At Euclid Middle School the mathematics teachers are Miss Germain, Mr. Newton, and Mrs. Young. There are 11 students in Mrs. Germain's class, 8 students in Mr. Newton's class, and 9 students in Mrs. Young's class taking the AMC 8 this year. How many mathematics students at Euclid Middle School are taking the contest?
The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price?
Alice needs to replace a light bulb located 10 centimeters below the ceiling in her kitchen. The ceiling is 2.4 meters above the floor. Alice is 1.5 meters tall and can reach 46 centimeters above the top of her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters?
Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar?
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Greedy: start with pennies (need 4 to cover 1–4 cents), then nickels, dimes, quarters — one of each value up to the next.
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After 4 pennies + 1 nickel, you can pay anything up to 9 cents. Each new coin should extend the reachable range as much as possible.
As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction 1/2 mile in front of her. After she passes him, she can see him in her rear mirror until he is 1/2 mile behind her. Emily rides at a constant rate of 12 miles per hour, and Emerson skates at a constant rate of 8 miles per hour. For how many minutes can Emily see Emerson?
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Use the gap-closing speed: Emily − Emerson = 4 mph. She sees him while the gap shrinks from 1/2 ahead to 1/2 behind — a total relative shift of 1 mile.
Show solution
Relative speed: 12 − 8 = 4 mph.
Emily must cover 1 mile of relative displacement (from 1/2 ahead to 1/2 behind).
Ryan got 80% of the problems correct on a 25-problem test, 90% on a 40-problem test, and 70% on a 10-problem test. What percent of all the problems did Ryan answer correctly?
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Count correct on each test, then divide by the total number of problems.
Six pepperoni circles will exactly fit across the diameter of a 12-inch pizza when placed. If a total of 24 circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered by pepperoni?
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Each pepperoni has diameter 12/6 = 2, so its area is π, compared to the pizza's 36π. Each pepperoni is 1/36 of the pizza.
Show solution
Pepperoni radius: 1. Pizza radius: 6. Area ratio: (1/6)2 = 1/36 per pepperoni.
The top of one tree is 16 feet higher than the top of another tree. The heights of the two trees are in the ratio 3 : 4. In feet, how tall is the taller tree?
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The ratio 3:4 means the difference (1 part) corresponds to 16 feet. So 1 part = 16 ft.
Of the 500 balls in a large bag, 80% are red and the rest are blue. How many of the red balls must be removed from the bag so that 75% of the remaining balls are red?
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Blue balls don't change. 75% red ⇒ 25% blue, so the 100 blue balls represent 25% of the new total.
Show solution
Initial: 400 red, 100 blue.
After removal, 25% blue means total = 100 / 0.25 = 400 balls.
The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shortest side is 30% of the perimeter. What is the length of the longest side?
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Let the smallest side be s. Then perimeter = 3s + 3. Set s = 0.3 · (3s + 3).
Show solution
Sides: s, s+1, s+2. Perimeter: 3s + 3.
s = 0.3(3s + 3) ⇒ s = 0.9s + 0.9 ⇒ 0.1s = 0.9 ⇒ s = 9.
A jar contains five different colors of gumdrops: 30% are blue, 20% are brown, 15% red, 10% yellow, and the other 30 gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown?
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Green = 100% − 30 − 20 − 15 − 10 = 25%, so 30 gumdrops = 25% ⇒ total = 120.
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Brown starts at 20% · 120 = 24. Add half the blue gumdrops (which switch color).
Show solution
Green % = 25 ⇒ total = 30 / 0.25 = 120.
Blue count: 30% · 120 = 36. Brown count: 20% · 120 = 24.
The diagram shows an octagon consisting of 10 unit squares. The portion below PQ is a unit square and a triangle with base 5. If PQ bisects the area of the octagon, what is the ratio XQQY?
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Total area = 10, so each half = 5. Below PQ: a unit square + a triangle of base 5. So the triangle has area 4 and base 5.
A decorative window is made up of a rectangle with semicircles on either end. The ratio of AD to AB is 3 : 2, and AB is 30 inches. What is the ratio of the area of the rectangle to the combined areas of the semicircles?
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Two semicircles of diameter 30 combine to a single circle of diameter 30 ⇒ area 225π.
The two circles pictured have the same center C. Chord AD is tangent to the inner circle at B, AC is 10, and chord AD has length 16. What is the area between the two circles?
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CB ⊥ AD at the tangent point and B bisects AD, so AB = 8.
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Annulus area = π(AC2 − CB2) = π · AB2 by Pythagoras.
Show solution
Tangent ⇒ CB ⊥ AD at B, so B is the midpoint of AD: AB = 8.
In a room, 2/5 of the people are wearing gloves, and 3/4 of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove?
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Total people must be divisible by 5 and 4 ⇒ multiple of 20. Smallest is 20.
Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read 1/5 of the pages plus 12 more, and on the second day she read 1/4 of the remaining pages plus 15 pages. On the third day she read 1/3 of the remaining pages plus 18 pages. She then realized that there were only 62 pages left to read, which she read the next day. How many pages are in this book?
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Work backwards from the 62 pages left. Day 3 left her with 62 pages after reading (1/3)R + 18 from R; so 2R/3 − 18 = 62.
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Repeat the inversion for day 2 and day 1.
Show solution
After day 3 there are 62 pages left. If R3 = pages at start of day 3: (2/3)R3 − 18 = 62 ⇒ R3 = 120.
The hundreds digit of a three-digit number is 2 more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?
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Let units = u, tens = t, hundreds = u + 2. Original − reversed simplifies to a constant.
Show hint (sharpest)
Reversing flips the hundreds and units digits. Difference = 99 · (hundreds − units).
Everyday at school, Jo climbs a flight of 6 stairs. Jo can take the stairs 1, 2, or 3 at a time. For example, Jo could climb 3, then 1, then 2. In how many ways can Jo climb the stairs?
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Let f(n) be the number of ways to climb n stairs. Each climb ends in a 1, 2, or 3 step: f(n) = f(n−1) + f(n−2) + f(n−3).