Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way?
A sign at the fish market says, "50% off, today only: half-pound packages for just $3 per package." What is the regular price for a full pound of fish, in dollars? (Assume that there are no deals for bulk.)
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First scale half-pound to full-pound; then undo the 50% discount.
Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill?
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The 7 friends together covered Judi's share: 7 × $2.50. That's one-eighth of the total.
Show solution
Judi's share = 7 × $2.50 = $17.50.
Everyone paid the same, so total = 8 × $17.50 = $140.
Hammie is in the 6th grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?
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Order the 5 weights to find the median. Mean = total / 5. The single outlier (106) pulls the mean way up.
The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, 30 = 6 × 5. What is the missing number in the top row?
Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?
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Set up a proportion: 6 cars per 10 sec. Convert 2 min 45 sec to seconds.
The Incredible Hulk can double the distance it jumps with each succeeding jump. If its first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will it first be able to jump more than 1 kilometer (1,000 meters)?
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Jump n is 2n−1 meters. Find the smallest n with 2n−1 > 1000.
Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?
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Time for 2 miles at rate r is 2/r hours. Convert each to minutes and compare with the all-4-mph plan.
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Monday: (2/5) hr = 24 min. Wednesday: (2/3) hr = 40 min. Friday: (2/4) hr = 30 min. Total: 94 min.
At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?
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Compute the dollar saved on the 2nd and 3rd pairs (the 1st is full price). Then divide by $150.
Show solution
2nd pair saves 40% of $50 = $20. 3rd pair saves half of $50 = $25.
Number Theoryplace-value-differencedivisibility-by-9
When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?
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If a score's last two digits are 10a + b, reversing gives 10b + a. The difference is 9(a − b): always a multiple of 9.
Show solution
(10a + b) − (10b + a) = 9(a − b).
The sum's error must therefore be a multiple of 9.
Abe holds 1 green and 1 red jelly bean in his hand. Bob holds 1 green, 1 yellow, and 2 red jelly beans in his hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?
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Match in two ways: both green, or both red. Multiply each person's color probability per case, then add.
A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of 8th-graders to 6th-graders is 5 : 3, and the ratio of 8th-graders to 7th-graders is 8 : 5. What is the smallest number of students that could be participating in the project?
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Both ratios link to the 8th-graders. The number of 8th-graders must be a multiple of 5 (first ratio) and 8 (second ratio).
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Smallest such number is lcm(5, 8) = 40.
Show solution
8th-graders divisible by both 5 and 8 ⇒ smallest is 40.
Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?
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Outer block-volume = 12 · 10 · 5. Subtract the hollow inside. Inside shrinks by 1 in each wall direction and 1 in the floor (no ceiling).
Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest score?
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Both Cassie and Bridget saw Hannah's score. Why can each be sure of her own ranking only after seeing Hannah's?
Show solution
Cassie's claim is only certain if her score > Hannah's (otherwise Hannah's lower score below Cassie's could be the lowest, and Cassie wouldn't know). So Cassie > Hannah.
Bridget's claim is only certain if her score < Hannah's. So Hannah > Bridget.
Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?
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Three independent legs: home → SW corner, the unique diagonal through the park, NE corner → school. Count lattice paths for each leg and multiply.
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Home to SW corner: 2 E + 1 N. NE corner to school: 2 E + 2 N.
Show solution
Home → SW corner: choose 1 of 3 step-orderings = C(3, 1) = 3.
Angle ABC of ▵ABC is a right angle. The sides of ▵ABC are the diameters of semicircles as shown. The area of the semicircle on AB equals 8π, and the arc of the semicircle on AC has length 8.5π. What is the radius of the semicircle on BC?
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Recover AB from the semicircle area, and AC from the semicircle arc length. Then use the Pythagorean theorem to find BC, then halve it.
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Semicircle area = (1/2)πr2. Semicircle arc length = πr. So radius is straightforward from each.
Show solution
Semicircle on AB: (1/2)πr2 = 8π ⇒ r = 4 ⇒ AB = 8.
Semicircle on AC: πr = 8.5π ⇒ r = 8.5 ⇒ AC = 17.
Right angle at B: BC = √(172 − 82) = √225 = 15. (8-15-17 triple.)
Squares ABCD, EFGH, and GHIJ are equal in area. Points C and D are the midpoints of sides IH and HE, respectively. What is the ratio of the area of the shaded pentagon AJICB to the sum of the areas of the three squares?
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Set each square's side to 1. Place coordinates: F = (0, 0), E = (0, 1), G = (1, 0), H = (1, 1), J = (2, 0), I = (2, 1).
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Then D = (0.5, 1), C = (1.5, 1), A = (0.5, 2), B = (1.5, 2). Use the shoelace formula on pentagon AJICB.
Show solution
Side length 1. Vertices: A = (0.5, 2), J = (2, 0), I = (2, 1), C = (1.5, 1), B = (1.5, 2).
A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are R1 = 100 inches, R2 = 60 inches, and R3 = 80 inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?
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The ball's center traces a path parallel to the track at distance = ball's radius (2 inches). On the outside of an arc the center's arc has radius R − 2; on the inside it has radius R + 2.