Problem-Solving & Data Analysis practice questions

Problem-Solving & Data Analysis questions

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57. 57PercentagesFoundation

    A jacket priced at 80 pounds is reduced by 35 percent. What is the sale price?A jacket priced at 80 pounds is reduced by 35 percent. What is the sale price?

    • A52 pounds52 pounds
    • B45 pounds45 pounds
    • C28 pounds28 pounds
    • D58 pounds58 pounds
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    Answer: 52 pounds52 pounds

    A 35 percent reduction leaves 65 percent, and 0.65 × 80 = 52 pounds. The choice 28 is the size of the discount rather than the price left after it.A 35 percent reduction leaves 65 percent, and 0.65 × 80 = 52 pounds. The choice 28 is the size of the discount rather than the price left after it.

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58. 58MeanStandard

    Rainfall at a weather station over five months was 42 mm, 55 mm, 38 mm, 61 mm and 44 mm. What is the mean monthly rainfall?Rainfall at a weather station over five months was 42 mm, 55 mm, 38 mm, 61 mm and 44 mm. What is the mean monthly rainfall?

    • A44 mm44 mm
    • B48 mm48 mm
    • C55 mm55 mm
    • D46 mm46 mm
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    Answer: 48 mm48 mm

    The five values total 240 mm, and 2405 = 48 mm. The choice 44 is the middle value when the list is sorted, which is the median, and 55 is one reading rather than a summary of all five.The five values total 240 mm, and 240/5 = 48 mm. The choice 44 is the middle value when the list is sorted, which is the median, and 55 is one reading rather than a summary of all five.

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59. 59ProbabilityStandard

    A box holds 5 red counters, 7 blue counters and 8 green counters. One counter is drawn at random. What is the probability that it is not green?A box holds 5 red counters, 7 blue counters and 8 green counters. One counter is drawn at random. What is the probability that it is not green?

    • A252/5
    • B151/5
    • C353/5
    • D7207/20
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    Answer: 353/5

    There are 20 counters in all and 12 that are not green, so the probability is 1220 = 35. The choice 25 is the probability that it is green, which is what you get by answering the question that was not asked.There are 20 counters in all and 12 that are not green, so the probability is 12/20 = 3/5. The choice 2/5 is the probability that it is green, which is what you get by answering the question that was not asked.

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60. 60Two-way frequenciesStandard

    Of 120 students surveyed, 45 cycle to school and 75 do not. Among those who cycle, 18 play a team sport; among those who do not cycle, 42 play a team sport. If a student who plays a team sport is chosen at random, what is the probability that the student cycles?Of 120 students surveyed, 45 cycle to school and 75 do not. Among those who cycle, 18 play a team sport; among those who do not cycle, 42 play a team sport. If a student who plays a team sport is chosen at random, what is the probability that the student cycles?

    • A252/5
    • B3203/20
    • C121/2
    • D3103/10
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    Answer: 3103/10

    The group being chosen from is the 18 + 42 = 60 students who play a team sport, and 18 of them cycle, so the probability is 1860 = 310. Dividing by 45 instead gives 25, which answers the reverse question: the chance a cyclist plays a sport.The group being chosen from is the 18 + 42 = 60 students who play a team sport, and 18 of them cycle, so the probability is 18/60 = 3/10. Dividing by 45 instead gives 2/5, which answers the reverse question: the chance a cyclist plays a sport.

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61. 61MedianChallenge

    A shop recorded umbrellas sold on nine days: 3, 7, 4, 12, 5, 7, 2, 9, 6. What is the median number sold?A shop recorded umbrellas sold on nine days: 3, 7, 4, 12, 5, 7, 2, 9, 6. What is the median number sold?

    • A66
    • B77
    • C55
    • D6.16.1
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    Answer: 66

    Sorted, the values are 2, 3, 4, 5, 6, 7, 7, 9, 12. With nine values the median is the fifth, which is 6. The choice 7 is the mode, and 6.1 is roughly the mean, which the single high value of 12 pulls upward.Sorted, the values are 2, 3, 4, 5, 6, 7, 7, 9, 12. With nine values the median is the fifth, which is 6. The choice 7 is the mode, and 6.1 is roughly the mean, which the single high value of 12 pulls upward.

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62. 62RatesStandard

    A printer produces 14 pages a minute. How many minutes does it take to print 630 pages?A printer produces 14 pages a minute. How many minutes does it take to print 630 pages?

    • A3535
    • B4545
    • C5050
    • D4444
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    Answer: 4545

    Dividing the total by the rate gives 63014 = 45 minutes. Multiplying instead of dividing, or dividing by the wrong figure, produces the other choices.Dividing the total by the rate gives 630/14 = 45 minutes. Multiplying instead of dividing, or dividing by the wrong figure, produces the other choices.

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63. 63Percent changeChallenge

    A town grew from 12,500 people to 14,000 over one decade, then lost 8 percent of its population over the next. What was the population at the end of the second decade?A town grew from 12,500 people to 14,000 over one decade, then lost 8 percent of its population over the next. What was the population at the end of the second decade?

    • A13,20013,200
    • B12,50012,500
    • C12,88012,880
    • D11,50011,500
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    Answer: 12,88012,880

    The 8 percent loss applies to 14,000, not to the original figure: 14,000 × 0.92 = 12,880. Applying it to 12,500 gives 11,500, and subtracting 8 percent of the growth instead gives 13,200. Note that the town ends above where it started.The 8 percent loss applies to 14,000, not to the original figure: 14,000 × 0.92 = 12,880. Applying it to 12,500 gives 11,500, and subtracting 8 percent of the growth instead gives 13,200. Note that the town ends above where it started.

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64. 64SamplingChallenge

    A researcher wants to estimate the average commute time of the 2,000 staff at a company with six sites. She surveys the 90 people who arrive at the main site before 8 a.m. What is the strongest objection to her method?A researcher wants to estimate the average commute time of the 2,000 staff at a company with six sites. She surveys the 90 people who arrive at the main site before 8 a.m. What is the strongest objection to her method?

    • AThe sample of 90 is too small to support any estimate.The sample of 90 is too small to support any estimate.
    • BCommute times are not the sort of quantity that can be averaged.Commute times are not the sort of quantity that can be averaged.
    • CThe survey should have asked about the cost of the journey as well.The survey should have asked about the cost of the journey as well.
    • DStaff who arrive early at one site are unlikely to resemble staff across all six.Staff who arrive early at one site are unlikely to resemble staff across all six.
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    Answer: Staff who arrive early at one site are unlikely to resemble staff across all six.Staff who arrive early at one site are unlikely to resemble staff across all six.

    The problem is how the sample was chosen, not how large it is: early arrivals at one site may well live nearer, and five sites are missing entirely. A sample of 90 can support an estimate when it is drawn at random from the whole group.The problem is how the sample was chosen, not how large it is: early arrivals at one site may well live nearer, and five sites are missing entirely. A sample of 90 can support an estimate when it is drawn at random from the whole group.

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65. 65Unit conversionFoundation

    A tank holds 4.5 cubic meters of water. Given that 1 cubic meter is 1,000 liters, how many liters does it hold?A tank holds 4.5 cubic meters of water. Given that 1 cubic meter is 1,000 liters, how many liters does it hold?

    • A4,5004,500
    • B450450
    • C45,00045,000
    • D4.54.5
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    Answer: 4,5004,500

    Multiplying by 1,000 moves the decimal point three places: 4.5 × 1,000 = 4,500 liters. Moving it two places or four gives the other choices, which is why it is worth naming the factor before converting.Multiplying by 1,000 moves the decimal point three places: 4.5 × 1,000 = 4,500 liters. Moving it two places or four gives the other choices, which is why it is worth naming the factor before converting.

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66. 66Percent decreaseStandard

    A subscription fell from 48 pounds a year to 36 pounds. What was the percentage decrease?A subscription fell from 48 pounds a year to 36 pounds. What was the percentage decrease?

    • A33 percent33 percent
    • B25 percent25 percent
    • C12 percent12 percent
    • D75 percent75 percent
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    Answer: 25 percent25 percent

    The fall is 12 pounds, and a percentage decrease is measured against the starting figure: 1248 = 0.25, so 25 percent. Dividing by 36 gives about 33 percent, which would be the increase needed to go back.The fall is 12 pounds, and a percentage decrease is measured against the starting figure: 12/48 = 0.25, so 25 percent. Dividing by 36 gives about 33 percent, which would be the increase needed to go back.

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67. 67RangeStandard

    Daily high temperatures for a week, in degrees Celsius, were 14, 19, 11, 22, 17, 13 and 20. What is the range?Daily high temperatures for a week, in degrees Celsius, were 14, 19, 11, 22, 17, 13 and 20. What is the range?

    • A22°22°
    • B16.6°16.6°
    • C11°11°
    • D9°9°
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    Answer: 11°11°

    The range is the highest value minus the lowest: 22 - 11 = 11°. The choice 22 is the maximum on its own and 16.6 is roughly the mean, neither of which measures spread.The range is the highest value minus the lowest: 22 - 11 = 11°. The choice 22 is the maximum on its own and 16.6 is roughly the mean, neither of which measures spread.

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68. 68Weighted averagesChallenge

    A class of 30 students averaged 72 on a test and a class of 20 averaged 82. What is the average score across both classes?A class of 30 students averaged 72 on a test and a class of 20 averaged 82. What is the average score across both classes?

    • A7777
    • B7575
    • C7878
    • D7676
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    Answer: 7676

    Totals must be combined before dividing: (30 × 72) + (20 × 82) = 2,160 + 1,640 = 3,800, and 3,80050 = 76. Averaging 72 and 82 directly gives 77, which would only be right if the classes were the same size.Totals must be combined before dividing: (30 × 72) + (20 × 82) = 2,160 + 1,640 = 3,800, and 3,800/50 = 76. Averaging 72 and 82 directly gives 77, which would only be right if the classes were the same size.

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69. 69Linear relationshipsStandard

    A candle burns at a constant rate. Two hours after it is lit it is 21 cm tall, and five hours after it is lit it is 15 cm tall. How tall was it when it was lit?A candle burns at a constant rate. Two hours after it is lit it is 21 cm tall, and five hours after it is lit it is 15 cm tall. How tall was it when it was lit?

    • A25 cm25 cm
    • B23 cm23 cm
    • C27 cm27 cm
    • D24 cm24 cm
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    Answer: 25 cm25 cm

    The height falls 6 cm over 3 hours, so the rate is 2 cm an hour. Working back two hours from 21 cm gives 21 + 4 = 25 cm. Working back only one hour, or forgetting to work back at all, produces the other choices.The height falls 6 cm over 3 hours, so the rate is 2 cm an hour. Working back two hours from 21 cm gives 21 + 4 = 25 cm. Working back only one hour, or forgetting to work back at all, produces the other choices.

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70. 70ProbabilityStandard

    Two fair six-sided dice are rolled. What is the probability that the two numbers total 9?Two fair six-sided dice are rolled. What is the probability that the two numbers total 9?

    • A161/6
    • B191/9
    • C4124/12
    • D1121/12
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    Answer: 191/9

    There are 36 equally likely outcomes and four give a total of 9: 3 and 6, 4 and 5, 5 and 4, 6 and 3. So the probability is 436 = 19. Counting 3 and 6 as the same outcome as 6 and 3 loses two of the four.There are 36 equally likely outcomes and four give a total of 9: 3 and 6, 4 and 5, 5 and 4, 6 and 3. So the probability is 4/36 = 1/9. Counting 3 and 6 as the same outcome as 6 and 3 loses two of the four.

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71. 71PercentagesChallenge

    A price rises by 20 percent and then falls by 20 percent. Compared with the original price, the final price isA price rises by 20 percent and then falls by 20 percent. Compared with the original price, the final price is

    • Athe same.the same.
    • B4 percent higher.4 percent higher.
    • C4 percent lower.4 percent lower.
    • D20 percent lower.20 percent lower.
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    Answer: 4 percent lower.4 percent lower.

    The two changes are percentages of different amounts. Multiplying gives 1.20 × 0.80 = 0.96, so the final price is 96 percent of the original, 4 percent lower. The fall is larger in money terms because it is taken from the raised price.The two changes are percentages of different amounts. Multiplying gives 1.20 × 0.80 = 0.96, so the final price is 96 percent of the original, 4 percent lower. The fall is larger in money terms because it is taken from the raised price.

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72. 72Two-way frequenciesChallenge

    In a survey of 200 people, 120 own a bicycle and 90 own a car. If 40 own both, how many own neither?In a survey of 200 people, 120 own a bicycle and 90 own a car. If 40 own both, how many own neither?

    • A1010
    • B5050
    • C7070
    • D3030
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    Answer: 3030

    Adding 120 and 90 counts the 40 who own both twice, so the number owning at least one is 120 + 90 - 40 = 170. That leaves 200 - 170 = 30 owning neither. Forgetting to subtract the overlap gives 210, which is more people than were surveyed — a useful sign that a step is missing.Adding 120 and 90 counts the 40 who own both twice, so the number owning at least one is 120 + 90 - 40 = 170. That leaves 200 - 170 = 30 owning neither. Forgetting to subtract the overlap gives 210, which is more people than were surveyed — a useful sign that a step is missing.

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73. 73Proportional relationshipsFoundation

    A car uses 6 liters of fuel for every 100 km driven. How much fuel does it use for 250 km?A car uses 6 liters of fuel for every 100 km driven. How much fuel does it use for 250 km?

    • A15 liters15 liters
    • B24 liters24 liters
    • C12 liters12 liters
    • D41.7 liters41.7 liters
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    Answer: 15 liters15 liters

    250 km is 2.5 lots of 100 km, so the fuel used is 2.5 × 6 = 15 liters. Dividing 250 by 6 gives 41.7, which answers how many 6-km stretches there are rather than how much fuel is burned.250 km is 2.5 lots of 100 km, so the fuel used is 2.5 × 6 = 15 liters. Dividing 250 by 6 gives 41.7, which answers how many 6-km stretches there are rather than how much fuel is burned.

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74. 74Percent decreaseStandard

    A machine that produced 480 units an hour was reconfigured and now produces 408. What was the percentage decrease?A machine that produced 480 units an hour was reconfigured and now produces 408. What was the percentage decrease?

    • A17.6 percent17.6 percent
    • B15 percent15 percent
    • C72 percent72 percent
    • D12 percent12 percent
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    Answer: 15 percent15 percent

    The fall of 72 is measured against the original 480: 72480 = 0.15. Dividing by the new figure of 408 gives 17.6 percent, which is the rise needed to get back rather than the fall that happened.The fall of 72 is measured against the original 480: 72/480 = 0.15. Dividing by the new figure of 408 gives 17.6 percent, which is the rise needed to get back rather than the fall that happened.

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75. 75SpreadStandard

    Class A scored 68, 70, 71, 72 and 74 on a test. Class B scored 55, 64, 71, 79 and 86. Both classes have a mean of 71. Which statement is true?Class A scored 68, 70, 71, 72 and 74 on a test. Class B scored 55, 64, 71, 79 and 86. Both classes have a mean of 71. Which statement is true?

    • AClass A's scores are more spread out than Class B's.Class A's scores are more spread out than Class B's.
    • BThe two classes have the same range.The two classes have the same range.
    • CClass B's scores are more spread out than Class A's.Class B's scores are more spread out than Class A's.
    • DSpread cannot be compared when two sets have the same mean.Spread cannot be compared when two sets have the same mean.
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    Answer: Class B's scores are more spread out than Class A's.Class B's scores are more spread out than Class A's.

    The means are identical and say nothing about spread. Class A runs from 68 to 74, a range of 6; class B runs from 55 to 86, a range of 31. Equal means with different spreads is the ordinary case, not a special one.The means are identical and say nothing about spread. Class A runs from 68 to 74, a range of 6; class B runs from 55 to 86, a range of 31. Equal means with different spreads is the ordinary case, not a special one.

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76. 76RangeStandard

    Six parcels had masses, in kg, of 2.4, 5.1, 3.8, 7.2, 4.0 and 6.5. What is the range of these masses?Six parcels had masses, in kg, of 2.4, 5.1, 3.8, 7.2, 4.0 and 6.5. What is the range of these masses?

    • A7.2 kg7.2 kg
    • B4.9 kg4.9 kg
    • C2.4 kg2.4 kg
    • D4.8 kg4.8 kg
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    Answer: 4.8 kg4.8 kg

    The range is the largest value minus the smallest: 7.2 - 2.4 = 4.8 kg. The choices 7.2 and 2.4 are those two values on their own, neither of which describes how far apart the masses are.The range is the largest value minus the smallest: 7.2 - 2.4 = 4.8 kg. The choices 7.2 and 2.4 are those two values on their own, neither of which describes how far apart the masses are.

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77. 77Linear relationshipsStandard

    A tank is filled at a constant rate. Three minutes after filling starts it holds 26 liters, and eight minutes after it starts it holds 66 liters. How much was in the tank when filling began?A tank is filled at a constant rate. Three minutes after filling starts it holds 26 liters, and eight minutes after it starts it holds 66 liters. How much was in the tank when filling began?

    • A2 liters2 liters
    • B0 liters0 liters
    • C8 liters8 liters
    • D10 liters10 liters
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    Answer: 2 liters2 liters

    The tank gains 40 liters in 5 minutes, a rate of 8 liters a minute. Working back three minutes from 26 liters gives 26 - 24 = 2 liters. Assuming it started empty ignores the figures given.The tank gains 40 liters in 5 minutes, a rate of 8 liters a minute. Working back three minutes from 26 liters gives 26 - 24 = 2 liters. Assuming it started empty ignores the figures given.

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78. 78ProbabilityChallenge

    A bag holds 4 white marbles and 6 black ones. Two marbles are drawn without replacement. What is the probability that both are white?A bag holds 4 white marbles and 6 black ones. Two marbles are drawn without replacement. What is the probability that both are white?

    • A4254/25
    • B2152/15
    • C151/5
    • D252/5
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    Answer: 2152/15

    The first draw is white with probability 410. The second is then drawn from 9 marbles of which 3 are white, so 39. Multiplying gives 1290 = 215. Using 410 twice gives 425 and quietly puts the first marble back.The first draw is white with probability 4/10. The second is then drawn from 9 marbles of which 3 are white, so 3/9. Multiplying gives 12/90 = 2/15. Using 4/10 twice gives 4/25 and quietly puts the first marble back.

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79. 79PercentagesStandard

    A jacket costs 64 pounds after a 20 percent discount. Enter its original price in pounds.A jacket costs 64 pounds after a 20 percent discount. Enter its original price in pounds.

    Student-produced response: write your own answer rather than choosing one.

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    Answer: 8080

    The sale price is 80 percent of the original, so the original is 640.8 = 80 pounds. Adding 20 percent to 64 gives 76.80, which is a different quantity: 20 percent of the smaller number.The sale price is 80 percent of the original, so the original is 64/0.8 = 80 pounds. Adding 20 percent to 64 gives 76.80, which is a different quantity: 20 percent of the smaller number.

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80. 80RatesStandard

    A tap fills a 240-liter tank in 16 minutes. Enter the number of liters it delivers per minute.A tap fills a 240-liter tank in 16 minutes. Enter the number of liters it delivers per minute.

    Student-produced response: write your own answer rather than choosing one.

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    Answer: 1515

    Dividing the volume by the time gives 24016 = 15 liters a minute.Dividing the volume by the time gives 240/16 = 15 liters a minute.

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81. 81PercentagesFoundation

    What is 15 percent of 240?What is 15 percent of 240?

    • A3636
    • B1616
    • C2424
    • D360360
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    Answer: 3636

    15 percent is 0.15, and 0.15 × 240 = 36. A useful check: 10 percent is 24 and 5 percent is 12, and 24 + 12 = 36.15 percent is 0.15, and 0.15 × 240 = 36. A useful check: 10 percent is 24 and 5 percent is 12, and 24 + 12 = 36.

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82. 82MeanStandard

    Six readings were 12, 15, 11, 18, 14 and 20. What is their mean?Six readings were 12, 15, 11, 18, 14 and 20. What is their mean?

    • A1414
    • B1515
    • C1616
    • D14.514.5
    Show the answer and explanationHide the explanation

    Answer: 1515

    The six values total 90, and 906 = 15. The choice 14.5 is the median — the average of the third and fourth values once sorted — which happens to differ here.The six values total 90, and 90/6 = 15. The choice 14.5 is the median — the average of the third and fourth values once sorted — which happens to differ here.

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83. 83RatesStandard

    A cyclist covers 54 km in 2 hours and 15 minutes. What is the average speed in kilometers per hour?A cyclist covers 54 km in 2 hours and 15 minutes. What is the average speed in kilometers per hour?

    • A2727
    • B21.621.6
    • C2424
    • D3636
    Show the answer and explanationHide the explanation

    Answer: 2424

    2 hours 15 minutes is 2.25 hours, and 542.25 = 24 km/h. Dividing by 2 instead gives 27, which treats the extra quarter-hour as free.2 hours 15 minutes is 2.25 hours, and 54/2.25 = 24 km/h. Dividing by 2 instead gives 27, which treats the extra quarter-hour as free.

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84. 84ProbabilityStandard

    A spinner has 8 equal sectors numbered 1 to 8. What is the probability of landing on a prime number?A spinner has 8 equal sectors numbered 1 to 8. What is the probability of landing on a prime number?

    • A383/8
    • B585/8
    • C141/4
    • D121/2
    Show the answer and explanationHide the explanation

    Answer: 121/2

    The primes from 1 to 8 are 2, 3, 5 and 7 — four of the eight, so the probability is 12. Counting 1 as prime gives 58; it is not, because a prime has exactly two distinct factors.The primes from 1 to 8 are 2, 3, 5 and 7 — four of the eight, so the probability is 1/2. Counting 1 as prime gives 5/8; it is not, because a prime has exactly two distinct factors.

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85. 85Two-way frequenciesStandard

    Of 250 people surveyed, 140 are under 40. Of those under 40, 98 use public transport daily. What proportion of the under-40s use public transport daily?Of 250 people surveyed, 140 are under 40. Of those under 40, 98 use public transport daily. What proportion of the under-40s use public transport daily?

    • A39.2 percent39.2 percent
    • B56 percent56 percent
    • C70 percent70 percent
    • D43 percent43 percent
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    Answer: 70 percent70 percent

    The group being asked about is the 140 under-40s, so the calculation is 98140 = 0.70. Dividing by the whole 250 gives 39.2 percent, which answers a question about the survey rather than about the under-40s.The group being asked about is the 140 under-40s, so the calculation is 98/140 = 0.70. Dividing by the whole 250 gives 39.2 percent, which answers a question about the survey rather than about the under-40s.

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86. 86Percent changeChallenge

    A quantity falls by 25 percent and then rises by 40 percent. What is the overall change?A quantity falls by 25 percent and then rises by 40 percent. What is the overall change?

    • A15 percent rise15 percent rise
    • B5 percent fall5 percent fall
    • CNo changeNo change
    • D5 percent rise5 percent rise
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    Answer: 5 percent rise5 percent rise

    Percentage changes multiply rather than add: 0.75 × 1.40 = 1.05, a 5 percent rise. Adding -25 and +40 gives 15, which would only be right if both percentages were of the same starting amount.Percentage changes multiply rather than add: 0.75 × 1.40 = 1.05, a 5 percent rise. Adding -25 and +40 gives 15, which would only be right if both percentages were of the same starting amount.

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87. 87MedianStandard

    Enter the median of the values 8, 3, 11, 6, 9 and 4.Enter the median of the values 8, 3, 11, 6, 9 and 4.

    Student-produced response: write your own answer rather than choosing one.

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    Answer: 77

    Sorted, the values are 3, 4, 6, 8, 9, 11. With six values the median is the average of the third and fourth: 6 + 82 = 7. Note that 7 is not one of the values, which is normal for an even-sized set.Sorted, the values are 3, 4, 6, 8, 9, 11. With six values the median is the average of the third and fourth: (6 + 8)/2 = 7. Note that 7 is not one of the values, which is normal for an even-sized set.

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88. 88Unit conversionStandard

    A recipe calls for 1.5 kg of flour. Enter that quantity in grams.A recipe calls for 1.5 kg of flour. Enter that quantity in grams.

    Student-produced response: write your own answer rather than choosing one.

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    Answer: 15001500

    One kilogram is 1,000 grams, so 1.5 kg is 1,500 g. Going to a smaller unit always multiplies, which is the check: 0.0015 would mean a gram is larger than a kilogram.One kilogram is 1,000 grams, so 1.5 kg is 1,500 g. Going to a smaller unit always multiplies, which is the check: 0.0015 would mean a gram is larger than a kilogram.

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89. 89Proportional relationshipsFoundation

    Two lengths are in the ratio 3 : 5, and together they measure 96 cm. What is the shorter length?Two lengths are in the ratio 3 : 5, and together they measure 96 cm. What is the shorter length?

    • A60 cm60 cm
    • B36 cm36 cm
    • C32 cm32 cm
    • D48 cm48 cm
    Show the answer and explanationHide the explanation

    Answer: 36 cm36 cm

    The ratio has 3 + 5 = 8 parts, so each part is 968 = 12 cm and the shorter length is 3 × 12 = 36 cm. The choice 60 is the longer length and 48 is half the total.The ratio has 3 + 5 = 8 parts, so each part is 96/8 = 12 cm and the shorter length is 3 × 12 = 36 cm. The choice 60 is the longer length and 48 is half the total.

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90. 90RatesChallenge

    A driver covers 60 km at 60 km/h and the next 60 km at 40 km/h. What is the average speed for the whole journey?A driver covers 60 km at 60 km/h and the next 60 km at 40 km/h. What is the average speed for the whole journey?

    • A50 km/h50 km/h
    • B52 km/h52 km/h
    • C48 km/h48 km/h
    • D45 km/h45 km/h
    Show the answer and explanationHide the explanation

    Answer: 48 km/h48 km/h

    Average speed is total distance over total time. The two legs take 1 hour and 1.5 hours, so 120 km in 2.5 hours is 48 km/h. Averaging the two speeds gives 50, which would only be right if equal time were spent at each.Average speed is total distance over total time. The two legs take 1 hour and 1.5 hours, so 120 km in 2.5 hours is 48 km/h. Averaging the two speeds gives 50, which would only be right if equal time were spent at each.

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91. 91ProbabilityStandard

    Two fair coins are tossed. What is the probability of getting at least one head?Two fair coins are tossed. What is the probability of getting at least one head?

    • A121/2
    • B141/4
    • C232/3
    • D343/4
    Show the answer and explanationHide the explanation

    Answer: 343/4

    The four equally likely outcomes are HH, HT, TH and TT, and three of them contain a head, so the probability is 34. It is quicker as 1 minus the probability of no heads: 1 - 14.The four equally likely outcomes are HH, HT, TH and TT, and three of them contain a head, so the probability is 3/4. It is quicker as 1 minus the probability of no heads: 1 - 1/4.

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92. 92PercentagesChallenge

    An account holds 2,000 pounds and earns 5 percent interest each year, added to the balance. What is the balance after two years?An account holds 2,000 pounds and earns 5 percent interest each year, added to the balance. What is the balance after two years?

    • A2,205 pounds2,205 pounds
    • B2,200 pounds2,200 pounds
    • C2,100 pounds2,100 pounds
    • D2,050 pounds2,050 pounds
    Show the answer and explanationHide the explanation

    Answer: 2,205 pounds2,205 pounds

    Each year multiplies the balance by 1.05: 2,000 × 1.05 × 1.05 = 2,205 pounds. Taking 5 percent of the original twice gives 2,200 and misses the interest earned on the first year's interest.Each year multiplies the balance by 1.05: 2,000 × 1.05 × 1.05 = 2,205 pounds. Taking 5 percent of the original twice gives 2,200 and misses the interest earned on the first year's interest.

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93. 93Percent changeStandard

    A shop's weekly takings rose from 3,400 pounds to 3,825 pounds. What was the percentage increase?A shop's weekly takings rose from 3,400 pounds to 3,825 pounds. What was the percentage increase?

    • A11.1 percent11.1 percent
    • B12.5 percent12.5 percent
    • C14.3 percent14.3 percent
    • D8.5 percent8.5 percent
    Show the answer and explanationHide the explanation

    Answer: 12.5 percent12.5 percent

    The rise is 425 pounds, measured against the original 3,400: 4253,400 = 0.125, so 12.5 percent. Dividing by the new figure gives 11.1 percent, which is the fall needed to go back.The rise is 425 pounds, measured against the original 3,400: 425/3,400 = 0.125, so 12.5 percent. Dividing by the new figure gives 11.1 percent, which is the fall needed to go back.

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94. 94Two-way frequenciesStandard

    In a club of 80 members, 52 play chess and 45 play bridge. If 30 play both, how many play neither?In a club of 80 members, 52 play chess and 45 play bridge. If 30 play both, how many play neither?

    • A1717
    • B2222
    • C1313
    • D88
    Show the answer and explanationHide the explanation

    Answer: 1313

    The number playing at least one is 52 + 45 - 30 = 67, so 80 - 67 = 13 play neither. Adding 52 and 45 without removing the overlap gives 97, more members than the club has.The number playing at least one is 52 + 45 - 30 = 67, so 80 - 67 = 13 play neither. Adding 52 and 45 without removing the overlap gives 97, more members than the club has.

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95. 95MeanStandard

    Five numbers have a mean of 18. Four of them are 12, 20, 15 and 22. Enter the fifth number.Five numbers have a mean of 18. Four of them are 12, 20, 15 and 22. Enter the fifth number.

    Student-produced response: write your own answer rather than choosing one.

    Show the answer and explanationHide the explanation

    Answer: 2121

    The five must total 5 × 18 = 90. The four given add to 69, so the fifth is 90 - 69 = 21. Working from the total rather than from the mean is what makes this straightforward.The five must total 5 × 18 = 90. The four given add to 69, so the fifth is 90 - 69 = 21. Working from the total rather than from the mean is what makes this straightforward.

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96. 96Unit conversionStandard

    A journey takes 2 hours and 40 minutes. Enter its length in minutes.A journey takes 2 hours and 40 minutes. Enter its length in minutes.

    Student-produced response: write your own answer rather than choosing one.

    Show the answer and explanationHide the explanation

    Answer: 160160

    Convert the hours and add the remainder: 2 × 60 = 120 minutes, and 120 + 40 = 160. Time is the one conversion that is not decimal, so 2.4 hours is not 2 hours 40.Convert the hours and add the remainder: 2 × 60 = 120 minutes, and 120 + 40 = 160. Time is the one conversion that is not decimal, so 2.4 hours is not 2 hours 40.

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97. 97PercentagesFoundation

    What is 18 percent of 350?What is 18 percent of 350?

    • A5353
    • B7070
    • C1818
    • D6363
    Show the answer and explanationHide the explanation

    Answer: 6363

    0.18 × 350 = 63. As a check, 10 percent is 35 and 8 percent is 28, and 35 + 28 = 63.0.18 × 350 = 63. As a check, 10 percent is 35 and 8 percent is 28, and 35 + 28 = 63.

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98. 98Proportional relationshipsStandard

    Two amounts are in the ratio 2 : 7 and total 108. What is the larger amount?Two amounts are in the ratio 2 : 7 and total 108. What is the larger amount?

    • A8484
    • B9696
    • C2424
    • D5454
    Show the answer and explanationHide the explanation

    Answer: 8484

    The ratio has 9 parts, so each part is 1089 = 12 and the larger amount is 7 × 12 = 84. The smaller is 24, and the two sum to 108 as a check.The ratio has 9 parts, so each part is 108/9 = 12 and the larger amount is 7 × 12 = 84. The smaller is 24, and the two sum to 108 as a check.

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99. 99ProbabilityStandard

    Two fair six-sided dice are rolled. What is the probability that the total is 7?Two fair six-sided dice are rolled. What is the probability that the total is 7?

    • A1121/12
    • B161/6
    • C7367/36
    • D191/9
    Show the answer and explanationHide the explanation

    Answer: 161/6

    Six of the 36 outcomes total 7 — (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — so the probability is 636 = 16. Seven is the likeliest total, which is worth remembering.Six of the 36 outcomes total 7 — (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — so the probability is 6/36 = 1/6. Seven is the likeliest total, which is worth remembering.

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100. 100Percent decreaseStandard

     A weekly figure fell from 250 to 200. What was the percentage decrease?A weekly figure fell from 250 to 200. What was the percentage decrease?

     • A20 percent20 percent
     • B25 percent25 percent
     • C50 percent50 percent
     • D80 percent80 percent
     Show the answer and explanationHide the explanation

     Answer: 20 percent20 percent

     The fall of 50 is measured against the starting figure: 50250 = 0.20. Dividing by 200 gives 25 percent, which is the rise needed to return.The fall of 50 is measured against the starting figure: 50/250 = 0.20. Dividing by 200 gives 25 percent, which is the rise needed to return.

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101. 101MeanStandard

     Six numbers have a mean of 14. Five of them are 9, 17, 12, 20 and 11. Enter the sixth.Six numbers have a mean of 14. Five of them are 9, 17, 12, 20 and 11. Enter the sixth.

     Student-produced response: write your own answer rather than choosing one.

     Show the answer and explanationHide the explanation

     Answer: 1515

     The six must total 6 × 14 = 84. The five given add to 69, so the sixth is 84 - 69 = 15.The six must total 6 × 14 = 84. The five given add to 69, so the sixth is 84 - 69 = 15.

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102. 102RatesStandard

     A train covers 45 km in 36 minutes. What is its average speed in kilometers per hour?A train covers 45 km in 36 minutes. What is its average speed in kilometers per hour?

     • A6060
     • B7575
     • C8080
     • D5454
     Show the answer and explanationHide the explanation

     Answer: 7575

     36 minutes is 0.6 hours, and 450.6 = 75 km/h. Dividing by 36 rather than converting to hours first gives a figure per minute, not per hour.36 minutes is 0.6 hours, and 45/0.6 = 75 km/h. Dividing by 36 rather than converting to hours first gives a figure per minute, not per hour.

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103. 103PercentagesChallenge

     An account of 4,000 pounds earns 3 percent interest a year, added to the balance. What is the balance after two years?An account of 4,000 pounds earns 3 percent interest a year, added to the balance. What is the balance after two years?

     • A4,240 pounds4,240 pounds
     • B4,120 pounds4,120 pounds
     • C4,243.60 pounds4,243.60 pounds
     • D4,360 pounds4,360 pounds
     Show the answer and explanationHide the explanation

     Answer: 4,243.60 pounds4,243.60 pounds

     Each year multiplies by 1.03: 4,000 × 1.03 × 1.03 = 4,243.60. Taking 3 percent of 4,000 twice gives 4,240 and misses the interest earned on the first year's interest.Each year multiplies by 1.03: 4,000 × 1.03 × 1.03 = 4,243.60. Taking 3 percent of 4,000 twice gives 4,240 and misses the interest earned on the first year's interest.

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104. 104Unit conversionStandard

     A plank measures 3.2 meters. Enter its length in centimeters.A plank measures 3.2 meters. Enter its length in centimeters.

     Student-produced response: write your own answer rather than choosing one.

     Show the answer and explanationHide the explanation

     Answer: 320320

     One meter is 100 centimeters, so 3.2 m is 3.2 × 100 = 320 cm. Going to a smaller unit multiplies; dividing would give 0.032 and a plank shorter than a finger.One meter is 100 centimeters, so 3.2 m is 3.2 × 100 = 320 cm. Going to a smaller unit multiplies; dividing would give 0.032 and a plank shorter than a finger.

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105. 105PercentagesFoundation

     What is 24 percent of 425?What is 24 percent of 425?

     • A9898
     • B106106
     • C120120
     • D102102
     Show the answer and explanationHide the explanation

     Answer: 102102

     0.24 × 425 = 102. As a check, 25 percent of 425 is 106.25, so the answer must be slightly below that.0.24 × 425 = 102. As a check, 25 percent of 425 is 106.25, so the answer must be slightly below that.

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106. 106Proportional relationshipsStandard

     Two amounts are in the ratio 4 : 5 and total 153. What is the larger amount?Two amounts are in the ratio 4 : 5 and total 153. What is the larger amount?

     • A8585
     • B6868
     • C76.576.5
     • D9292
     Show the answer and explanationHide the explanation

     Answer: 8585

     The ratio has 9 parts, so each part is 1539 = 17 and the larger amount is 5 × 17 = 85. The smaller is 68, and the two sum to 153 as a check.The ratio has 9 parts, so each part is 153/9 = 17 and the larger amount is 5 × 17 = 85. The smaller is 68, and the two sum to 153 as a check.

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107. 107ProbabilityStandard

     Two fair six-sided dice are rolled. What is the probability that the total is 5?Two fair six-sided dice are rolled. What is the probability that the total is 5?

     • A1121/12
     • B191/9
     • C5365/36
     • D161/6
     Show the answer and explanationHide the explanation

     Answer: 191/9

     Four of the 36 outcomes total 5 — (1,4), (2,3), (3,2) and (4,1) — so the probability is 436 = 19. Counting (1,4) and (4,1) as one outcome would halve it wrongly.Four of the 36 outcomes total 5 — (1,4), (2,3), (3,2) and (4,1) — so the probability is 4/36 = 1/9. Counting (1,4) and (4,1) as one outcome would halve it wrongly.

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108. 108Percent decreaseStandard

     A weekly figure fell from 640 to 544. What was the percentage decrease?A weekly figure fell from 640 to 544. What was the percentage decrease?

     • A17.6 percent17.6 percent
     • B96 percent96 percent
     • C15 percent15 percent
     • D12 percent12 percent
     Show the answer and explanationHide the explanation

     Answer: 15 percent15 percent

     The fall of 96 is measured against the starting figure: 96640 = 0.15. Dividing by 544 gives 17.6 percent, which is the rise needed to return.The fall of 96 is measured against the starting figure: 96/640 = 0.15. Dividing by 544 gives 17.6 percent, which is the rise needed to return.

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109. 109MeanStandard

     Seven numbers have a mean of 22. Six of them are 19, 25, 18, 27, 21 and 20. Enter the seventh.Seven numbers have a mean of 22. Six of them are 19, 25, 18, 27, 21 and 20. Enter the seventh.

     Student-produced response: write your own answer rather than choosing one.

     Show the answer and explanationHide the explanation

     Answer: 2424

     The seven must total 7 × 22 = 154. The six given add to 130, so the seventh is 154 - 130 = 24.The seven must total 7 × 22 = 154. The six given add to 130, so the seventh is 154 - 130 = 24.

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110. 110RatesStandard

     A coach covers 84 km in 1 hour and 12 minutes. What is its average speed in kilometers per hour?A coach covers 84 km in 1 hour and 12 minutes. What is its average speed in kilometers per hour?

     • A6262
     • B7575
     • C8484
     • D7070
     Show the answer and explanationHide the explanation

     Answer: 7070

     1 hour 12 minutes is 1.2 hours, and 841.2 = 70 km/h. Treating the 12 minutes as 0.12 of an hour gives about 75, which is the commonest conversion slip.1 hour 12 minutes is 1.2 hours, and 84/1.2 = 70 km/h. Treating the 12 minutes as 0.12 of an hour gives about 75, which is the commonest conversion slip.

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111. 111PercentagesChallenge

     An account of 5,000 pounds earns 4 percent interest a year, added to the balance. What is the balance after two years?An account of 5,000 pounds earns 4 percent interest a year, added to the balance. What is the balance after two years?

     • A5,408 pounds5,408 pounds
     • B5,400 pounds5,400 pounds
     • C5,200 pounds5,200 pounds
     • D5,416 pounds5,416 pounds
     Show the answer and explanationHide the explanation

     Answer: 5,408 pounds5,408 pounds

     Each year multiplies by 1.04: 5,000 × 1.04 × 1.04 = 5,408. Taking 4 percent of 5,000 twice gives 5,400 and misses the interest earned on the first year's interest.Each year multiplies by 1.04: 5,000 × 1.04 × 1.04 = 5,408. Taking 4 percent of 5,000 twice gives 5,400 and misses the interest earned on the first year's interest.

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112. 112Unit conversionStandard

     A bag of flour weighs 2.75 kg. Enter its mass in grams.A bag of flour weighs 2.75 kg. Enter its mass in grams.

     Student-produced response: write your own answer rather than choosing one.

     Show the answer and explanationHide the explanation

     Answer: 27502750

     One kilogram is 1,000 grams, so 2.75 kg is 2.75 × 1,000 = 2,750 g. Going to a smaller unit multiplies, so the number gets bigger, never smaller.One kilogram is 1,000 grams, so 2.75 kg is 2.75 × 1,000 = 2,750 g. Going to a smaller unit multiplies, so the number gets bigger, never smaller.

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