, { "id": "d", "label": "12" } ] }, { "id": "numerical-reasoning-s02-q07", "prompt": "What is 18/24 written in its simplest form?", "choices": [ { "id": "a", "label": "2/3" }, { "id": "b", "label": "4/6" }, { "id": "c", "label": "6/8" }, { "id": "d", "label": "3/4" } ] }, { "id": "numerical-reasoning-s02-q08", "prompt": "Two thirds of a class of 27 pupils walk to school. How many pupils walk?", "choices": [ { "id": "a", "label": "12" }, { "id": "b", "label": "18" }, { "id": "c", "label": "21" }, { "id": "d", "label": "24" } ] }, { "id": "numerical-reasoning-s02-q09", "prompt": "What is 0.3 × 0.4?", "choices": [ { "id": "a", "label": "0.12" }, { "id": "b", "label": "0.7" }, { "id": "c", "label": "1.2" }, { "id": "d", "label": "12" } ] }, { "id": "numerical-reasoning-s02-q10", "prompt": "A recipe needs 3/4 of a cup of flour. If you triple the recipe, how many cups of flour do you need?", "choices": [ { "id": "a", "label": "1 1/2" }, { "id": "b", "label": "2" }, { "id": "c", "label": "2 1/4" }, { "id": "d", "label": "2 3/4" } ] } ] }, { "id": "numerical-reasoning-s03", "title": "Percentages", "instructions": "Find percentages of amounts, and express changes as percentages.", "items": [ { "id": "numerical-reasoning-q1", "prompt": "A jacket normally costs $80. It is discounted by 25%. What is the sale price?", "choices": [ { "id": "a", "label": "$55" }, { "id": "b", "label": "$60" }, { "id": "c", "label": "$65" }, { "id": "d", "label": "$70" } ] }, { "id": "numerical-reasoning-s03-q02", "prompt": "What is 15% of 60?", "choices": [ { "id": "a", "label": "3" }, { "id": "b", "label": "6" }, { "id": "c", "label": "7.5" }, { "id": "d", "label": "9" } ] }, { "id": "numerical-reasoning-s03-q03", "prompt": "A pupil scores 45 marks out of 60. What percentage is this?", "choices": [ { "id": "a", "label": "75%" }, { "id": "b", "label": "80%" }, { "id": "c", "label": "85%" }, { "id": "d", "label": "90%" } ] }, { "id": "numerical-reasoning-s03-q04", "prompt": "A price rises from £50 to £58. What is the percentage increase?", "choices": [ { "id": "a", "label": "8%" }, { "id": "b", "label": "16%" }, { "id": "c", "label": "18%" }, { "id": "d", "label": "20%" } ] }, { "id": "numerical-reasoning-s03-q05", "prompt": "A restaurant meal costs £40, and a 12.5% service charge is added. What is the total bill?", "choices": [ { "id": "a", "label": "£42" }, { "id": "b", "label": "£44" }, { "id": "c", "label": "£45" }, { "id": "d", "label": "£47.50" } ] }, { "id": "numerical-reasoning-q3", "prompt": "In a class of 30 students, 40% are boys. How many girls are in the class?", "choices": [ { "id": "a", "label": "12" }, { "id": "b", "label": "15" }, { "id": "c", "label": "18" }, { "id": "d", "label": "20" } ] }, { "id": "numerical-reasoning-s03-q07", "prompt": "30% of a number is 21. What is the number?", "choices": [ { "id": "a", "label": "70" }, { "id": "b", "label": "72" }, { "id": "c", "label": "75" }, { "id": "d", "label": "90" } ] }, { "id": "numerical-reasoning-s03-q08", "prompt": "A test is marked out of 25. A pupil scores 68%. How many marks did they get?", "choices": [ { "id": "a", "label": "14" }, { "id": "b", "label": "15" }, { "id": "c", "label": "16" }, { "id": "d", "label": "17" } ] }, { "id": "numerical-reasoning-s03-q09", "prompt": "A village's population falls from 2,500 to 2,300 over a decade. What is the percentage decrease?", "choices": [ { "id": "a", "label": "6%" }, { "id": "b", "label": "8%" }, { "id": "c", "label": "9%" }, { "id": "d", "label": "10%" } ] }, { "id": "numerical-reasoning-s03-q10", "prompt": "Which is greater: 25% of 80, or 30% of 70?", "choices": [ { "id": "a", "label": "30% of 70" }, { "id": "b", "label": "25% of 80" }, { "id": "c", "label": "They are equal" }, { "id": "d", "label": "It cannot be determined" } ] } ] }, { "id": "numerical-reasoning-s04", "title": "Ratios and proportion", "instructions": "Share amounts in a given ratio and scale quantities proportionally.", "items": [ { "id": "numerical-reasoning-s04-q01", "prompt": "£60 is shared between two people in the ratio 2 : 1. How much does the person with the larger share receive?", "choices": [ { "id": "a", "label": "£30" }, { "id": "b", "label": "£40" }, { "id": "c", "label": "£45" }, { "id": "d", "label": "£50" } ] }, { "id": "numerical-reasoning-s04-q02", "prompt": "A shelter houses 12 cats and 18 dogs. What is the ratio of cats to dogs in its simplest form?", "choices": [ { "id": "a", "label": "3 : 2" }, { "id": "b", "label": "6 : 9" }, { "id": "c", "label": "2 : 3" }, { "id": "d", "label": "4 : 6" } ] }, { "id": "numerical-reasoning-s04-q03", "prompt": "A map uses a scale of 1 : 50,000. Two towns are 4 cm apart on the map. What is the real distance between them?", "choices": [ { "id": "a", "label": "0.2 km" }, { "id": "b", "label": "0.5 km" }, { "id": "c", "label": "1 km" }, { "id": "d", "label": "2 km" } ] }, { "id": "numerical-reasoning-s04-q04", "prompt": "A squash drink is mixed from concentrate and water in the ratio 1 : 4. How much water is needed for 250 ml of concentrate?", "choices": [ { "id": "a", "label": "1,000 ml" }, { "id": "b", "label": "1,250 ml" }, { "id": "c", "label": "1,500 ml" }, { "id": "d", "label": "2,000 ml" } ] }, { "id": "numerical-reasoning-q4", "prompt": "A recipe uses a ratio of 2 cups flour to 3 cups sugar. If you use 9 cups of sugar, how many cups of flour do you need?", "choices": [ { "id": "a", "label": "4" }, { "id": "b", "label": "5" }, { "id": "c", "label": "6" }, { "id": "d", "label": "7" } ] }, { "id": "numerical-reasoning-s04-q06", "prompt": "45 sweets are shared between two children in the ratio 4 : 5. How many sweets does the child with the smaller share receive?", "choices": [ { "id": "a", "label": "18" }, { "id": "b", "label": "20" }, { "id": "c", "label": "22" }, { "id": "d", "label": "25" } ] }, { "id": "numerical-reasoning-s04-q07", "prompt": "In a car park, the ratio of red cars to silver cars is 3 : 7. If there are 21 red cars, how many silver cars are there?", "choices": [ { "id": "a", "label": "35" }, { "id": "b", "label": "42" }, { "id": "c", "label": "45" }, { "id": "d", "label": "49" } ] }, { "id": "numerical-reasoning-s04-q08", "prompt": "Three identical printers together produce 360 pages in an hour. At the same rate, how many pages would five such printers produce in an hour?", "choices": [ { "id": "a", "label": "480" }, { "id": "b", "label": "540" }, { "id": "c", "label": "600" }, { "id": "d", "label": "720" } ] }, { "id": "numerical-reasoning-s04-q09", "prompt": "A biscuit mix uses flour, butter, and sugar in the ratio 3 : 2 : 1. If the mix weighs 480 g in total, how much butter does it contain?", "choices": [ { "id": "a", "label": "80 g" }, { "id": "b", "label": "160 g" }, { "id": "c", "label": "240 g" }, { "id": "d", "label": "320 g" } ] }, { "id": "numerical-reasoning-s04-q10", "prompt": "8 identical pens cost £6. How much would 12 of the same pens cost?", "choices": [ { "id": "a", "label": "£9" }, { "id": "b", "label": "£10" }, { "id": "c", "label": "£10.50" }, { "id": "d", "label": "£12" } ] } ] }, { "id": "numerical-reasoning-s05", "title": "Rates, time, and conversions", "instructions": "Work with speed, unit rates, clock arithmetic, and unit conversions.", "items": [ { "id": "numerical-reasoning-q2", "prompt": "A car travels 180 km in 3 hours at a constant speed. How far will it travel in 5 hours at the same speed?", "choices": [ { "id": "a", "label": "240 km" }, { "id": "b", "label": "270 km" }, { "id": "c", "label": "300 km" }, { "id": "d", "label": "320 km" } ] }, { "id": "numerical-reasoning-s05-q02", "prompt": "Convert 3,200 metres into kilometres.", "choices": [ { "id": "a", "label": "3.2 km" }, { "id": "b", "label": "32 km" }, { "id": "c", "label": "320 km" }, { "id": "d", "label": "3,200 km" } ] }, { "id": "numerical-reasoning-s05-q03", "prompt": "A cyclist covers 45 km in 2.5 hours. What is her average speed?", "choices": [ { "id": "a", "label": "12 km/h" }, { "id": "b", "label": "15 km/h" }, { "id": "c", "label": "16 km/h" }, { "id": "d", "label": "18 km/h" } ] }, { "id": "numerical-reasoning-s05-q04", "prompt": "A coach departs at 09:40 and the journey takes 1 hour 35 minutes. At what time does it arrive?", "choices": [ { "id": "a", "label": "11:05" }, { "id": "b", "label": "11:15" }, { "id": "c", "label": "11:25" }, { "id": "d", "label": "11:35" } ] }, { "id": "numerical-reasoning-s05-q05", "prompt": "A tap fills a paddling pool at 4 litres per minute. How long will it take to fill a 90-litre pool?", "choices": [ { "id": "a", "label": "20 minutes" }, { "id": "b", "label": "21.5 minutes" }, { "id": "c", "label": "22.5 minutes" }, { "id": "d", "label": "24 minutes" } ] }, { "id": "numerical-reasoning-s05-q06", "prompt": "Walking at a steady 5 km/h, how long does a 12 km walk take?", "choices": [ { "id": "a", "label": "2 hours 24 minutes" }, { "id": "b", "label": "2 hours 30 minutes" }, { "id": "c", "label": "2 hours 36 minutes" }, { "id": "d", "label": "2 hours 48 minutes" } ] }, { "id": "numerical-reasoning-s05-q07", "prompt": "The exchange rate is £1 = €1.15. How many euros do you receive for £60?", "choices": [ { "id": "a", "label": "€63" }, { "id": "b", "label": "€65" }, { "id": "c", "label": "€66" }, { "id": "d", "label": "€69" } ] }, { "id": "numerical-reasoning-s05-q08", "prompt": "Express 45 minutes as a decimal fraction of an hour.", "choices": [ { "id": "a", "label": "0.45" }, { "id": "b", "label": "0.6" }, { "id": "c", "label": "0.75" }, { "id": "d", "label": "0.8" } ] }, { "id": "numerical-reasoning-s05-q09", "prompt": "A bottling machine fills 240 bottles in 20 minutes. At the same rate, how many bottles does it fill per hour?", "choices": [ { "id": "a", "label": "480" }, { "id": "b", "label": "720" }, { "id": "c", "label": "960" }, { "id": "d", "label": "1,200" } ] }, { "id": "numerical-reasoning-s05-q10", "prompt": "A lorry travels at a steady 64 km/h for 2 hours 15 minutes. How far does it travel?", "choices": [ { "id": "a", "label": "144 km" }, { "id": "b", "label": "148 km" }, { "id": "c", "label": "152 km" }, { "id": "d", "label": "160 km" } ] } ] }, { "id": "numerical-reasoning-s06", "title": "Money and financial arithmetic", "instructions": "Handle change, wages, unit pricing, interest, and simple profit.", "items": [ { "id": "numerical-reasoning-s06-q01", "prompt": "You buy items costing £7.35 and £4.20, and pay with a £20 note. How much change should you receive?", "choices": [ { "id": "a", "label": "£8.35" }, { "id": "b", "label": "£8.45" }, { "id": "c", "label": "£8.55" }, { "id": "d", "label": "£9.45" } ] }, { "id": "numerical-reasoning-s06-q02", "prompt": "A 500 g pack of rice costs £2.40 and a 750 g pack of the same rice costs £3.30. Which is the better value for money?", "choices": [ { "id": "a", "label": "The 750 g pack" }, { "id": "b", "label": "The 500 g pack" }, { "id": "c", "label": "Both offer identical value" }, { "id": "d", "label": "It cannot be determined" } ] }, { "id": "numerical-reasoning-s06-q03", "prompt": "£800 is saved in an account paying 5% simple interest per year. How much interest is earned over 3 years?", "choices": [ { "id": "a", "label": "£40" }, { "id": "b", "label": "£80" }, { "id": "c", "label": "£100" }, { "id": "d", "label": "£120" } ] }, { "id": "numerical-reasoning-s06-q04", "prompt": "A worker earns £2,400 per month and spends 35% of it on rent. How much is the rent?", "choices": [ { "id": "a", "label": "£760" }, { "id": "b", "label": "£800" }, { "id": "c", "label": "£840" }, { "id": "d", "label": "£960" } ] }, { "id": "numerical-reasoning-q5", "prompt": "A $500 investment grows by 10% in year one, then by 10% of the new amount in year two. What is the value after two years?", "choices": [ { "id": "a", "label": "$590" }, { "id": "b", "label": "$600" }, { "id": "c", "label": "$605" }, { "id": "d", "label": "$610" } ] }, { "id": "numerical-reasoning-s06-q06", "prompt": "A restaurant bill of £86.40 is split equally among 6 diners. How much does each pay?", "choices": [ { "id": "a", "label": "£13.40" }, { "id": "b", "label": "£14.40" }, { "id": "c", "label": "£14.60" }, { "id": "d", "label": "£15.40" } ] }, { "id": "numerical-reasoning-s06-q07", "prompt": "A washing machine costs £640. A buyer pays a 15% deposit. How much is left to pay afterwards?", "choices": [ { "id": "a", "label": "£544" }, { "id": "b", "label": "£560" }, { "id": "c", "label": "£576" }, { "id": "d", "label": "£600" } ] }, { "id": "numerical-reasoning-s06-q08", "prompt": "An employee works 38 hours in a week at £12.50 per hour. What is the weekly wage?", "choices": [ { "id": "a", "label": "£455" }, { "id": "b", "label": "£462.50" }, { "id": "c", "label": "£470" }, { "id": "d", "label": "£475" } ] }, { "id": "numerical-reasoning-s06-q09", "prompt": "A shop runs a 'buy three, get one free' offer on drinks costing £2.50 each. How much do you pay if you take 8 drinks?", "choices": [ { "id": "a", "label": "£12.50" }, { "id": "b", "label": "£15" }, { "id": "c", "label": "£17.50" }, { "id": "d", "label": "£20" } ] }, { "id": "numerical-reasoning-s06-q10", "prompt": "A trader buys 40 plants at £1.20 each and sells them all at £2 each. What is the total profit?", "choices": [ { "id": "a", "label": "£32" }, { "id": "b", "label": "£38" }, { "id": "c", "label": "£40" }, { "id": "d", "label": "£48" } ] } ] }, { "id": "numerical-reasoning-s07", "title": "Averages and simple data", "instructions": "Calculate and interpret means, medians, modes, ranges, and weighted averages.", "items": [ { "id": "numerical-reasoning-s07-q01", "prompt": "What is the mean of 4, 8, 9, and 11?", "choices": [ { "id": "a", "label": "6" }, { "id": "b", "label": "7" }, { "id": "c", "label": "8" }, { "id": "d", "label": "9" } ] }, { "id": "numerical-reasoning-s07-q02", "prompt": "What is the median of 3, 7, 8, 12, and 15?", "choices": [ { "id": "a", "label": "7" }, { "id": "b", "label": "8" }, { "id": "c", "label": "9" }, { "id": "d", "label": "12" } ] }, { "id": "numerical-reasoning-s07-q03", "prompt": "What is the mode of the data set 2, 4, 4, 5, 7, 4, 9?", "choices": [ { "id": "a", "label": "4" }, { "id": "b", "label": "5" }, { "id": "c", "label": "7" }, { "id": "d", "label": "9" } ] }, { "id": "numerical-reasoning-s07-q04", "prompt": "What is the range of the data set 12, 5, 19, 8, 23?", "choices": [ { "id": "a", "label": "11" }, { "id": "b", "label": "15" }, { "id": "c", "label": "18" }, { "id": "d", "label": "23" } ] }, { "id": "numerical-reasoning-s07-q05", "prompt": "The mean of five test scores is 14. Four of the scores are 12, 15, 10, and 18. What is the fifth score?", "choices": [ { "id": "a", "label": "11" }, { "id": "b", "label": "12" }, { "id": "c", "label": "14" }, { "id": "d", "label": "15" } ] }, { "id": "numerical-reasoning-s07-q06", "prompt": "Class A has 20 pupils with a mean score of 60. Class B has 10 pupils with a mean score of 75. What is the mean score across both classes combined?", "choices": [ { "id": "a", "label": "65" }, { "id": "b", "label": "66" }, { "id": "c", "label": "67.5" }, { "id": "d", "label": "70" } ] }, { "id": "numerical-reasoning-s07-q07", "prompt": "A van drives 120 km in the first 2 hours, then 60 km in the next hour. What is its average speed over the whole journey?", "choices": [ { "id": "a", "label": "55 km/h" }, { "id": "b", "label": "60 km/h" }, { "id": "c", "label": "65 km/h" }, { "id": "d", "label": "67.5 km/h" } ] }, { "id": "numerical-reasoning-s07-q08", "prompt": "A stall's sales were 40 cakes on Monday, 55 on Tuesday, and 25 on Wednesday. What was the mean daily sale?", "choices": [ { "id": "a", "label": "35" }, { "id": "b", "label": "38" }, { "id": "c", "label": "40" }, { "id": "d", "label": "45" } ] }, { "id": "numerical-reasoning-s07-q09", "prompt": "The mean of six numbers is 10. A seventh number, 24, is added to the set. What is the new mean?", "choices": [ { "id": "a", "label": "10.5" }, { "id": "b", "label": "11" }, { "id": "c", "label": "11.5" }, { "id": "d", "label": "12" } ] }, { "id": "numerical-reasoning-s07-q10", "prompt": "A quiz team of five players has a mean score of 50 points per player. What is the highest score achieved by any single player?", "choices": [ { "id": "a", "label": "It cannot be determined from the information given" }, { "id": "b", "label": "Exactly 50" }, { "id": "c", "label": "Exactly 250" }, { "id": "d", "label": "At least 60" } ] } ] }, { "id": "numerical-reasoning-s08", "title": "Number sequences and relationships", "instructions": "Find the rule behind each sequence, then apply it to the missing term.", "items": [ { "id": "numerical-reasoning-s08-q01", "prompt": "What is the next number in the sequence 5, 9, 13, 17, ...?", "choices": [ { "id": "a", "label": "18" }, { "id": "b", "label": "19" }, { "id": "c", "label": "20" }, { "id": "d", "label": "21" } ] }, { "id": "numerical-reasoning-s08-q02", "prompt": "What is the next number in the sequence 2, 6, 18, 54, ...?", "choices": [ { "id": "a", "label": "108" }, { "id": "b", "label": "148" }, { "id": "c", "label": "162" }, { "id": "d", "label": "216" } ] }, { "id": "numerical-reasoning-s08-q03", "prompt": "What is the next number in the sequence 64, 32, 16, 8, ...?", "choices": [ { "id": "a", "label": "2" }, { "id": "b", "label": "4" }, { "id": "c", "label": "5" }, { "id": "d", "label": "6" } ] }, { "id": "numerical-reasoning-s08-q04", "prompt": "What is the next number in the sequence 1, 4, 9, 16, 25, ...?", "choices": [ { "id": "a", "label": "36" }, { "id": "b", "label": "42" }, { "id": "c", "label": "49" }, { "id": "d", "label": "50" } ] }, { "id": "numerical-reasoning-s08-q05", "prompt": "What is the next number in the sequence 3, 4, 7, 11, 18, ...?", "choices": [ { "id": "a", "label": "29" }, { "id": "b", "label": "30" }, { "id": "c", "label": "32" }, { "id": "d", "label": "36" } ] }, { "id": "numerical-reasoning-s08-q06", "prompt": "What is the next number in the sequence 40, 37, 33, 28, 22, ...?", "choices": [ { "id": "a", "label": "14" }, { "id": "b", "label": "15" }, { "id": "c", "label": "16" }, { "id": "d", "label": "17" } ] }, { "id": "numerical-reasoning-s08-q07", "prompt": "What is the next number in the sequence 1, 10, 3, 8, 5, 6, 7, ...?", "choices": [ { "id": "a", "label": "4" }, { "id": "b", "label": "5" }, { "id": "c", "label": "6" }, { "id": "d", "label": "8" } ] }, { "id": "numerical-reasoning-s08-q08", "prompt": "What is the next number in the sequence 5, 6, 8, 11, 15, 20, ...?", "choices": [ { "id": "a", "label": "24" }, { "id": "b", "label": "25" }, { "id": "c", "label": "26" }, { "id": "d", "label": "27" } ] }, { "id": "numerical-reasoning-s08-q09", "prompt": "Which number does not belong with the others: 8, 27, 64, 100, 125?", "choices": [ { "id": "a", "label": "27" }, { "id": "b", "label": "64" }, { "id": "c", "label": "100" }, { "id": "d", "label": "125" } ] }, { "id": "numerical-reasoning-s08-q10", "prompt": "A sequence follows the rule 'double the previous term, then subtract 1': 3, 5, 9, 17, ... What is the next term?", "choices": [ { "id": "a", "label": "31" }, { "id": "b", "label": "33" }, { "id": "c", "label": "34" }, { "id": "d", "label": "35" } ] } ] }, { "id": "numerical-reasoning-s09", "title": "Multi-step word problems", "instructions": "Combine two or more operations: plan the steps before you calculate.", "items": [ { "id": "numerical-reasoning-s09-q01", "prompt": "A hall has 18 rows of 22 seats. If 315 tickets are sold for a concert, how many seats remain empty?", "choices": [ { "id": "a", "label": "71" }, { "id": "b", "label": "79" }, { "id": "c", "label": "81" }, { "id": "d", "label": "91" } ] }, { "id": "numerical-reasoning-s09-q02", "prompt": "A plumber charges a £40 call-out fee plus £15 per hour of work. If a bill comes to £145, how many hours did the job take?", "choices": [ { "id": "a", "label": "6" }, { "id": "b", "label": "7" }, { "id": "c", "label": "8" }, { "id": "d", "label": "9" } ] }, { "id": "numerical-reasoning-s09-q03", "prompt": "A car uses 6 litres of fuel per 100 km, and fuel costs £1.50 per litre. How much does the fuel for a 450 km trip cost?", "choices": [ { "id": "a", "label": "£36.00" }, { "id": "b", "label": "£37.50" }, { "id": "c", "label": "£40.00" }, { "id": "d", "label": "£40.50" } ] }, { "id": "numerical-reasoning-s09-q04", "prompt": "A jumper priced at £45 is reduced by 20% in a sale, and a £5 voucher is then taken off the sale price. What do you pay?", "choices": [ { "id": "a", "label": "£31" }, { "id": "b", "label": "£32" }, { "id": "c", "label": "£35" }, { "id": "d", "label": "£36" } ] }, { "id": "numerical-reasoning-s09-q05", "prompt": "Amir is twice as old as Beth, and their ages add up to 36. How old is Amir?", "choices": [ { "id": "a", "label": "18" }, { "id": "b", "label": "20" }, { "id": "c", "label": "24" }, { "id": "d", "label": "27" } ] }, { "id": "numerical-reasoning-s09-q06", "prompt": "A number is multiplied by 4 and then 6 is added, giving 46. What was the original number?", "choices": [ { "id": "a", "label": "7" }, { "id": "b", "label": "8" }, { "id": "c", "label": "9" }, { "id": "d", "label": "10" } ] }, { "id": "numerical-reasoning-s09-q07", "prompt": "A warehouse holds 1,200 units. On Monday it ships out 15% of its stock, and on Tuesday it receives a delivery of 250 units. How many units does it hold after Tuesday?", "choices": [ { "id": "a", "label": "1,270" }, { "id": "b", "label": "1,330" }, { "id": "c", "label": "1,370" }, { "id": "d", "label": "1,450" } ] }, { "id": "numerical-reasoning-s09-q08", "prompt": "A floor measures 4 m by 3 m. Each tile covers 0.25 m², and tiles are sold in boxes of 10. How many boxes are needed to cover the floor exactly?", "choices": [ { "id": "a", "label": "4" }, { "id": "b", "label": "5" }, { "id": "c", "label": "6" }, { "id": "d", "label": "48" } ] }, { "id": "numerical-reasoning-s09-q09", "prompt": "Adult tickets cost £9 and child tickets cost £5. A family buys 2 adult and 3 child tickets, paying with a £50 note. How much change do they receive?", "choices": [ { "id": "a", "label": "£14" }, { "id": "b", "label": "£15" }, { "id": "c", "label": "£16" }, { "id": "d", "label": "£17" } ] }, { "id": "numerical-reasoning-s09-q10", "prompt": "A shop's June sales were 20% higher than its May sales. June sales were 660 units. How many units were sold in May?", "choices": [ { "id": "a", "label": "550" }, { "id": "b", "label": "572" }, { "id": "c", "label": "605" }, { "id": "d", "label": "792" } ] } ] }, { "id": "numerical-reasoning-s10", "title": "Compound change and combined rates", "instructions": "The toughest set: repeated percentage change, reverse percentages, and rates working together.", "items": [ { "id": "numerical-reasoning-s10-q01", "prompt": "A machine worth £2,000 loses 10% of its value each year. What is it worth after two years?", "choices": [ { "id": "a", "label": "£1,600" }, { "id": "b", "label": "£1,620" }, { "id": "c", "label": "£1,640" }, { "id": "d", "label": "£1,800" } ] }, { "id": "numerical-reasoning-s10-q02", "prompt": "A coat priced at £150 is reduced by 20%, and the sale price is later reduced by a further 10%. What is the final price?", "choices": [ { "id": "a", "label": "£102" }, { "id": "b", "label": "£104" }, { "id": "c", "label": "£105" }, { "id": "d", "label": "£108" } ] }, { "id": "numerical-reasoning-s10-q03", "prompt": "After a 20% price increase, a ticket costs £96. What did it cost before the increase?", "choices": [ { "id": "a", "label": "£80" }, { "id": "b", "label": "£84" }, { "id": "c", "label": "£88" }, { "id": "d", "label": "£92" } ] }, { "id": "numerical-reasoning-s10-q04", "prompt": "Working alone, machine A completes a job in 3 hours. Working together, machines A and B complete the same job in 2 hours. How long would machine B take alone?", "choices": [ { "id": "a", "label": "4 hours" }, { "id": "b", "label": "5 hours" }, { "id": "c", "label": "6 hours" }, { "id": "d", "label": "12 hours" } ] }, { "id": "numerical-reasoning-s10-q05", "prompt": "A driver travels from town P to town Q at 60 km/h and returns along the same road at 40 km/h. What is the average speed for the whole round trip?", "choices": [ { "id": "a", "label": "45 km/h" }, { "id": "b", "label": "48 km/h" }, { "id": "c", "label": "50 km/h" }, { "id": "d", "label": "52 km/h" } ] }, { "id": "numerical-reasoning-s10-q06", "prompt": "In a company, 30% of the staff work part-time, and 40% of those part-time staff work at weekends. What percentage of all staff are part-time weekend workers?", "choices": [ { "id": "a", "label": "8%" }, { "id": "b", "label": "10%" }, { "id": "c", "label": "11%" }, { "id": "d", "label": "12%" } ] }, { "id": "numerical-reasoning-s10-q07", "prompt": "Without calculating exactly, which is the best estimate of 41,872 ÷ 68.9?", "choices": [ { "id": "a", "label": "About 600" }, { "id": "b", "label": "About 6,000" }, { "id": "c", "label": "About 60,000" }, { "id": "d", "label": "About 600,000" } ] }, { "id": "numerical-reasoning-q6", "prompt": "Pipe A fills a tank alone in 4 hours; pipe B fills it alone in 6 hours. Working together, about how many hours will it take?", "choices": [ { "id": "a", "label": "2 hours" }, { "id": "b", "label": "2.4 hours" }, { "id": "c", "label": "3 hours" }, { "id": "d", "label": "5 hours" } ] }, { "id": "numerical-reasoning-s10-q09", "prompt": "A bag contains red and blue counters in the ratio 3 : 2, and holds 30 counters in total. If 6 more blue counters are added, what is the new ratio of red to blue?", "choices": [ { "id": "a", "label": "3 : 2" }, { "id": "b", "label": "6 : 5" }, { "id": "c", "label": "1 : 1" }, { "id": "d", "label": "2 : 3" } ] }, { "id": "numerical-reasoning-s10-q10", "prompt": "An investment rises in value by 25% in its first year and then falls by 20% in its second year. What is the overall effect on its value?", "choices": [ { "id": "a", "label": "A 5% overall increase" }, { "id": "b", "label": "No overall change" }, { "id": "c", "label": "A 5% overall decrease" }, { "id": "d", "label": "A 10% overall increase" } ] } ] } ] }, "authoredHints": { "title": "Authored hints", "policy": "This cognitive practice source does not publish authored hints. 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"numerical-reasoning-s02-q08", "correctChoiceId": "b", "correctChoiceLabel": "18" }, { "itemId": "numerical-reasoning-s02-q09", "correctChoiceId": "a", "correctChoiceLabel": "0.12" }, { "itemId": "numerical-reasoning-s02-q10", "correctChoiceId": "c", "correctChoiceLabel": "2 1/4" }, { "itemId": "numerical-reasoning-q1", "correctChoiceId": "b", "correctChoiceLabel": "$60" }, { "itemId": "numerical-reasoning-s03-q02", "correctChoiceId": "d", "correctChoiceLabel": "9" }, { "itemId": "numerical-reasoning-s03-q03", "correctChoiceId": "a", "correctChoiceLabel": "75%" }, { "itemId": "numerical-reasoning-s03-q04", "correctChoiceId": "b", "correctChoiceLabel": "16%" }, { "itemId": "numerical-reasoning-s03-q05", "correctChoiceId": "c", "correctChoiceLabel": "£45" }, { "itemId": "numerical-reasoning-q3", "correctChoiceId": "c", "correctChoiceLabel": "18" }, { "itemId": "numerical-reasoning-s03-q07", "correctChoiceId": "a", "correctChoiceLabel": "70" }, { "itemId": "numerical-reasoning-s03-q08", "correctChoiceId": "d", "correctChoiceLabel": "17" }, { "itemId": "numerical-reasoning-s03-q09", "correctChoiceId": "b", "correctChoiceLabel": "8%" }, { "itemId": "numerical-reasoning-s03-q10", "correctChoiceId": "a", "correctChoiceLabel": "30% of 70" }, { "itemId": "numerical-reasoning-s04-q01", "correctChoiceId": "b", "correctChoiceLabel": "£40" }, { "itemId": "numerical-reasoning-s04-q02", "correctChoiceId": "c", "correctChoiceLabel": "2 : 3" }, { "itemId": "numerical-reasoning-s04-q03", "correctChoiceId": "d", "correctChoiceLabel": "2 km" }, { "itemId": "numerical-reasoning-s04-q04", "correctChoiceId": "a", "correctChoiceLabel": "1,000 ml" }, { "itemId": "numerical-reasoning-q4", "correctChoiceId": "c", "correctChoiceLabel": "6" }, { "itemId": "numerical-reasoning-s04-q06", "correctChoiceId": "b", "correctChoiceLabel": "20" }, { "itemId": "numerical-reasoning-s04-q07", "correctChoiceId": "d", "correctChoiceLabel": "49" }, { "itemId": "numerical-reasoning-s04-q08", "correctChoiceId": "c", "correctChoiceLabel": "600" }, { "itemId": "numerical-reasoning-s04-q09", "correctChoiceId": "b", "correctChoiceLabel": "160 g" }, { "itemId": "numerical-reasoning-s04-q10", "correctChoiceId": "a", "correctChoiceLabel": "£9" }, { "itemId": "numerical-reasoning-q2", "correctChoiceId": "c", "correctChoiceLabel": "300 km" }, { "itemId": "numerical-reasoning-s05-q02", "correctChoiceId": "a", "correctChoiceLabel": "3.2 km" }, { "itemId": "numerical-reasoning-s05-q03", "correctChoiceId": "d", "correctChoiceLabel": "18 km/h" }, { "itemId": "numerical-reasoning-s05-q04", "correctChoiceId": "b", "correctChoiceLabel": "11:15" }, { "itemId": "numerical-reasoning-s05-q05", "correctChoiceId": "c", "correctChoiceLabel": "22.5 minutes" }, { "itemId": "numerical-reasoning-s05-q06", "correctChoiceId": "a", "correctChoiceLabel": "2 hours 24 minutes" }, { "itemId": "numerical-reasoning-s05-q07", "correctChoiceId": "d", "correctChoiceLabel": "€69" }, { "itemId": "numerical-reasoning-s05-q08", "correctChoiceId": "c", "correctChoiceLabel": "0.75" }, { "itemId": "numerical-reasoning-s05-q09", "correctChoiceId": "b", "correctChoiceLabel": "720" }, { "itemId": "numerical-reasoning-s05-q10", "correctChoiceId": "a", "correctChoiceLabel": "144 km" }, { "itemId": "numerical-reasoning-s06-q01", "correctChoiceId": "b", "correctChoiceLabel": "£8.45" }, { "itemId": "numerical-reasoning-s06-q02", "correctChoiceId": "a", "correctChoiceLabel": "The 750 g pack" }, { "itemId": "numerical-reasoning-s06-q03", "correctChoiceId": "d", "correctChoiceLabel": "£120" }, { "itemId": "numerical-reasoning-s06-q04", "correctChoiceId": "c", "correctChoiceLabel": "£840" }, { "itemId": "numerical-reasoning-q5", "correctChoiceId": "c", "correctChoiceLabel": "$605" }, { "itemId": "numerical-reasoning-s06-q06", "correctChoiceId": "b", "correctChoiceLabel": "£14.40" }, { "itemId": "numerical-reasoning-s06-q07", "correctChoiceId": "a", "correctChoiceLabel": "£544" }, { "itemId": "numerical-reasoning-s06-q08", "correctChoiceId": "d", "correctChoiceLabel": "£475" }, { "itemId": "numerical-reasoning-s06-q09", "correctChoiceId": "b", "correctChoiceLabel": "£15" }, { "itemId": "numerical-reasoning-s06-q10", "correctChoiceId": "a", "correctChoiceLabel": "£32" }, { "itemId": "numerical-reasoning-s07-q01", "correctChoiceId": "c", "correctChoiceLabel": "8" }, { "itemId": "numerical-reasoning-s07-q02", "correctChoiceId": "b", "correctChoiceLabel": "8" }, { "itemId": "numerical-reasoning-s07-q03", "correctChoiceId": "a", "correctChoiceLabel": "4" }, { "itemId": "numerical-reasoning-s07-q04", "correctChoiceId": "c", "correctChoiceLabel": "18" }, { "itemId": "numerical-reasoning-s07-q05", "correctChoiceId": "d", "correctChoiceLabel": "15" }, { "itemId": "numerical-reasoning-s07-q06", "correctChoiceId": "a", "correctChoiceLabel": "65" }, { "itemId": "numerical-reasoning-s07-q07", "correctChoiceId": "b", "correctChoiceLabel": "60 km/h" }, { "itemId": "numerical-reasoning-s07-q08", "correctChoiceId": "c", "correctChoiceLabel": "40" }, { "itemId": "numerical-reasoning-s07-q09", "correctChoiceId": "d", "correctChoiceLabel": "12" }, { "itemId": "numerical-reasoning-s07-q10", "correctChoiceId": "a", "correctChoiceLabel": "It cannot be determined from the information given" }, { "itemId": "numerical-reasoning-s08-q01", "correctChoiceId": "d", "correctChoiceLabel": "21" }, { "itemId": "numerical-reasoning-s08-q02", "correctChoiceId": "c", "correctChoiceLabel": "162" }, { "itemId": "numerical-reasoning-s08-q03", "correctChoiceId": "b", "correctChoiceLabel": "4" }, { "itemId": "numerical-reasoning-s08-q04", "correctChoiceId": "a", "correctChoiceLabel": "36" }, { "itemId": "numerical-reasoning-s08-q05", "correctChoiceId": "a", "correctChoiceLabel": "29" }, { "itemId": "numerical-reasoning-s08-q06", "correctChoiceId": "b", "correctChoiceLabel": "15" }, { "itemId": "numerical-reasoning-s08-q07", "correctChoiceId": "a", "correctChoiceLabel": "4" }, { "itemId": "numerical-reasoning-s08-q08", "correctChoiceId": "c", "correctChoiceLabel": "26" }, { "itemId": "numerical-reasoning-s08-q09", "correctChoiceId": "c", "correctChoiceLabel": "100" }, { "itemId": "numerical-reasoning-s08-q10", "correctChoiceId": "b", "correctChoiceLabel": "33" }, { "itemId": "numerical-reasoning-s09-q01", "correctChoiceId": "c", "correctChoiceLabel": "81" }, { "itemId": "numerical-reasoning-s09-q02", "correctChoiceId": "b", "correctChoiceLabel": "7" }, { "itemId": "numerical-reasoning-s09-q03", "correctChoiceId": "d", "correctChoiceLabel": "£40.50" }, { "itemId": "numerical-reasoning-s09-q04", "correctChoiceId": "a", "correctChoiceLabel": "£31" }, { "itemId": "numerical-reasoning-s09-q05", "correctChoiceId": "c", "correctChoiceLabel": "24" }, { "itemId": "numerical-reasoning-s09-q06", "correctChoiceId": "d", "correctChoiceLabel": "10" }, { "itemId": "numerical-reasoning-s09-q07", "correctChoiceId": "a", "correctChoiceLabel": "1,270" }, { "itemId": "numerical-reasoning-s09-q08", "correctChoiceId": "b", "correctChoiceLabel": "5" }, { "itemId": "numerical-reasoning-s09-q09", "correctChoiceId": "d", "correctChoiceLabel": "£17" }, { "itemId": "numerical-reasoning-s09-q10", "correctChoiceId": "a", "correctChoiceLabel": "550" }, { "itemId": "numerical-reasoning-s10-q01", "correctChoiceId": "b", "correctChoiceLabel": "£1,620" }, { "itemId": "numerical-reasoning-s10-q02", "correctChoiceId": "d", "correctChoiceLabel": "£108" }, { "itemId": "numerical-reasoning-s10-q03", "correctChoiceId": "a", "correctChoiceLabel": "£80" }, { "itemId": "numerical-reasoning-s10-q04", "correctChoiceId": "c", "correctChoiceLabel": "6 hours" }, { "itemId": "numerical-reasoning-s10-q05", "correctChoiceId": "b", "correctChoiceLabel": "48 km/h" }, { "itemId": "numerical-reasoning-s10-q06", "correctChoiceId": "d", "correctChoiceLabel": "12%" }, { "itemId": "numerical-reasoning-s10-q07", "correctChoiceId": "a", "correctChoiceLabel": "About 600" }, { "itemId": "numerical-reasoning-q6", "correctChoiceId": "b", "correctChoiceLabel": "2.4 hours" }, { "itemId": "numerical-reasoning-s10-q09", "correctChoiceId": "c", "correctChoiceLabel": "1 : 1" }, { "itemId": "numerical-reasoning-s10-q10", "correctChoiceId": "b", "correctChoiceLabel": "No overall change" } ] }, "workedExplanations": { "title": "Worked explanations", "items": [ { "itemId": "numerical-reasoning-s01-q01", "explanation": "Add the two shelf counts directly: 24 + 38 = 62." }, { "itemId": "numerical-reasoning-s01-q02", "explanation": "Equal sharing is division: 96 ÷ 8 = 12 pencils each." }, { "itemId": "numerical-reasoning-s01-q03", "explanation": "Repeated equal amounts call for multiplication: 7 × £9 = £63." }, { "itemId": "numerical-reasoning-s01-q04", "explanation": "Subtract the drained amount from the full tank: 500 − 137 = 363 litres." }, { "itemId": "numerical-reasoning-s01-q05", "explanation": "Division is carried out before subtraction: 12 ÷ 3 = 4, so 18 − 4 = 14. Working left to right first would wrongly give 2." }, { "itemId": "numerical-reasoning-s01-q06", "explanation": "Multiply carriages by seats per carriage: 6 × 72 = 432." }, { "itemId": "numerical-reasoning-s01-q07", "explanation": "Multiply the number of boxes by apples per box: 12 × 15 = 180." }, { "itemId": "numerical-reasoning-s01-q08", "explanation": "9 × 8 = 72 is the largest multiple of 9 that fits within 75, leaving a remainder of 75 − 72 = 3." }, { "itemId": "numerical-reasoning-s01-q09", "explanation": "Multiplication comes before subtraction: 7 × 8 = 56, then 56 − 6 = 50." }, { "itemId": "numerical-reasoning-s01-q10", "explanation": "From 9:00 am to 5:00 pm is 8 hours, plus the extra 30 minutes makes 8.5 hours." }, { "itemId": "numerical-reasoning-s02-q01", "explanation": "Convert to a common denominator first: 1/2 = 2/4, so 2/4 + 1/4 = 3/4. Adding tops and bottoms separately is the classic trap." }, { "itemId": "numerical-reasoning-s02-q02", "explanation": "Find one fifth first: 40 ÷ 5 = 8, then multiply by three: 3 × 8 = 24." }, { "itemId": "numerical-reasoning-s02-q03", "explanation": "0.25 is one quarter, so multiplying by it is dividing by 4: 80 ÷ 4 = 20." }, { "itemId": "numerical-reasoning-s02-q04", "explanation": "Converting to decimals makes the comparison easy: 3/5 = 0.6, 7/12 ≈ 0.583, 2/3 ≈ 0.667, and 5/8 = 0.625, so 2/3 is largest." }, { "itemId": "numerical-reasoning-s02-q05", "explanation": "Line up the decimal places by writing 7.2 as 7.20, then subtract: 7.20 − 3.85 = 3.35." }, { "itemId": "numerical-reasoning-s02-q06", "explanation": "Divide the total length by the piece length: 3.6 ÷ 0.4 = 36 ÷ 4 = 9 pieces." }, { "itemId": "numerical-reasoning-s02-q07", "explanation": "Divide top and bottom by their highest common factor, 6: 18 ÷ 6 = 3 and 24 ÷ 6 = 4, giving 3/4. Equivalent fractions like 6/8 are not fully simplified." }, { "itemId": "numerical-reasoning-s02-q08", "explanation": "One third of 27 is 9, so two thirds is 2 × 9 = 18 pupils." }, { "itemId": "numerical-reasoning-s02-q09", "explanation": "Multiply as whole numbers (3 × 4 = 12), then place two decimal digits because the factors have one each: 0.12." }, { "itemId": "numerical-reasoning-s02-q10", "explanation": "Multiply the fraction by three: 3 × 3/4 = 9/4, which is 2 1/4 cups." }, { "itemId": "numerical-reasoning-q1", "explanation": "25% of $80 is $20, so the sale price is $80 − $20 = $60." }, { "itemId": "numerical-reasoning-s03-q02", "explanation": "10% of 60 is 6 and 5% is 3, so 15% is 6 + 3 = 9." }, { "itemId": "numerical-reasoning-s03-q03", "explanation": "Divide the score by the total and multiply by 100: 45 ÷ 60 = 0.75, which is 75%." }, { "itemId": "numerical-reasoning-s03-q04", "explanation": "Percentage change is the change divided by the original amount: 8 ÷ 50 = 0.16, so a 16% increase." }, { "itemId": "numerical-reasoning-s03-q05", "explanation": "12.5% is one eighth, so the charge is £40 ÷ 8 = £5, giving £40 + £5 = £45." }, { "itemId": "numerical-reasoning-q3", "explanation": "40% of 30 is 12 boys, so the remaining 30 − 12 = 18 students are girls." }, { "itemId": "numerical-reasoning-s03-q07", "explanation": "If 30% is 21, then 10% is 21 ÷ 3 = 7, so the whole number (100%) is 7 × 10 = 70." }, { "itemId": "numerical-reasoning-s03-q08", "explanation": "Convert the percentage to a decimal and multiply: 0.68 × 25 = 17 marks." }, { "itemId": "numerical-reasoning-s03-q09", "explanation": "The fall is 200 people; divide by the original population: 200 ÷ 2,500 = 0.08, an 8% decrease." }, { "itemId": "numerical-reasoning-s03-q10", "explanation": "Work both out: 25% of 80 is 20, while 30% of 70 is 21, so 30% of 70 is slightly greater." }, { "itemId": "numerical-reasoning-s04-q01", "explanation": "The ratio 2 : 1 has 3 equal parts, each worth £60 ÷ 3 = £20; the larger share is 2 × £20 = £40." }, { "itemId": "numerical-reasoning-s04-q02", "explanation": "Divide both sides by their highest common factor, 6: 12 : 18 becomes 2 : 3. Keep the order as asked: cats first." }, { "itemId": "numerical-reasoning-s04-q03", "explanation": "Multiply the map distance by the scale: 4 cm × 50,000 = 200,000 cm, and converting units gives 2,000 m = 2 km." }, { "itemId": "numerical-reasoning-s04-q04", "explanation": "Every 1 part of concentrate needs 4 parts of water, so 250 ml needs 4 × 250 = 1,000 ml of water." }, { "itemId": "numerical-reasoning-q4", "explanation": "9 cups of sugar is 3× the ratio's 3 cups, so flour scales the same way: 2 × 3 = 6 cups." }, { "itemId": "numerical-reasoning-s04-q06", "explanation": "The ratio 4 : 5 has 9 parts, each worth 45 ÷ 9 = 5 sweets; the smaller share is 4 × 5 = 20." }, { "itemId": "numerical-reasoning-s04-q07", "explanation": "21 red cars is 7 times the ratio's 3, so scale the silver side the same way: 7 × 7 = 49." }, { "itemId": "numerical-reasoning-s04-q08", "explanation": "Each printer produces 360 ÷ 3 = 120 pages per hour, so five produce 5 × 120 = 600." }, { "itemId": "numerical-reasoning-s04-q09", "explanation": "The ratio has 3 + 2 + 1 = 6 parts, each worth 480 ÷ 6 = 80 g; butter is 2 parts, so 160 g." }, { "itemId": "numerical-reasoning-s04-q10", "explanation": "One pen costs £6 ÷ 8 = £0.75, so 12 pens cost 12 × £0.75 = £9." }, { "itemId": "numerical-reasoning-q2", "explanation": "180 km ÷ 3 h = 60 km/h. At 60 km/h for 5 hours: 60 × 5 = 300 km." }, { "itemId": "numerical-reasoning-s05-q02", "explanation": "There are 1,000 metres in a kilometre, so divide by 1,000: 3,200 m = 3.2 km." }, { "itemId": "numerical-reasoning-s05-q03", "explanation": "Average speed is distance divided by time: 45 ÷ 2.5 = 18 km/h." }, { "itemId": "numerical-reasoning-s05-q04", "explanation": "Break the journey into easy steps: 09:40 plus 20 minutes reaches 10:00, and the remaining 1 hour 15 minutes reaches 11:15." }, { "itemId": "numerical-reasoning-s05-q05", "explanation": "Divide the volume by the rate: 90 ÷ 4 = 22.5 minutes." }, { "itemId": "numerical-reasoning-s05-q06", "explanation": "Time is distance divided by speed: 12 ÷ 5 = 2.4 hours, and 0.4 of an hour is 0.4 × 60 = 24 minutes." }, { "itemId": "numerical-reasoning-s05-q07", "explanation": "Multiply the pounds by the rate per pound: 60 × 1.15 = €69." }, { "itemId": "numerical-reasoning-s05-q08", "explanation": "Divide by the 60 minutes in an hour: 45 ÷ 60 = 0.75. Writing 45 minutes as 0.45 hours is the common slip." }, { "itemId": "numerical-reasoning-s05-q09", "explanation": "An hour contains three 20-minute periods, so the hourly total is 3 × 240 = 720 bottles." }, { "itemId": "numerical-reasoning-s05-q10", "explanation": "Convert the time to a decimal: 2 hours 15 minutes = 2.25 hours, then multiply: 64 × 2.25 = 144 km." }, { "itemId": "numerical-reasoning-s06-q01", "explanation": "The total spend is £7.35 + £4.20 = £11.55, so the change is £20 − £11.55 = £8.45." }, { "itemId": "numerical-reasoning-s06-q02", "explanation": "Compare a common unit: per 100 g, the small pack costs £2.40 ÷ 5 = £0.48 while the large pack costs £3.30 ÷ 7.5 = £0.44, so the larger pack is cheaper per gram." }, { "itemId": "numerical-reasoning-s06-q03", "explanation": "Simple interest pays the same amount each year: 5% of £800 is £40, so three years earn 3 × £40 = £120." }, { "itemId": "numerical-reasoning-s06-q04", "explanation": "10% of £2,400 is £240, so 35% is 3.5 × £240 = £840." }, { "itemId": "numerical-reasoning-q5", "explanation": "$500 × 1.10 = $550 after year one; $550 × 1.10 = $605 after year two." }, { "itemId": "numerical-reasoning-s06-q06", "explanation": "Divide the bill by the number of diners: £86.40 ÷ 6 = £14.40." }, { "itemId": "numerical-reasoning-s06-q07", "explanation": "The deposit is 15% of £640 = £96, leaving £640 − £96 = £544, or simply take 85% of the price directly." }, { "itemId": "numerical-reasoning-s06-q08", "explanation": "Multiply hours by the hourly rate: 38 × £12.50 = £475. Doubling £12.50 to £25 per 2 hours makes the mental sum easier: 19 × £25 = £475." }, { "itemId": "numerical-reasoning-s06-q09", "explanation": "Eight drinks form two groups of four, and in each group one is free, so you pay for 6 drinks: 6 × £2.50 = £15." }, { "itemId": "numerical-reasoning-s06-q10", "explanation": "The profit per plant is £2 − £1.20 = £0.80, so 40 plants earn 40 × £0.80 = £32." }, { "itemId": "numerical-reasoning-s07-q01", "explanation": "Add the values and divide by how many there are: (4 + 8 + 9 + 11) ÷ 4 = 32 ÷ 4 = 8." }, { "itemId": "numerical-reasoning-s07-q02", "explanation": "With the five values already in order, the median is the middle (third) value: 8." }, { "itemId": "numerical-reasoning-s07-q03", "explanation": "The mode is the most frequent value, and 4 appears three times, more than any other." }, { "itemId": "numerical-reasoning-s07-q04", "explanation": "The range is the largest value minus the smallest: 23 − 5 = 18." }, { "itemId": "numerical-reasoning-s07-q05", "explanation": "A mean of 14 across five scores means a total of 5 × 14 = 70; the four known scores sum to 55, so the fifth is 70 − 55 = 15." }, { "itemId": "numerical-reasoning-s07-q06", "explanation": "Weight each mean by its class size: total marks are 20 × 60 + 10 × 75 = 1,950, spread over 30 pupils gives 65. Simply averaging 60 and 75 ignores the different class sizes." }, { "itemId": "numerical-reasoning-s07-q07", "explanation": "Average speed is total distance over total time: (120 + 60) ÷ (2 + 1) = 180 ÷ 3 = 60 km/h." }, { "itemId": "numerical-reasoning-s07-q08", "explanation": "Sum the three days and divide by three: (40 + 55 + 25) ÷ 3 = 120 ÷ 3 = 40." }, { "itemId": "numerical-reasoning-s07-q09", "explanation": "The original total is 6 × 10 = 60; adding 24 gives 84 across seven numbers, and 84 ÷ 7 = 12." }, { "itemId": "numerical-reasoning-s07-q10", "explanation": "A mean only fixes the total (5 × 50 = 250 points), not how the points are spread. Every player could have scored exactly 50, or one could have scored far more." }, { "itemId": "numerical-reasoning-s08-q01", "explanation": "Each term increases by 4, so the next term is 17 + 4 = 21." }, { "itemId": "numerical-reasoning-s08-q02", "explanation": "Each term is multiplied by 3, so the next term is 54 × 3 = 162." }, { "itemId": "numerical-reasoning-s08-q03", "explanation": "Each term is halved, so the next term is 8 ÷ 2 = 4." }, { "itemId": "numerical-reasoning-s08-q04", "explanation": "These are the square numbers 1², 2², 3², 4², 5², so the next is 6² = 36." }, { "itemId": "numerical-reasoning-s08-q05", "explanation": "Each term is the sum of the two before it: 11 + 18 = 29." }, { "itemId": "numerical-reasoning-s08-q06", "explanation": "The amount subtracted grows by one each step (3, 4, 5, 6), so the next step subtracts 7: 22 − 7 = 15." }, { "itemId": "numerical-reasoning-s08-q07", "explanation": "Two sequences alternate: the odd positions climb 1, 3, 5, 7 while the even positions fall 10, 8, 6, so the next even-position term is 4." }, { "itemId": "numerical-reasoning-s08-q08", "explanation": "The gaps grow by one each time (1, 2, 3, 4, 5), so the next gap is 6: 20 + 6 = 26." }, { "itemId": "numerical-reasoning-s08-q09", "explanation": "8, 27, 64, and 125 are cube numbers (2³, 3³, 4³, 5³); 100 is a square but not a cube." }, { "itemId": "numerical-reasoning-s08-q10", "explanation": "Apply the rule to 17: 2 × 17 = 34, then 34 − 1 = 33." }, { "itemId": "numerical-reasoning-s09-q01", "explanation": "First find the capacity: 18 × 22 = 396 seats; then subtract the tickets sold: 396 − 315 = 81." }, { "itemId": "numerical-reasoning-s09-q02", "explanation": "Remove the fixed fee first: £145 − £40 = £105 of hourly work, then divide by the rate: 105 ÷ 15 = 7 hours." }, { "itemId": "numerical-reasoning-s09-q03", "explanation": "The trip uses 4.5 × 6 = 27 litres, and 27 × £1.50 = £40.50." }, { "itemId": "numerical-reasoning-s09-q04", "explanation": "The 20% reduction takes £45 down to £36, and the £5 voucher then brings it to £31. The order of the steps matters." }, { "itemId": "numerical-reasoning-s09-q05", "explanation": "Their combined age is three equal 'Beth-sized' parts (Beth plus twice Beth), so Beth is 36 ÷ 3 = 12 and Amir is 24." }, { "itemId": "numerical-reasoning-s09-q06", "explanation": "Work backwards, undoing each step in reverse: 46 − 6 = 40, then 40 ÷ 4 = 10." }, { "itemId": "numerical-reasoning-s09-q07", "explanation": "Monday's shipment is 15% of 1,200 = 180 units, leaving 1,020; Tuesday's delivery brings it to 1,020 + 250 = 1,270." }, { "itemId": "numerical-reasoning-s09-q08", "explanation": "The floor area is 4 × 3 = 12 m², needing 12 ÷ 0.25 = 48 tiles, which is 48 ÷ 10 = 4.8, so 5 full boxes must be bought." }, { "itemId": "numerical-reasoning-s09-q09", "explanation": "The tickets cost 2 × £9 + 3 × £5 = £18 + £15 = £33, so the change is £50 − £33 = £17." }, { "itemId": "numerical-reasoning-s09-q10", "explanation": "June is 120% of May, so divide rather than take 20% off: 660 ÷ 1.2 = 550. Subtracting 20% of June's figure would give the wrong base." }, { "itemId": "numerical-reasoning-s10-q01", "explanation": "Apply the multiplier twice: £2,000 × 0.9 × 0.9 = £1,620. The second year's 10% is taken from the reduced value, not the original." }, { "itemId": "numerical-reasoning-s10-q02", "explanation": "Chain the multipliers: £150 × 0.8 = £120, then £120 × 0.9 = £108. Two successive discounts of 20% and 10% are not the same as one 30% discount." }, { "itemId": "numerical-reasoning-s10-q03", "explanation": "£96 represents 120% of the original price, so divide by 1.2: 96 ÷ 1.2 = £80. Taking 20% off £96 would use the wrong base." }, { "itemId": "numerical-reasoning-s10-q04", "explanation": "Work in rates: together they do 1/2 of the job per hour and A alone does 1/3, so B's rate is 1/2 − 1/3 = 1/6 per hour. A full job takes B 6 hours." }, { "itemId": "numerical-reasoning-s10-q05", "explanation": "More time is spent at the slower speed, so the answer is below the midpoint. For a 120 km leg each way: 2 hours out plus 3 hours back is 240 km in 5 hours = 48 km/h." }, { "itemId": "numerical-reasoning-s10-q06", "explanation": "A percentage of a percentage multiplies: 40% of 30% is 0.4 × 0.3 = 0.12, which is 12% of all staff." }, { "itemId": "numerical-reasoning-s10-q07", "explanation": "Round to friendly numbers: 42,000 ÷ 70 = 600, so the quotient is about 600. Estimation is mainly about keeping track of the size of the answer." }, { "itemId": "numerical-reasoning-q6", "explanation": "Combined rate = 1/4 + 1/6 = 5/12 of the tank per hour, so time = 12/5 = 2.4 hours." }, { "itemId": "numerical-reasoning-s10-q09", "explanation": "Start from the actual counts: 3 : 2 across 30 counters means 18 red and 12 blue; adding 6 blue gives 18 : 18, which simplifies to 1 : 1." }, { "itemId": "numerical-reasoning-s10-q10", "explanation": "Multiply the two changes: 1.25 × 0.8 = 1, so the value ends exactly where it started. Percentage rises and falls of different sizes can cancel because they act on different bases." } ] } }