Number Sense
What is 3/4 of 28?
21
Divide 28 by 4, then multiply by 3: 7 × 3 = 21.
If 3x + 5 = 20, what is x?
5
Subtract 5 to get 3x = 15, then divide by 3.
How many prime numbers are greater than 10 and less than 20?
4
The primes are 11, 13, 17, and 19.
A rectangle is 8 cm long and 5 cm wide. What is its perimeter, in centimeters?
26 centimeters
Perimeter = 2(8 + 5) = 26.
Maya has 3 different shirts and 2 different pairs of pants. How many outfits can she make using one shirt and one pair of pants?
6 outfits
Each of 3 shirts can be paired with either of 2 pairs of pants: 3 × 2 = 6.
What is the sum 1 + 2 + 3 + ... + 20?
210
Pair 1 with 20, 2 with 19, and so on. There are 10 pairs, each with sum 21: 10 × 21 = 210.
The mean of five numbers is 12. Four of the numbers have a sum of 43. What is the fifth number?
17
All five numbers sum to 5 × 12 = 60. The missing number is 60 − 43 = 17.
What is the greatest common factor of 24 and 36?
12
24 = 2 × 12 and 36 = 3 × 12. No larger positive integer divides both.
A triangle has a base of 12 cm and a perpendicular height of 7 cm. What is its area, in square centimeters?
42 square centimeters
Area = (1/2) × 12 × 7 = 42.
A coin is flipped 4 times. How many different sequences of heads and tails are possible?
16 sequences
There are 2 choices for each flip: 2 × 2 × 2 × 2 = 16. Order matters.
A jacket costs $60. Its price is reduced by 20%, then the reduced price is increased by 10%. What is the final price?
$52.80
After the discount: $60 × 0.80 = $48. Then increase $48 by 10%: $48 × 1.10 = $52.80.
A jar contains 8 creatures: beetles with 6 legs each and spiders with 8 legs each. There are 54 legs in total. How many spiders are in the jar?
3 spiders
If all 8 were beetles, there would be 48 legs. Each spider adds 2 extra legs, so (54 − 48) ÷ 2 = 3.
Three distinct prime numbers have a sum of 16. What is their product?
66
Three odd primes would have an odd sum, so one prime must be 2. The other two distinct primes must sum to 14: they are 3 and 11. Product = 2 × 3 × 11 = 66.
A right triangle has legs of lengths 20 cm and 21 cm. What is the length of its hypotenuse, in centimeters?
29 centimeters
By the Pythagorean theorem, c² = 20² + 21² = 400 + 441 = 841 = 29².
How many different arrangements can be made using each of the letters A, B, C, D, and E exactly once?
120 arrangements
There are 5 choices for the first letter, then 4, 3, 2, and 1: 5 × 4 × 3 × 2 × 1 = 120.
Express 0.123123123... as a fraction in simplest form. The block 123 repeats forever.
41/333
Let x = 0.123123... . Then 1000x − x = 123, so x = 123/999 = 41/333.
The ratio of red marbles to blue marbles in a bag is 3:5. After 6 red marbles are added, the ratio is 3:4. How many blue marbles are in the bag?
40 blue marbles
Use 20 equal units for blue. Red changes from 12 units to 15 units. Those 3 units equal 6 marbles, so each unit is 2 marbles. Blue = 20 × 2 = 40.
What is the units digit of 7^2026?
9
The units digits of powers of 7 repeat 7, 9, 3, 1. Since 2026 has remainder 2 when divided by 4, use the second digit: 9.
A square has side length 10 cm. Connecting the midpoints of consecutive sides forms a smaller square. What is the area of the smaller square, in square centimeters?
50 square centimeters
The original area is 100. Remove four corner triangles, each with area (1/2) × 5 × 5 = 12.5. Remaining area = 100 − 4 × 12.5 = 50.
Two fair six-sided dice, one red and one blue, are rolled. What is the probability that their sum is 8? Express your answer as a fraction in simplest form.
5/36
There are 36 equally likely ordered outcomes. The 5 favorable outcomes are (2,6), (3,5), (4,4), (5,3), and (6,2).
What is the value of (1 − 1/2)(1 − 1/3)(1 − 1/4) ... (1 − 1/2025)? Express your answer as a fraction in simplest form.
1/2025
Rewrite the product as (1/2)(2/3)(3/4) ... (2024/2025). All intermediate factors cancel, leaving 1/2025.
Alice and Bob start at opposite ends of a straight path at the same time and run toward each other at constant speeds. After they meet, Alice needs 9 more minutes to reach Bob's starting point, and Bob needs 16 more minutes to reach Alice's starting point. How many minutes after starting do they meet?
12 minutes
Let t be the meeting time. From the two path segments, Alice's speed / Bob's speed = 16/t = t/9. Thus t² = 16 × 9 = 144, so t = 12.
How many integers from 1 through 100, inclusive, are relatively prime to 100? Relatively prime means their greatest common factor with 100 is 1.
40
Exclude multiples of 2 or 5. There are 50 multiples of 2, 20 multiples of 5, and 10 counted in both groups. Count = 100 − 50 − 20 + 10 = 40.
A rectangle has all four vertices on a circle of radius 6 cm. What is the greatest possible area of the rectangle, in square centimeters?
72 square centimeters
The rectangle's diagonal is the diameter, 12. If its sides are a and b, then a² + b² = 144. Since (a − b)² ≥ 0, 2ab ≤ 144. The maximum area is 72, achieved by a square.
How many ways can 9 identical cookies be distributed among Alice, Bob, and Claire if each person receives at least 1 cookie and no cookies are left over?
28 ways
If Alice gets 1 cookie, Bob and Claire can split the rest in 7 positive ways. For Alice getting 2, 3, ..., 7 cookies, there are 6, 5, ..., 1 ways. Total = 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28.