Arithmetic Reasoning test
20 questions at test level, the last 5 from earlier parts so you do not forget them. One at a time: pick an answer and read why. At the end you see your score and the lesson to review for every miss.
A rental shop charges a $56 delivery fee plus $28 a day for a pressure washer. Ben’s bill came to $392. For how many days did he rent it?
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A. Take the one-time fee out first: 392 − 56 = 336. That is what the days cost, so divide by the daily price: 336 ÷ 28 = 12 (check: 28 × 12 = 336). 14 forgets the fee and divides all of 392 by 28, 16 adds the fee instead of taking it away (448 ÷ 28), and 13 counts the delivery fee as one more day. Ben rented it for 12.
Lesson: Word problems
A school play sold 40 tickets for $395 in all. Adult tickets cost $12 and child tickets cost $7. How many adult tickets were sold?
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C. Pretend all 40 were child tickets: 40 × 7 = $280. The real total is 395 − 280 = $115 more. Each adult ticket in place of a child ticket adds 12 − 7 = $5, so 115 ÷ 5 = 23 adult tickets. Check: 23 × 12 = 276 and 17 × 7 = 119, and 276 + 119 = 395. 17 is the number of child tickets, 20 just splits the 40 tickets in half, and 33 divides 395 by 12 as if every ticket were an adult ticket (32.9, rounded up). Adult tickets sold: 23.
Lesson: Word problems
At a motor pool, the ratio of drivers to mechanics is 7 : 3. There are 21 drivers. How many drivers and mechanics are there in all?
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D. The 7 parts of drivers are 21 people, so one part is 21 ÷ 7 = 3 people. Mechanics are 3 parts: 3 × 3 = 9. In all, 21 + 9 = 30 (the same as 10 parts of 3 people each). 9 counts only the mechanics, 10 is just the number of parts in the ratio, and 24 adds the 3 from the ratio to the 21 drivers. In all there are 30.
Lesson: Ratios and proportions
A family’s monthly electric bill went from $250 in July to $210 in September. By what percent did the bill drop?
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A. The drop is 250 − 210 = $40. Compare it with where the bill started, $250: 1% of 250 is 2.50, and 40 ÷ 2.50 = 16, so the drop is 16%. 19% divides by the new bill, 210, 40% takes the $40 as if it were a percent, and 84% is the September bill as a percent of the July bill. The bill dropped 16%.
Lesson: Percent problems
After a 15% raise, Jo earns $23 an hour. What did she earn per hour before the raise?
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D. $23 is the old pay plus 15% of the old pay, so it is 115% of the old pay. Dividing by 1.15 by hand is slow, so test a choice: 10% of $20 is $2 and 5% is $1, so a 15% raise is $3, and 20 + 3 = 23. It works. $19.55 takes 15% of the new pay, $23, which is the wrong base, $3.45 is that 15% of $23 on its own, and $8.00 subtracts 15 dollars instead of 15 percent. Before the raise she earned $20.00.
Lesson: Percent problems
Ana borrows $3,600 at 6% simple interest for 18 months. How much does she pay back in all, interest included?
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D. The rate is per year, so change the time to years: 18 months is 1 1/2 years. One year of interest is 6% of 3,600: 1% is 36, so 6% is 6 × 36 = 216. Half a year more is half of 216, which is 108, so the interest is 216 + 108 = 324. She pays back the loan plus the interest: 3,600 + 324 = 3,924. $216 is one year of interest only, $324 is the interest without the loan, and $3,816 counts only one year. She pays back $3,924.
Lesson: Simple interest
A delivery van drives 1 hour at 30 mph through a city, then 150 miles at 50 mph on the highway. What is its average speed for the whole trip?
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C. Average speed is total distance ÷ total time. In the city, 1 hour at 30 mph is 30 miles. On the highway, 150 ÷ 50 = 3 hours. In all: 30 + 150 = 180 miles in 1 + 3 = 4 hours, and 180 ÷ 4 = 45. 40 mph just averages the two speeds, 37.5 mph leaves out the 30 city miles, and 60 mph leaves out the hour in the city. The average speed is 45 mph.
Lesson: Distance, rate and time
Kim can set up the chairs for a graduation in 12 hours. Lu can do it in 18 hours. They work together for 6 hours. What fraction of the job is still left?
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A. Add the rates, not the times. In 1 hour Kim does 1/12 of the job and Lu does 1/18. With a common denominator of 36 (36 ÷ 12 = 3 and 36 ÷ 18 = 2), that is 3/36 + 2/36 = 5/36 per hour. In 6 hours they do 6 × 5 = 30 of those parts, 30/36, which is 5/6. What is left is 1 − 5/6 = 1/6. 5/6 is the part already done, 31/36 is what is left after only one hour, and 3/5 adds the rates wrong (1/12 + 1/18 is not 2/30). Still left: 1/6.
Lesson: Shared work
Six boxes have an average weight of 14 pounds. Two more boxes are added, and the average of all eight becomes 15 pounds. What is the average weight of the two new boxes?
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B. Work with totals. Before: 6 × 14 = 84 pounds. After: 8 × 15 = 120 pounds. The two new boxes weigh 120 − 84 = 36 pounds together, so each one averages 36 ÷ 2 = 18. 36 pounds is both new boxes together, 15 pounds is the new average of all eight, and 19 pounds spreads the 1-pound rise over all 8 boxes instead of the 6 old ones (15 + 8 ÷ 2). The new boxes average 18 pounds.
Lesson: Averages, remainders and units
Six campers are 12, 15, 11, 15, 14 and 17 years old. What is the median age?
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C. Put the ages in order first: 11, 12, 14, 15, 15, 17. With six numbers there is no single middle number, so take the two in the middle, 14 and 15, and find the number halfway between them: (14 + 15) ÷ 2 = 14.5. 13 takes the middle of the list without sorting it (11 and 15), 14 is the mean (84 ÷ 6), and 15 is the mode. The median age is 14.5.
Lesson: Mean, median and mode
Ana puts $35 in a jar in week 1 and adds $15 every week after that. In which week does the jar first hold more than $300?
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C. Week 1 has $35, and each week after it adds $15. The jar must grow past 300 − 35 = 265 dollars above the start, and 265 ÷ 15 is about 17.7, so 17 additions are not enough and 18 are needed: 35 + 270 = 305 dollars. Those 18 additions come after week 1, so they end in week 19. Check week 18: it has only 17 additions, and 35 + 255 = 290, still under 300. Week 18 forgets that week 1 had no addition, Week 17 rounds 17.7 down and also forgets that week 1 had no addition, and Week 20 divides 300 by 15 and ignores the $35 start. The jar first holds more than $300 in Week 19.
Lesson: Number sequences
A table shows a family’s monthly spending: rent $1,350, food $540, car $360, utilities $270 and other $180. What percent of the total is food?
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A. First the total of every row: 1,350 + 540 + 360 + 270 + 180 = 2,700. Food is 540 of that 2,700. Simplify: 540 ÷ 2,700 is the same as 54 ÷ 270, which is 1/5, and 1/5 is 20%. 40% compares food with rent instead of with the total, 25% leaves food out of the total (540 ÷ 2,160), and 80% is everything except food. Food is 20%.
Lesson: Charts, graphs and tables
A cook has 30 cups of broth that is 4% salt. He lets water boil off until the broth is 6% salt. How many cups of water boil off?
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B. Boiling takes out water, never salt. The salt is 4% of 30 cups: 1% is 0.3, so 4% is 1.2 cups. At the end, those same 1.2 cups must be 6% of the broth: 1% of the broth is 1.2 ÷ 6 = 0.2 cup, so all of it is 100 × 0.2 = 20 cups. It started at 30, so 30 − 20 = 10 cups boiled off. 20 is the broth that is left, 2 just subtracts the two percents, and 15 works it as if water were added (45 cups − 30). The cups of water that boil off: 10.
Lesson: Mixture problems
A mother is 28 years older than her daughter. In 4 years, she will be 3 times as old as her daughter. How old is the daughter now?
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A. The gap between their ages never changes: it is 28 years now and in 4 years. When the mother is 3 times as old, she is the daughter’s age plus two more copies of it, so the 28-year gap is 2 times the daughter’s age at that time: 28 ÷ 2 = 14. That is her age in 4 years, so now she is 14 − 4 = 10. Check: 10 and 38 now; 14 and 42 in 4 years, and 3 × 14 = 42. 14 is her age in 4 years, 38 is the mother’s age now, and 16 adds the 4 years to the mother only. The daughter is 10.
Lesson: Age problems
A painter buys 3 gallons of paint at $34 each and 2 rollers at $8.50 each. Sales tax of 5% is added to everything. She pays with $150. How much change does she get?
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A. Step 1, the items: 3 × 34 = 102 and 2 × 8.50 = 17, so 102 + 17 = 119. Step 2, the tax: 10% of 119 is 11.90, and 5% is half of that, 5.95. The total is 119 + 5.95 = 124.95. Step 3, the change, which is what the last sentence asks for: 150 − 124.95 = 25.05. $31.00 forgets the tax, $25.90 puts the tax on the paint only, and $124.95 is what she pays, not her change. Her change is $25.05.
Lesson: Long problems, taken apart
A gardener plants 48 tomato plants and 72 pepper plants in rows. Each row has only one kind of plant, every row has the same number of plants, and the rows are as long as possible. How many rows are there?
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B. Rows as long as possible means the greatest common factor of 48 and 72. 48 is 2 × 24 and 72 is 3 × 24, and 2 and 3 share no factor, so the GCF is 24 plants per row. Tomatoes fill 48 ÷ 24 = 2 rows and peppers fill 72 ÷ 24 = 3 rows: 2 + 3 = 5. 24 is the length of a row, not the number of rows, 10 uses 12, a common factor but not the greatest one (4 rows plus 6 rows), and 2 counts only the tomato rows. The number of rows: 5.
Lesson: Primes, factors, GCF and LCM
A rug is 15 feet wide and 21 feet long. What is its area in square yards?
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B. Change each side to yards first, 3 feet to a yard: 15 ÷ 3 = 5 yards and 21 ÷ 3 = 7 yards. Area: 5 × 7 = 35 square yards. (Or 15 × 21 = 315 square feet, and one square yard is 3 × 3 = 9 square feet, so 315 ÷ 9 = 35.) 315 is the area in square feet, 105 divides the square feet by 3 instead of 9, and 12 adds the two sides instead of multiplying them. The area in square yards: 35.
Lesson: Units and measurement
A mine elevator starts 240 feet below ground, written −240 feet. It goes up 95 feet, then down 130 feet. Where is it now?
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B. Up is +, down is −. Start at −240. Going up 95 feet brings it closer to zero: −240 + 95 = −145. Going down 130 feet: −145 − 130 = −275. It is 275 feet below ground. −465 subtracts the 95 too, as if the elevator went down, −205 swaps the two moves, and 205 feet drops the minus sign from the start. It is now at −275 feet.
Lesson: Negative numbers
A shuttle bus has 40 seats, and 5/8 of them are taken. At the next stop, 7 more people sit down. What percent of the seats are taken now?
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D. First the people already seated: 1/8 of 40 is 5, so 5/8 is 5 × 5 = 25 seats. Add the new riders: 25 + 7 = 32 seats. As a percent of all 40 seats: 32 ÷ 40 = 0.8, which is 80%. 62.5% is the 5/8 from before the stop, 17.5% counts only the 7 new riders (7 ÷ 40), and 69.5% adds 7 percentage points instead of 7 people. Now taken: 80%.
Lesson: Fractions, decimals and percents
Rosa earns $15.60 an hour. She works 7.5 hours and then spends $23.45 on gas. How much of her day’s pay is left?
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B. Break 7.5 hours into 7 hours and a half hour. 7 × 15.60 = 109.20, and half of 15.60 is 7.80, so her pay is 109.20 + 7.80 = 117.00. Then take away the gas: 117.00 − 23.45 = 93.55. $85.75 forgets the half hour (109.20 − 23.45), $140.45 adds the gas instead of subtracting it, and $1,146.55 puts the decimal point one place off in her pay (1,170 instead of 117). She has left $93.55.
Lesson: Decimals