Problem-Solving & Data Analysis practice questions
Rates, percentages, probability, and the centre and spread of a data set. The arithmetic is rarely hard; the work is deciding which quantity the question actually asks for.
Problem-Solving & Data Analysis questions
- 1Percent changeFoundation
A quantity increases from 48 to 60. What is the percent increase?
- A125%
- B30%
- C20%
- D25%
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Answer: 25%
The increase is 12. Divide by the original amount: 12 / 48 = 0.25, or 25%.
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What is the mean of 4, 6, 7, 9, and 14?
- A8
- B10
- C7
- D9
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Answer: 8
The sum is 40. Divide by 5 values to get a mean of 8.
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A bag has 3 red, 5 blue, and 2 green counters. What is the probability of choosing a counter that is not blue?
- A1/5
- B4/5
- C3/5
- D1/2
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Answer: 1/2
There are 5 counters that are not blue out of 10 total. That is 5/10, which simplifies to 1/2.
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A car travels 180 miles in 3 hours at a constant rate. Enter its speed in miles per hour.
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Answer: 60
Rate equals distance divided by time: 180 / 3 = 60 miles per hour.
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What is the median of 2, 5, 6, 10, and 12?
- A5
- B8
- C6
- D7
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Answer: 6
The values are already ordered. The middle value is 6.
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A jacket is discounted by 20% and then costs $72. What was its original price?
- A$86.40
- B$92
- C$80
- D$90
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Answer: $90
After a 20% discount, the price is 80% of the original. 72 / 0.8 = 90.
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In one group, 12 of 30 students joined a club. In another, 18 of 45 joined. Which statement is true?
- AThere is not enough information.
- BBoth groups have the same percentage.
- CThe second group has the greater percentage.
- DThe first group has the greater percentage.
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Answer: Both groups have the same percentage.
12 / 30 = 0.4 and 18 / 45 = 0.4, so both groups have 40% participation.
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What is the range of 4, 7, 7, 9, and 13?
- A7
- B6
- C9
- D13
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Answer: 9
Range is largest minus smallest: 13 - 4 = 9.
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A table has pairs (1, 5), (2, 8), and (3, 11). By how much does y change when x increases by 1?
- A5
- B2
- C3
- D1
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Answer: 3
The y-values increase by 3 each time: 5 to 8 to 11.
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A class has 20 students and 5 are absent. What percent of the class is present?
- A80%
- B50%
- C25%
- D75%
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Answer: 75%
15 of 20 students are present. 15 / 20 = 0.75, or 75%.
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What is the median of 3, 4, 6, 8, 10, and 13?
- A6
- B9
- C8
- D7
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Answer: 7
With six values, take the mean of the two middle values: (6 + 8) / 2 = 7.
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A printer produces 45 pages in 3 minutes at a constant rate. Enter the number of pages it produces in 8 minutes.
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Answer: 120
The rate is 45 / 3 = 15 pages per minute, and 15 x 8 = 120 pages.
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A store raises a $50 price by 10 percent and then lowers the new price by 10 percent. What is the final price?
- A$49.50
- B$45.00
- C$50.50
- D$50.00
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Answer: $49.50
The increase gives 50 x 1.10 = $55. The decrease is taken from $55, not $50: 55 x 0.90 = $49.50. The two percentages apply to different amounts, so they do not cancel.
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Five numbers have a mean of 12. Enter their sum.
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Answer: 60
The mean is the sum divided by the count, so the sum is 12 x 5 = 60.
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A spinner has 8 equal sections, 3 of which are shaded. What is the probability of landing on an unshaded section?
- A5/3
- B3/8
- C5/8
- D1/8
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Answer: 5/8
There are 8 - 3 = 5 unshaded sections out of 8 equally likely outcomes, so the probability is 5/8.
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Of 40 surveyed students, 24 ride the bus, and 10 of those bus riders also sometimes walk. What fraction of the bus riders sometimes walk?
- A1/4
- B5/24
- C5/12
- D3/5
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Answer: 5/12
The question asks for a fraction of bus riders, not of all students: 10 / 24, which reduces to 5/12.
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The list 4, 8, 9, 11, 13 gains a sixth number, which raises the mean of the list to 10. Enter the sixth number.
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Answer: 15
The five numbers sum to 45. Six numbers with a mean of 10 sum to 60, so the new number is 60 - 45 = 15.
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A map uses 1 inch to represent 25 miles. Enter the number of miles represented by 3.5 inches.
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Answer: 87.5
Multiply the scale by the measured length: 25 x 3.5 = 87.5 miles.
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After a 25 percent increase, a membership costs $80. What was the original price?
- A$100
- B$64
- C$60
- D$65
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Answer: $64
The new price is 1.25 times the original, so the original is 80 / 1.25 = $64. Subtracting 25 percent of $80 would be a different, incorrect calculation.
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A data set of home prices in one area includes a few homes that sold for far more than the rest. Which measure best represents a typical price in that area?
- AThe mean
- BThe median
- CThe maximum
- DThe range
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Answer: The median
A few very large values pull the mean upward, while the median depends on position rather than size and is not moved much by extreme values.
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One tap fills a tank in 6 minutes and a second tap fills the same tank in 12 minutes. Enter the number of minutes the two taps take to fill it together.
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Answer: 4
In one minute the taps fill 1/6 and 1/12 of the tank, so together they fill 1/6 + 1/12 = 3/12 = 1/4 of the tank per minute. Filling the whole tank takes 4 minutes.
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Of 250 survey responses, 60 selected the first of four listed choices. What percent selected it?
- A24%
- B40%
- C20%
- D30%
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Answer: 24%
Divide and convert: 60 / 250 = 0.24, which is 24 percent.
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A class of 10 students averaged 80 on a test and a class of 15 students averaged 90. Enter the average for all 25 students.
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Answer: 86
Averages cannot simply be averaged when the groups differ in size. Total points are 10 x 80 + 15 x 90 = 800 + 1350 = 2150, and 2150 / 25 = 86.
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Two data sets have the same mean. Set A is 48, 49, 50, 51, 52 and Set B is 30, 40, 50, 60, 70. Which statement is true?
- ASet B has the greater standard deviation.
- BSet A has the greater standard deviation.
- CThe two have equal standard deviations.
- DStandard deviation cannot be compared without more information.
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Answer: Set B has the greater standard deviation.
Standard deviation measures spread around the mean. Both sets centre on 50, but Set B's values sit much further from it, so its standard deviation is larger. No calculation is needed to see this.
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A tank holds 3.5 litres. If 1 litre is about 1000 millilitres, enter the capacity in millilitres.
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Answer: 3500
Multiply by the conversion factor: 3.5 x 1000 = 3500 millilitres.
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A scatterplot of study hours against test score has a line of best fit with a positive slope of 4. What does the slope indicate?
- AStudents who study score 4 points in total.
- BEach additional hour of study is associated with about 4 more points.
- CFour students studied for the test.
- DStudy time causes scores to rise by exactly 4 points.
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Answer: Each additional hour of study is associated with about 4 more points.
The slope of a line of best fit gives the average change in the predicted y-value per unit increase in x. It describes an association, not a guaranteed or causal effect.
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A drawer holds 4 blue socks and 6 grey socks. One sock is taken out and kept. What is the probability that the first two socks taken are both grey?
- A9/25
- B3/5
- C1/3
- D2/5
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Answer: 1/3
The first grey sock has probability 6/10. With that sock removed, 5 of the remaining 9 are grey, so the second has probability 5/9. Multiplying gives 30/90 = 1/3.
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A shop's takings rose from $2,400 to $2,760. What was the percent increase?
- A13%
- B36%
- C15%
- D8.7%
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Answer: 15%
The increase is 360. Divide by the original amount, not the new one: 360 / 2400 = 0.15, which is 15 percent.
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A cyclist rides 12 kilometres at 24 km/h and then 12 kilometres at 12 km/h. What is the average speed for the whole ride?
- A20 km/h
- B16 km/h
- C15 km/h
- D18 km/h
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Answer: 16 km/h
Average speed is total distance over total time, not the average of the speeds. The legs take 0.5 and 1 hour, so 24 kilometres in 1.5 hours gives 16 km/h. The tempting 18 km/h is the average of the two speeds.
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Enter the median of 7, 3, 11, 5, and 9.
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Answer: 7
Order the values first: 3, 5, 7, 9, 11. With five values the median is the third, which is 7.
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A researcher surveys shoppers leaving one supermarket on a Tuesday morning and concludes that most people in the city shop mid-week. What is the main problem with this conclusion?
- AThe researcher should have surveyed on more than one Tuesday.
- BSupermarkets are not a valid place to conduct surveys.
- CThe sample size is too small to calculate a percentage.
- DThe sample only includes people who already shop at that time.
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Answer: The sample only includes people who already shop at that time.
Everyone surveyed was, by construction, shopping mid-week, so the sample cannot say anything about people who shop at other times. This is a selection problem, not a size problem.
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Of 120 respondents, 45 own a bicycle and 18 of those also own a car. What percent of bicycle owners also own a car?
- A15%
- B26.7%
- C40%
- D37.5%
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Answer: 40%
The denominator is bicycle owners, not all respondents: 18 / 45 = 0.40, which is 40 percent. Dividing by 120 answers a different question.
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A population falls by 20 percent and then rises by 25 percent. What is the net change?
- AA 5 percent fall
- BA 45 percent rise
- CA 5 percent rise
- DNo net change
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Answer: No net change
Multiply rather than add: 0.80 x 1.25 = 1.00, so the population returns to its starting value. The percentages apply to different amounts, which is why they do not simply combine.
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A box holds 5 red, 3 green and 2 blue tokens. What is the probability that a token drawn at random is not red?
- A1/2
- B2/5
- C3/10
- D5/10
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Answer: 1/2
Five of the ten tokens are not red, so the probability is 5/10 = 1/2. Working with the complement is quicker than adding the green and blue cases separately, though both give the same answer.
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A pump moves 4,200 litres in 3.5 hours. Enter its rate in litres per hour.
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Answer: 1200
Divide the volume by the time: 4200 / 3.5 = 1200 litres per hour.
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Six numbers have a mean of 14. If the largest number is removed, the mean of the remaining five is 12. What was the largest number?
- A24
- B18
- C26
- D20
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Answer: 24
Six numbers at a mean of 14 sum to 84; five at a mean of 12 sum to 60. The removed value is the difference, 84 - 60 = 24.
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A recipe for 6 servings uses 300 grams of rice. How much rice is needed for 9 servings?
- A450 grams
- B350 grams
- C400 grams
- D500 grams
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Answer: 450 grams
The rate is 50 grams per serving, and 50 x 9 = 450 grams.
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Adding the value 12 to the data set 10, 11, 12, 13, 14 will change which of the following?
- ABoth the mean and the median
- BNeither the mean nor the median
- CThe mean only
- DThe median only
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Answer: Neither the mean nor the median
The set already has mean and median 12, and adding another 12 leaves both at 12. Adding a value equal to the centre moves neither measure.
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After a 15 percent discount, a coat costs $102. What was the price before the discount?
- A$117
- B$120
- C$125
- D$115
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Answer: $120
The discounted price is 85 percent of the original, so the original is 102 / 0.85 = $120. Adding 15 percent to $102 gives $117.30, which answers a different question.
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A council emails a survey about a proposed cycle lane to everyone on its cycling newsletter list. What is the main limitation of the result?
- AThe sample is too small to reach a conclusion.
- BEmail surveys always have low response rates.
- CThe council should have used a paper survey instead.
- DThe people surveyed are more likely than the general public to favour the lane.
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Answer: The people surveyed are more likely than the general public to favour the lane.
Everyone asked has already signalled an interest in cycling, so the sample is not representative of the population the decision affects. That is a selection problem, and no response rate or sample size fixes it.
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A test has 45 questions. A student answers 36 correctly. What percentage did they answer correctly?
- A75%
- B85%
- C90%
- D80%
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Answer: 80%
36 / 45 = 0.80, which is 80 percent.
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Two fair coins are tossed. What is the probability that at least one shows heads?
- A3/4
- B2/3
- C1/2
- D1/4
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Answer: 3/4
The four equally likely outcomes are HH, HT, TH, TT. Three contain at least one head. Working with the complement is faster: the only outcome without a head is TT, so 1 - 1/4 = 3/4.
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A train travels 210 kilometres in 2 hours 30 minutes. Enter its average speed in kilometres per hour.
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Answer: 84
Convert the time to hours first: 2 hours 30 minutes is 2.5 hours. Then 210 / 2.5 = 84 kilometres per hour.
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A student's four test scores average 78. What score on a fifth test would raise the average to 80?
- A86
- B90
- C88
- D82
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Answer: 88
Four scores at 78 total 312. Five scores at 80 total 400. The fifth score must be 400 - 312 = 88.
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Of 80 members, 50 attend the evening session and 30 of those also attend the morning one. What fraction of evening attendees also attend in the morning?
- A3/5
- B3/8
- C3/10
- D5/8
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Answer: 3/5
The denominator is evening attendees: 30 / 50 = 3/5. Dividing by all 80 members answers a different question.
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A data set has mean 50 and every value is increased by 10. What happens to the mean and the standard deviation?
- AThe mean is unchanged; the standard deviation rises by 10.
- BThe mean rises to 60; the standard deviation falls.
- CBoth rise by 10.
- DThe mean rises to 60; the standard deviation is unchanged.
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Answer: The mean rises to 60; the standard deviation is unchanged.
Adding a constant shifts every value equally, so the centre moves but the distances between values do not change. Standard deviation measures spread, which a shift leaves alone.
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A subscription rises from $12 to $15 a month. What is the percent increase?
- A25%
- B3%
- C20%
- D30%
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Answer: 25%
The increase is $3. Divide by the original price, not the new one: 3 / 12 = 0.25, which is 25 percent.
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A scatterplot of rainfall against crop yield has a line of best fit with a negative slope. What does that indicate?
- AThere is no relationship between rainfall and yield.
- BRainfall causes crop yield to fall.
- CHigher rainfall is associated with lower yield in this data.
- DThe data contains an error.
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Answer: Higher rainfall is associated with lower yield in this data.
A negative slope describes the direction of an association within the data shown. It does not establish a cause, and a real association is not an error.
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A jacket costs $84 after a 30 percent discount. What was the original price?
- A$120
- B$114
- C$112
- D$109.20
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Answer: $120
The sale price is 70 percent of the original, so the original is 84 / 0.70 = $120. Adding 30 percent to $84 gives $109.20, which answers a different question.
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A bag holds 4 white and 6 black marbles. Two are drawn without replacement. What is the probability both are white?
- A1/5
- B2/15
- C8/45
- D4/25
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Answer: 2/15
The first is white with probability 4/10. With it removed, 3 of the remaining 9 are white, so the second is 3/9. Multiplying gives 12/90 = 2/15.
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A cyclist covers 45 kilometres at 15 km/h and returns along the same road at 30 km/h. Enter the average speed for the whole journey in km/h.
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Answer: 20
Average speed is total distance over total time. The legs take 3 hours and 1.5 hours, so 90 kilometres in 4.5 hours gives 20 km/h. Averaging the two speeds would give 22.5, which is wrong because more time is spent at the slower speed.
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The mean of five numbers is 20. Two of them are 14 and 26. What is the mean of the other three?
- A18
- B20
- C24
- D22
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Answer: 20
The five total 100, and 14 + 26 = 40, so the other three total 60 and average 20. The two removed happen to average 20 themselves, which is why the rest are unaffected.
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Of 200 respondents, 120 own a pet and 42 of those own more than one. What percentage of pet owners own more than one?
- A60%
- B35%
- C58%
- D21%
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Answer: 35%
The denominator is pet owners: 42 / 120 = 0.35, which is 35 percent. Dividing by all 200 gives 21 percent.
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Every value in a data set is multiplied by 3. What happens to the mean and the standard deviation?
- AThe mean is unchanged; the standard deviation is multiplied by 3.
- BThe mean is multiplied by 3; the standard deviation is unchanged.
- CBoth are multiplied by 9.
- DBoth are multiplied by 3.
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Answer: Both are multiplied by 3.
Multiplying scales the values and the distances between them equally, so both the centre and the spread are multiplied by 3. Adding a constant would move only the mean.
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Attendance fell from 250 to 200. What was the percent decrease?
- A80%
- B20%
- C25%
- D50%
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Answer: 20%
The decrease is 50. Divide by the original figure: 50 / 250 = 0.20, which is 20 percent. Dividing by 200 gives 25 percent, which measures the wrong thing.
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A researcher wants to estimate the average commute time in a city and surveys people waiting at a train station at 8 a.m. What is the main limitation?
- ACommute times cannot be averaged.
- BTrain stations are too noisy for accurate surveys.
- CPeople who commute by other means, or at other times, are excluded.
- DThe sample size will inevitably be too small.
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Answer: People who commute by other means, or at other times, are excluded.
Everyone surveyed commutes by train at peak time, so the sample cannot represent drivers, cyclists or off-peak travellers. That is a coverage problem no sample size fixes.
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