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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Add the two battery fractions to see what share of a full charge covered the 40 miles.
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Then scale up from that share to a whole battery.
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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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74% of the group must be a whole number of people.
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Reduce 74/100 to lowest terms; the denominator is the smallest possible group size.
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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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Place everyone on a timeline measured from Pedro.
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Then read off who is fourth from the front.
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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.
Sekou writes down the numbers 15, 16, 17, 18, 19. After he erases one of his numbers, the sum of the remaining four numbers is a multiple of 4. Which number did he erase?
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Adding the five up and testing each removal is slow. What does each number have in common with 4?
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Look at each number's remainder when divided by 4. The remainder of the whole sum tells you which one to erase.
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Remainders mod 4: 15→3, 16→0, 17→1, 18→2, 19→3. Their sum is 9, which leaves remainder 1 mod 4.
To make the remaining four sum divisible by 4, erase the one whose remainder is 1 — that's 17.
When two shapes overlap, adding their areas counts the overlap twice. What's the shape of that overlap?
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Inclusion–exclusion: (area of one) + (area of the other) − (area of overlap). The overlap is a square of side 2.5.
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Each rectangle has area 5 × 3 = 15.
Rotated 90° about the midpoint of DC, the second rectangle's lower-left quarter overlaps the first rectangle's lower-right quarter — a 2.5 by 2.5 square (half of DC = 2.5), area 2.52 = 6.25.
Use the plain oval P as a baseline. Which paths cut corners (shorter) and which add diagonals (longer)?
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R uses straight shortcuts across the rounded ends — shorter than P. S and Q swap edges for diagonals; a diagonal is a hypotenuse, always longer than the legs it replaces.
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R cuts the rounded ends with straight chords, so R is strictly shorter than P. Shortest.
S adds one diagonal X inside the oval — the diagonal is a hypotenuse, longer than the legs P would take. So S > P.
The 2×2 and 1×4 tiles each cover 4 squares. What does that tell you about the number of 1×1 tiles needed?
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21 squares total, and the 2×2 + 1×4 tiles fill a multiple of 4. So the 1×1 count is 1, 5, 9, … Try the smallest that actually fits.
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Each 2×2 and 1×4 tile covers 4 unit squares, so together they fill a multiple of 4. The 1×1 count must equal 21 − (multiple of 4) ≡ 1 (mod 4): possible values 1, 5, 9, …
Can we cover 20 of the 21 squares with 2×2 and 1×4 tiles, leaving just one 1×1? Trying shows it doesn't fit (a 3×7 rectangle with one cell removed can't be partitioned into 2×2's and 1×4's).
5 works: place four 2×2 and 1×4 tiles covering 16 squares, with the 5 remaining cells filled by 1×1's. Minimum = 5.
On Monday Taye has $2. Every day, he either gains $3 or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, 3 days later?
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Build a tree: from each daily amount, branch on +$3 or ×2. Some branches collide — just list the unique values.
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Tuesday: {$4, $5}. Wednesday: {$7, $8, $10} (the $8 from two paths). Thursday will land on 6 distinct values.
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From $2 each day choose +$3 or ×2. Tuesday: 2+3=5 or 2×2=4 → {$4, $5}.
All of the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles and twice as many blue marbles as green marbles. Which of the following could be the total number of marbles in Maria's collection?
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Pick one color to stand for a variable, then write the others in terms of it. The total reveals a hidden factor.
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Let r = red count. Green = 2r, blue = 4r. Total = 7r — it must be a multiple of 7.
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Let r be the number of red marbles. Half as many red as green → green = 2r. Twice as many blue as green → blue = 4r.
Total = r + 2r + 4r = 7r — always a multiple of 7.
Among the choices, only 28 = 7 × 4 is a multiple of 7.
In January 1980 the Mauna Loa Observatory recorded carbon dioxide CO2 levels of 338 ppm (parts per million). Over the years the average CO2 reading has increased by about 1.515 ppm each year. What is the expected CO2 level in ppm in January 2030? Round your answer to the nearest integer.
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Multiply the per-year rate by the number of years, then add to the starting value.
Each line has a simple equation. Plug in x = 15 and x = 16 (the rectangle's left and right sides) and see whether the resulting y falls between 3 and 5.
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Line AB: y = x/3. Line CD: y = 10 − x/2. Test each at x = 15 and 16.
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Rectangle: x ∈ [15, 16], y ∈ [3, 5].
Line AB: y = x/3. At x = 15: y = 5 ✓ (hits the corner (15, 5)). At x = 16: y ≈ 5.33, above the rectangle.
Line CD: y = 10 − x/2. At x = 15: y = 2.5, below. At x = 16: y = 2, below.
Harold made a plum pie to take on a picnic. He was able to eat only 14 of the pie, and he left the rest for his friends. A moose came by and ate 13 of what Harold left behind. After that, a porcupine ate 13 of what the moose left behind. How much of the original pie still remained after the porcupine left?
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Each eater leaves behind a fraction of what they found. Multiply those leftovers together.
