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?
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?
Show answer
Answer: B — 10 coins.
Show hints
Hint 1 of 2
Greedy: start with pennies (need 4 to cover 1–4 cents), then nickels, dimes, quarters — one of each value up to the next.
Still stuck? Show hint 2 →
Hint 2 of 2
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.
Show solution
Approach: build up the reachable range coin by coin
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?
Show answer
Answer: D — 15 minutes.
Show hint
Hint 1
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
Approach: relative-speed reasoning
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?
Show answer
Answer: D — 84%.
Show hint
Hint 1
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?
Show answer
Answer: B — 2/3.
Show hint
Hint 1
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
Approach: ratio of pepperoni area to pizza area
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?
Show answer
Answer: B — 64 feet.
Show hint
Hint 1
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?
Show answer
Answer: D — 100 red balls.
Show hint
Hint 1
Blue balls don't change. 75% red ⇒ 25% blue, so the 100 blue balls represent 25% of the new total.
Show solution
Approach: blue stays constant, so use blue to find new total
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?
Show answer
Answer: E — 11 inches.
Show hint
Hint 1
Let the smallest side be s. Then perimeter = 3s + 3. Set s = 0.3 · (3s + 3).
Show solution
Approach: translate the percent condition
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?
Show answer
Answer: C — 42.
Show hints
Hint 1 of 2
Green = 100% − 30 − 20 − 15 − 10 = 25%, so 30 gumdrops = 25% ⇒ total = 120.
Still stuck? Show hint 2 →
Hint 2 of 2
Brown starts at 20% · 120 = 24. Add half the blue gumdrops (which switch color).
Show solution
Approach: find the total, then update brown
Green % = 25 ⇒ total = 30 / 0.25 = 120.
Blue count: 30% · 120 = 36. Brown count: 20% · 120 = 24.
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?
Show answer
Answer: A — 3.
Show hints
Hint 1 of 2
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?
Show answer
Answer: C — 240 pages.
Show hints
Hint 1 of 2
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.
Still stuck? Show hint 2 →
Hint 2 of 2
Repeat the inversion for day 2 and day 1.
Show solution
Approach: work backwards day by day
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?
Show answer
Answer: E — 8.
Show hints
Hint 1 of 2
Let units = u, tens = t, hundreds = u + 2. Original − reversed simplifies to a constant.
Still stuck? Show hint 2 →
Hint 2 of 2
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?
Show answer
Answer: E — 24 ways.
Show hints
Hint 1 of 2
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).
