Brynn's savings decreased by 20% in July, then increased by 50% of the new amount in August. Brynn's savings are now what percent of the original amount?
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Answer: E — 120%.
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Hint 1 of 2
Don't pick a starting amount — that's extra work. Each percent change is something you can just multiply by.
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Hint 2 of 2
Each percent change is just a multiplier — you don't even need a starting amount. Multiply the two.
Casey went on a road trip that covered 100 miles, stopping only for a lunch break along the way. The trip took 3 hours in total and her average speed while driving was 40 miles per hour. In minutes, how long was the lunch break?
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Answer: B — 30 minutes.
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Hint 1 of 2
The 3 hours includes the break. Find the driving time first.
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Hint 2 of 2
Time = distance ÷ speed gives only the driving time. Whatever's left of the 3 hours is the break.
Mika wants to estimate how far a new electric bike goes on a full charge. She made two trips totaling 40 miles: the first used 12 of the battery and the second used 310 of the battery. How many miles can the bike go on a fully charged battery?
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Answer: C — 50 miles.
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Hint 1 of 2
Add the two battery fractions to see what share of a full charge covered the 40 miles.
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Hint 2 of 2
Then scale up from that share to a whole battery.
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Approach: scale up from the fraction used
The two trips used ½ + 3/10 = 4/5 of the battery for 40 miles.
A poll asked some people whether they liked solving mathematics problems, and exactly 74% answered "yes." What is the fewest possible number of people who could have been asked?
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Answer: D — 50 people.
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Hint 1 of 2
74% of the group must be a whole number of people.
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Hint 2 of 2
Reduce 74/100 to lowest terms; the denominator is the smallest possible group size.
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Approach: reduce the percentage to lowest terms
74% = 74/100 = 37/50 in lowest terms, so the number of people must be a multiple of 50.
Five runners finished a race: Luke, Melina, Nico, Olympia, and Pedro. Nico finished 11 minutes behind Pedro. Olympia finished 2 minutes ahead of Melina but 3 minutes behind Pedro. Olympia finished 6 minutes ahead of Luke. Which runner finished fourth?
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Answer: A — Luke.
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Hint 1 of 2
Place everyone on a timeline measured from Pedro.
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Hint 2 of 2
Then read off who is fourth from the front.
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Approach: place everyone relative to Pedro
Measuring minutes behind Pedro: Olympia +3, Melina +5 (2 behind Olympia), Luke +9 (6 behind Olympia), Nico +11.
The order is Pedro, Olympia, Melina, Luke, Nico, so fourth is Luke.
Jami picked three equally spaced integers on the number line. The sum of the first and second is 40, and the sum of the second and third is 60. What is the sum of all three numbers?
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Answer: B — 75.
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Hint 1 of 2
For equally spaced numbers, the middle one is the average of the other two.
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Hint 2 of 2
Add the two given sums and see how many times the middle number appears.
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Approach: the middle number carries everything
Since the numbers are equally spaced, first + third = 2 × second. Adding the two given sums: 40 + 60 = first + 2·second + third = 4·second, so the middle is 25.
A cube's two shaded faces share an edge, so both must be glued to neighbors to hide them.
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Hint 2 of 2
Look for an arrangement where every cube has two glued faces that meet at an edge.
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Approach: every cube needs two adjacent glued faces
To hide a cube's two shaded faces — which meet at an edge — it must be glued to neighbors on two faces sharing an edge.
Four cubes arranged in a 2 × 2 square give each cube exactly two such adjacent glued faces; with three or fewer, some cube has only one glued face (or two opposite ones), so a shaded face shows.
Consider all positive four-digit integers whose digits are all even. What fraction of these integers are divisible by 4?
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Answer: D — 3/5.
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Hint 1 of 2
Divisibility by 4 depends only on the last two digits.
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Hint 2 of 2
With an even tens digit, the tens part is already a multiple of 4, so only the units digit decides it.
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Approach: reduce divisibility by 4 to the units digit
A number is divisible by 4 exactly when its last two digits are. With an even tens digit, 10·(tens) is already a multiple of 4, so it comes down to the units digit being 0, 4, or 8.
That's 3 of the 5 even units digits, and it holds for every choice of the other digits, so the fraction is 3/5.
Four students sit in a row and chat with the people next to them. They then rearrange themselves so that no one is seated next to anyone they sat next to before. How many such rearrangements are possible?
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Answer: A — 2.
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Hint 1 of 2
Label the seats 1, 2, 3, 4; the forbidden neighbor-pairs are 1-2, 2-3, and 3-4.
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Hint 2 of 2
Try to build a valid order — the options turn out to be very tight.
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Approach: avoid all three original neighbor-pairs
No two originally-adjacent students may be neighbors, so none of 1-2, 2-3, 3-4 can touch.
The only orders that work are 2-4-1-3 and its reverse 3-1-4-2, giving 2 rearrangements.
Miguel and his dog Luna start together at a park entrance. Miguel throws a ball straight ahead to a tree and keeps walking at a steady pace. Luna sprints to the ball and immediately brings it back to Miguel. Luna runs 5 times as fast as Miguel walks. What fraction of the entrance-to-tree distance does Miguel cover by the time Luna brings him the ball?
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Answer: D — 1/3.
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Hint 1 of 2
In the same time, Luna covers 5 times the distance Miguel does.
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Hint 2 of 2
Luna's path is entrance → tree → Miguel; set its length to 5 times Miguel's walk.
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Approach: equal time means Luna's path is 5× Miguel's
Let the entrance-to-tree distance be 1 and Miguel's distance be d. Luna runs to the tree and back to Miguel: 1 + (1 − d) = 2 − d.
Since Luna covers 5 times Miguel's distance, 5d = 2 − d, so 6d = 2 and d = 1/3.
The land of Catania uses gold coins (1 mm thick) and silver coins (3 mm thick). In how many ways can Taylor make a stack exactly 8 mm tall using any arrangement of gold and silver coins, where order matters?
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Answer: D — 13.
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Hint 1 of 2
Let f(n) count stacks of height n; the top coin is either gold (leaving n − 1) or silver (leaving n − 3).
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Hint 2 of 2
Build the counts up from small heights using f(n) = f(n−1) + f(n−3).
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Approach: count by the top coin: f(n) = f(n−1) + f(n−3)
