Averages and Weighted Thinking Lesson Test

Answer: B

Correct answer: 8. Add the values:

Divide by the count:

oil changes per day.

• A: This is the middle value when the data are sorted, which is the median, not the mean.
• B (correct): The five values add to 40, and 40 divided by 5 is 8.
• C: This is halfway between the smallest and largest values, which ignores the three middle values.
• D: This comes from dividing the total of 40 by 4 instead of by the 5 values.

Answer: D

Correct answer: 96 minutes. Turn the average into a total:

• A: This multiplies 12 by 7, using one call too few.
• B: This multiplies 12 by 9, using one call too many.
• C: This adds the count and the average instead of multiplying them.
• D (correct): Average times count gives the total: 8 times 12 is 96 minutes.

Answer: B

Correct answer: 15 pounds. Required total:

Known total:

Fifth package:

• A: This assumes the missing package equals the target average, but the known weights already total 55, not 56.
• B (correct): The five must total 70 pounds; the four known weights total 55, leaving 15.
• C: This is the average of the four known weights, not the missing weight.
• D: This uses a required total of 4 times 14, or 56, forgetting that there are five packages.

Answer: C

Correct answer: 82. Weighted average:

• A: This divides the weighted sum by 2; when percent weights total 100, no extra division is needed.
• B: This is the simple average of 70 and 90, which ignores that the practical test counts more.
• C (correct): Multiplying each score by its weight gives 28 + 54 = 82.
• D: This swaps the weights, giving 40 percent to the 90 and 60 percent to the 70.

Answer: A

Correct answer: 3 liters. Pure antifreeze = concentration times volume:

• A (correct): Pure amount equals concentration times volume: 0.20 times 15 is 3 liters.
• B: This divides 15 by 20 instead of multiplying 15 by 0.20.
• C: This treats 20 percent as 2 instead of 0.20, a decimal slip that gives more antifreeze than the whole solution holds.
• D: This is the amount of water in the solution, the other 80 percent.

Answer: D

Correct answer: 34. Totals:

Combined:

deliveries. The result is closer to 30 because that group is larger.

• A: This divides the combined total of 680 by 2 groups instead of by 20 drivers.
• B: This attaches each average to the wrong group size, multiplying 12 by 40 and 8 by 30.
• C: This averages the two averages, ignoring that the 30-delivery group is larger.
• D (correct): The group totals are 360 and 320, and 680 divided by 20 drivers is 34.

Answer: D

Correct answer: 7.6 hours. Weight each value by how often it occurs:

• A: This swaps the counts, treating 10 hours as occurring on 3 days.
• B: This averages 6 and 10 as if each happened on the same number of days.
• C: This divides the 38 total hours by 2 instead of by the 5 days worked.
• D (correct): Using the day counts as weights, 18 + 20 = 38 hours over 5 days is 7.6 hours.

Answer: C

Correct answer: $6.80. Total cost:

Price per pound: $68 divided by 10 pounds = $6.80.

• A: This swaps the quantities, pricing 4 pounds at $8 and 6 pounds at $5.
• B: This averages the two prices, ignoring that there are more pounds of the pricier nut.
• C (correct): Total cost is $20 + $48 = $68 for 10 pounds, which is $6.80 per pound.
• D: This adds the two prices per pound instead of dividing total cost by total weight.

Answer: A

Correct answer: $600. Needed total:

Current total:

Sixth day:

• A (correct): Six days at $450 require $2,700; the first five total $2,100, so day six needs $600.
• B: This finds only how far the first five days fall short of $450 each and forgets the sixth day must also reach $450.
• C: This assumes the sixth day just matches the target average, which would leave the overall average below $450.
• D: This adds only $30 to the target, but each of the first five days fell $30 short, so the sixth day must make up $150 above $450.

Answer: D

Correct answer: 36 miles per hour. Times:

Average speed:

per hour.

• A: This averages the two speeds, ignoring that more time was spent at 20 miles per hour.
• B: This divides 180 miles by 2 legs instead of by the 5 hours of driving.
• C: This uses the equal-distance shortcut, but the two legs are not the same length.
• D (correct): The trip is 180 miles in 2 + 3 = 5 hours, and 180 divided by 5 is 36.

Answer: A

Correct answer: 27 percent. Pure amounts:

L and

Total pure: 5.4 L in 20 L, and

or 27 percent.

• A (correct): The pure salt is 0.6 + 4.8 = 5.4 liters in 20 liters, which is 27 percent.
• B: This adds the two concentrations; a mixture can never be stronger than its strongest part.
• C: This averages 12 and 32 percent, ignoring that there is three times as much of the stronger solution.
• D: This swaps the volumes, treating the weaker solution as the 15-liter one.

Answer: C

Correct answer: 88. Known parts:

The final must supply

points, so

and F = 88.

• A: This assumes the final only needs to match the target, but the quiz score pulls the average down.
• B: This treats the three parts as equal weights and subtracts from 3 times 85, ignoring that the final counts double.
• C (correct): Homework and quizzes contribute 22 + 19 = 41 points, so the final must contribute 44, and 44 divided by 0.50 is 88.
• D: This is the number of points the final must contribute; it still has to be divided by the 0.50 weight.

Answer: C

Correct answer: 45 miles per hour. Total time allowed:

Outbound time:

Return: 90 miles in 2 hours, or

per hour.

• A: This assumes the average of the two speeds must be 36, but average speed is total distance over total time.
• B: This divides 90 miles by the full 5 hours, forgetting that 3 hours were already used on the way out.
• C (correct): The round trip must take 180 divided by 36, or 5 hours; 3 hours are used going out, so 90 miles in 2 hours is 45 miles per hour.
• D: Driving back at 36 would keep the overall average below 36, because the outbound leg was slower.

Answer: B

Correct answer: 5 liters. Let x be the liters added. Pure coolant:

Then 0.2x = 1 and x = 5 liters.

• A: This forgets that the added liquid also increases the total volume; 2 liters would give only 25 percent.
• B (correct): With 5 liters added, the pure coolant is 2 + 2.5 = 4.5 liters in 15 liters, which is 30 percent.
• C: This stops at the extra liter of pure coolant the mix needs instead of solving for the volume of solution to add.
• D: Equal volumes would give the midpoint, 35 percent, which is stronger than needed.

Answer: A

Correct answer: 5. Let n be the evening count:

giving 10n = 50 and n = 5 workers.

• A (correct): With 5 evening workers, the total is 500 + 325 = 825 units for 15 workers, an average of 55.
• B: This is the total number of workers after the shifts are combined, not the size of the evening shift.
• C: Equal shift sizes would put the combined average at 57.5, halfway between 50 and 65.
• D: This flips the balance: the larger group sits closer to the combined average, so the evening group must be smaller, not larger.

Answer: D

Correct answer: 6 liters. Water adds volume but no cleaner, so the pure amount stays at

At 30 percent, the total volume must be

Water to add:

• A: 18 liters is the total volume the 30 percent solution must have, not the amount of water to add.
• B: This multiplies 12 by the 15-point drop in percent, which does not find the volume that holds 5.4 liters at 30 percent.
• C: Adding 15 liters gives 5.4 liters of cleaner in 27 liters, which is only 20 percent.
• D (correct): The 5.4 liters of pure cleaner must make up 30 percent of the new volume, so the total is 18 liters and 6 liters of water are added.

Answer: D

Correct answer: 72 degrees. Add the values:

Divide by the number of days:

• A: 71 is the middle value when the temperatures are put in order, which is the median, not the mean.
• B: This divides the total of 360 by 4 instead of by the 5 days.
• C: 72.5 is halfway between the lowest and highest readings, which ignores the three middle values.
• D (correct): The five readings add to 360, and 360 divided by 5 is 72.

Answer: A

Correct answer: 161 pounds. Turn the average into a total by multiplying by the count:

• A (correct): Average times count gives the total: 7 times 23 is 161 pounds.
• B: This adds the count and the average instead of multiplying them.
• C: This multiplies 23 by 8, using one crate too many.
• D: This comes from a multiplication slip: 7 times 20 is 140, but the 3 must also be multiplied by 7.

Answer: D

Correct answer: 165. Needed total:

Current total:

Fifth game:

• A: Scoring the target average would only work if the first four games already averaged 160, but they average 158.75.
• B: 158.75 is the average of the first four games, not the score needed in the fifth.
• C: This uses a total of 4 times 160 instead of 5 times 160, forgetting to count the new game.
• D (correct): Five games averaging 160 must total 800, and 800 minus the 635 already bowled leaves 165.

Answer: B

Correct answer: 82. Work with totals. All 40 recruits:

The platoon of 16:

The other 24 recruits total

so their average is

• A: This averages the averages, assuming 78 is halfway between 72 and the unknown, which ignores that the groups have different sizes.
• B (correct): The other recruits' total is 3,120 minus 1,152, or 1,968, and 1,968 divided by 24 is 82.
• C: 75 is halfway between 72 and 78; it does not use the group totals at all.
• D: This swaps the group sizes, multiplying 72 by 24 and dividing the remainder by 16.

Answer: A

Correct answer: 15 gallons. Add the amounts:

Divide by the 4 weeks:

• A (correct): The four amounts add to 60 gallons, and 60 divided by 4 is 15.
• B: 60 gallons is the total for all four weeks, not the mean.
• C: This divides the total of 60 by 5 instead of by the 4 weeks.
• D: 14.5 is halfway between the smallest and largest amounts, which ignores the other two weeks.

Answer: B

Correct answer: 76. Find each group's total:

Combined mean:

It lies closer to 72 because that group is larger.

• A: 77 is the simple average of 72 and 82, which treats the groups as if they were the same size.
• B (correct): The totals are 1,296 and 984, which add to 2,280, and 2,280 divided by 30 is 76.
• C: This gives the larger weight to the 82 group, mixing up which group has 18 sailors.
• D: 2,280 is the combined total of all the scores; it still has to be divided by 30.

Answer: D

Correct answer: 81. Multiply each score by its weight and add:

The weights total 100 percent, so no further division is needed.

• A: 80 is the simple average of the three scores, which ignores that the run counts twice as much as each other event.
• B: This puts the 50 percent weight on the push-up score instead of on the run.
• C: 84 is just the run score; the other two events still count for half of the grade.
• D (correct): Weighting gives 42 + 23 + 16, which is 81.

Answer: C

Correct answer: 37 inches. Required total:

Known boards:

Third board:

• A: The third board does not have to equal the average; the known boards do not balance around 40.
• B: 41.5 is the average of the two known boards, not the length of the missing one.
• C (correct): The three boards must total 120 inches, and 120 minus 83 leaves 37.
• D: 83 inches is the sum of the two known boards; it still has to be subtracted from the 120-inch total.

Answer: A

Correct answer: 14.4 miles per hour. Times:

Average speed = total distance over total time:

per hour.

• A (correct): The trip is 72 miles in 3 + 2 = 5 hours, and 72 divided by 5 is 14.4.
• B: 15 is the simple average of the two speeds, but the boat spends more time at the slower speed.
• C: This divides only the one-way distance of 36 miles by the total time.
• D: This divides the total distance by the average of the two times instead of the total time.

Answer: B

Correct answer: 55 miles per hour. Distance:

in 2 hours. Average speed:

per hour.

When equal time is spent at each speed, the simple average of the speeds gives the same answer.

• A: 110 is the total distance in miles; it must be divided by the 2 hours.
• B (correct): Equal times were spent at each speed, so the simple average works: 110 miles in 2 hours is 55.
• C: This treats the legs as equal distances; here the times are equal, so the speeds are not weighted differently.
• D: 65 is only the speed of the second hour, not the average for the whole drive.

Answer: C

Correct answer: 50 percent. Pure amounts:

gallon and

for 6 gallons in 12 gallons.

Concentration:

or 50 percent.

• A: 40 percent averages the two concentrations without weighting by volume; there is three times as much of the 60 percent solution.
• B: This gives the larger volume to the 20 percent solution, reversing the amounts.
• C (correct): The pure amounts are 0.6 and 5.4 gallons, giving 6 gallons in 12, which is 50 percent.
• D: This adds the two percentages, but a mixture cannot be stronger than its strongest part.

Answer: A

Correct answer: $4.00. Total cost:

Total weight: 8 pounds.

Average price: $32 divided by 8 is $4.00 per pound.

• A (correct): The blend costs $18 + $14 = $32 for 8 pounds, which is $4.00 per pound.
• B: $5.00 averages the two prices without weighting; there is three times as much of the cheaper seed.
• C: This multiplies the 6 pounds by the $7 price, matching the amounts with the wrong prices.
• D: $32.00 is the total cost of the blend, not the cost per pound.

Answer: B

Correct answer: 43.5. Each count acts as a weight:

Divide by the total count:

• A: 45 is the average of the three different scores, ignoring how many soldiers earned each.
• B (correct): The weighted total is 200 + 135 + 100 = 435, and 435 divided by 10 is 43.5.
• C: This divides the total of 435 by the 3 score levels instead of by the 10 soldiers.
• D: This matches the counts with the wrong scores, giving the most weight to 50 instead of 40.

Answer: A

Correct answer: 60. Total for 8:

Total for 7:

Removed score:

A low score was removed, which is why the average went up.

• A (correct): The 8 scores total 592 and the remaining 7 total 532, so the removed score is 60.
• B: This assumes the removed score is 2 points below 74, but removing one score must shift the whole total, not just the average.
• C: This moves 2 points in the wrong direction; removing a high score would lower the average, not raise it.
• D: This multiplies both averages by 8, forgetting that only 7 trainees remain.

Answer: C

Correct answer: 3.375. Multiply each grade by its credits:

The weights are credits, not percents, so divide by their sum:

• A: 3.0 is the simple average of 4.0, 3.0, and 2.0, which ignores that the courses carry different credits.
• B: This divides the 27 weighted points by the 3 courses instead of by the 8 credits.
• C (correct): The weighted points are 16 + 9 + 2 = 27, and 27 divided by 8 credits is 3.375.
• D: This pairs the 4 credits with the C and the 1 credit with the A, reversing the weights.

Answer: C

Correct answer: 12 miles per hour. Times:

Average speed:

per hour, below the simple average because more time is spent at 10 miles per hour.

• A: 12.5 averages the two speeds, but she spends 3 hours at the slower speed and only 2 at the faster one.
• B: This divides one 30-mile leg by the total time instead of the full 60 miles.
• C (correct): She rides 60 miles in 2 + 3 = 5 hours, and 60 divided by 5 is 12.
• D: This adds the two speeds instead of finding total distance over total time.

Answer: B

Correct answer: 6 gallons. Write the percent as a decimal and multiply by the volume:

• A: This divides 40 by 15 instead of multiplying 40 by 0.15.
• B (correct): Pure amount equals concentration times volume: 0.15 times 40 is 6 gallons.
• C: This uses 1.5 instead of 0.15; the pure amount cannot exceed the 40 gallons in the tank.
• D: 34 gallons is the amount of the tank that is not fertilizer.

Answer: A

Correct answer: 24 percent. Pure amount:

Water adds none, and the total volume becomes 15 liters.

Concentration:

or 24 percent.

• A (correct): The 3.6 liters of disinfectant are now spread through 15 liters, and 3.6 divided by 15 is 24 percent.
• B: This averages 40 percent and 0 percent as if the volumes were equal, but there is more of the 40 percent solution.
• C: Adding water does not change the amount of disinfectant, but it does increase the volume, so the strength drops.
• D: This weights 40 percent by the 6 liters of water instead of by the 9 liters of solution.

Answer: B

Correct answer: 60 pounds. Weight each value by its count:

Divide by the 10 shipments: 60 pounds.

• A: 62.5 averages 50 and 75 without accounting for there being more 50-pound shipments.
• B (correct): The total weight is 300 + 300 = 600 pounds, and 600 divided by 10 is 60.
• C: 600 pounds is the total weight of all 10 shipments, not the average.
• D: This multiplies 6 by 75 and 4 by 50, matching the counts with the wrong weights.

Answer: C

Correct answer: 202. Differences from 200: +3, −3, +5, −1, +6, which sum to +10.

Average difference:

Mean:

• A: This adds the full sum of the differences, +10, to the baseline without dividing it by 5.
• B: 203 is the middle value when the counts are put in order, which is the median, not the mean.
• C (correct): The differences from 200 are +3, −3, +5, −1, and +6, which total +10; 10 divided by 5 is +2, so the mean is 202.
• D: 200 is only the baseline; the differences above it outweigh those below it.

Answer: A

Correct answer: 48 miles per hour. Distances:

for 240 miles.

Total time including the stop:

Average speed:

per hour.

• A (correct): The convoy covers 180 + 60 = 240 miles in 3 + 0.5 + 1.5 = 5 hours, and 240 divided by 5 is 48.
• B: 50 averages the two driving speeds, ignoring both the different times and the stop.
• C: This divides 240 miles by only the 4.5 driving hours, leaving out the 30-minute stop the question says to count.
• D: This averages 60, 0, and 40 as if the stop and both legs lasted the same amount of time.

Answer: B

Correct answer: $14.60. Group totals:

for $730.

Average wage: $730 divided by 50 is $14.60 per hour.

• A: $14.00 averages the three wage rates without weighting by how many laborers earn each.
• B (correct): The hourly totals are $120 + $210 + $400 = $730, and $730 divided by 50 is $14.60.
• C: This divides the $730 total by the 3 wage groups instead of by the 50 laborers.
• D: This pairs the 25 laborers with $12 and the 10 laborers with $16, reversing the weights.

Answer: C

Correct answer: 26 years. Crew total:

Four members:

Other two:

so their average is

• A: This assumes the other two are as far above 22 as the four are below it, ignoring the different group sizes.
• B: 52 years is the combined age of the other two members; it still has to be divided by 2.
• C (correct): The crew totals 132 years and the four total 80, leaving 52 years for 2 people, or 26 each.
• D: 21 is the average of 22 and 20, which does not use the totals at all.

Answer: B

Correct answer: $3.40. Total cost:

for 20 gallons.

Average price: $68.00 divided by 20 is $3.40 per gallon.

• A: $3.45 averages the two prices, but more gallons were bought at the lower price.
• B (correct): The total cost is $38.40 + $29.60 = $68.00 for 20 gallons, which is $3.40 per gallon.
• C: This multiplies 12 gallons by the higher price, matching the amounts with the wrong prices.
• D: $68.00 is the total cost of the gas, not the price per gallon.

Answer: D

Correct answer: 81. Multiply each score by its weight and add:

The weights are counts, so divide by their sum,

• A: 80 is the simple average of the three scores, which ignores that the final counts three times.
• B: This gives the weight of 3 to homework and 1 to the final, reversing the weights.
• C: This divides the weighted total of 486 by 3 scores instead of by the sum of the weights, 6.
• D (correct): The weighted total is 80 + 148 + 258 = 486, and 486 divided by the total weight of 6 is 81.

Answer: C

Correct answer: 5 pounds. Let x be the pounds of $10 tea. Total cost must equal $6 times the total weight:

So

giving

• A: This reverses the ratio; with 20 pounds added the blend would cost $240 for 30 pounds, or $8 per pound.
• B: Equal amounts would give a blend at $7 per pound, the plain average of the two prices.
• C (correct): With 5 pounds added, the blend costs $40 + $50 = $90 for 15 pounds, which is $6 per pound.
• D: This uses the $2 gap between $4 and $6 as the number of pounds; the blend would cost $60 for 12 pounds, or $5 per pound.

Answer: A

Correct answer: 21 miles. Turn each average into a total: 5 days

first 3

last 3

The two 3-day totals together count day 3 twice:

Day 3:

• A (correct): The first 3 days total 45 and the last 3 total 66, which is 111; the 5 days total only 90, so day 3, counted twice, is 21.
• B: 18 is the overall average; nothing requires the middle day to equal it.
• C: 7 is the difference between the two 3-day averages, which does not use the totals.
• D: 24 is 90 minus the last-three total of 66, which is the sum of days 1 and 2, not day 3.

Answer: C

Correct answer: 500 miles. The first leg takes

If d is the second distance, its time is $d÷250$.

So

Check: 800 miles in 4 hours is 200 miles per hour.

• A: Equal distances put more time at the slower speed: 600 miles in 3.2 hours averages only 187.5 miles per hour.
• B: Flying 200 more miles gives 500 miles in 2.8 hours, about 178.6 miles per hour.
• C (correct): Then she flies 800 miles in 2 + 2 = 4 hours, and 800 divided by 4 is 200 miles per hour.
• D: Flying 400 more miles gives 700 miles in 3.6 hours, about 194.4 miles per hour, still short of 200.

Answer: D

Correct answer: 36 percent. Pure acid:

Total volume: 10 liters. Concentration:

or 36 percent.

• A: 30 percent averages the three concentrations without weighting by volume; most of the mixture is the 50 percent solution.
• B: This pairs the 5 liters with 10 percent and the 2 liters with 50 percent, reversing the volumes.
• C: 3.6 is the number of liters of pure acid; it still has to be divided by the 10-liter total.
• D (correct): The pure amounts are 0.2 + 0.9 + 2.5 = 3.6 liters in 10 liters, which is 36 percent.

Answer: D

Correct answer: 73. Total of 8 scores:

Total of the middle 6:

Highest plus lowest:

Lowest:

• A: This assumes the highest and lowest average 82, which ignores that the other six average 81.
• B: 170 is the sum of the highest and lowest scores, not the lowest score alone.
• C: This assumes the highest and lowest average 81, the same as the remaining six.
• D (correct): The 8 scores total 656 and the 6 total 486, so the two dropped scores total 170; 170 minus 97 is 73.

Answer: C

Correct answer: 153 pounds. Differences from 150: −2, +3, +6, −1, +2, +10, which sum to +18.

Average difference:

Mean:

• A: This adds the whole +18 sum of differences to 150 without dividing by the 6 pallets.
• B: 152.5 is the median, halfway between the two middle weights, not the mean.
• C (correct): The differences from 150 are −2, +3, +6, −1, +2, and +10, totaling +18; 18 divided by 6 is +3, so the mean is 153.
• D: 154 is halfway between the lightest and heaviest pallets, which ignores the four other weights.

Answer: A

Correct answer: 51 miles per hour. Distances:

for 255 miles.

Average speed:

per hour. The time spent at each speed acts as its weight.

• A (correct): The convoy covers 90 + 165 = 255 miles in 5 hours, and 255 divided by 5 is 51.
• B: 50 averages the two speeds, but more time was spent at 55 miles per hour.
• C: This gives the 3 hours to the 45-mile-per-hour leg, reversing the times.
• D: 255 is the total distance in miles; it still has to be divided by the 5 hours.

Answer: D

Correct answer: $49.00. Add the tips:

Divide by the 4 shifts: $196 divided by 4 is $49.00.

• A: $196.00 is the total of the four shifts, not the mean.
• B: This divides the $196 total by 3 instead of by the 4 shifts.
• C: $49.50 is halfway between the smallest and largest tips, which ignores the other two shifts.
• D (correct): The tips add to $196, and $196 divided by 4 shifts is $49.00.

Answer: D

Correct answer: 84.4. Multiply each score by its weight and add:

The weights total 100 percent, so no further division is needed.

• A: 87 is the simple average of the three scores, which ignores that quality counts for half the review.
• B: This puts the 50 percent weight on attendance and the 20 percent weight on quality, swapping them.
• C: This adds the three scores without applying the weights at all.
• D (correct): Weighting gives 19 + 39 + 26.4, which is 84.4.