Section A
Multiple Choice Questions — 15 Questions
(+3)
1
measurement
easy
A caterpillar crawled on top of a ruler. The caterpillar's head is about 2 centimeters above the top of the ruler's edge. What is the length of the whole caterpillar, in centimeters, from head to tail?
Answer: B
The horizontal part extends from about 1 cm to 9 cm, a length of about 8 cm. The raised part is about 2 cm long. The total length is about 8 + 2 = 10 cm.
2
number-sense
medium
There are three whole numbers: A, B, and C. A is less than B; B is less than C. If A + B + C = 25 and A = 2, what is the largest possible value of C?
Answer: B
To make C as large as possible, make B the smallest whole number greater than A = 2, so B = 3. Then C = 25 − 2 − 3 = 20.
3
number-line
easy
There is a number that is halfway between 3 and 12 on the number line. What is the value of that number?
Answer: C
The halfway value is the average: (3 + 12) ÷ 2 = 15 ÷ 2 = 7 1/2.
4
fractions
easy
Jason had saved some money. Earlier today, he received an additional 12 dollars. He then went to a bookstore and spent one third of all his money on a book. Now he has 18 dollars left. How much money did Jason spend on the book?
Answer: C
After buying the book, 2/3 of Jason's money remained. If 2/3 is 18 dollars, the total before buying the book was 27 dollars. He spent 1/3 of 27, which is 9 dollars.
5
geometry-area
easy
A square has an area of four square centimeters. It is divided into two equal triangles along one of its diagonals. What is the area of each triangle in square centimeters?
Answer: C
A diagonal divides a square into two equal triangles. Each triangle has half of 4 square centimeters, so its area is 2 square centimeters.
6
fractions
easy
Which of the following numbers has the smallest value?
Compare 3/2, 2/3, 4/3, and 5/4.
Answer: D
The values are 5/4 = 1.25, 4/3 ≈ 1.33, 3/2 = 1.5, and 2/3 ≈ 0.67. Therefore 2/3 is the smallest.
7
counting-triangles
medium
A triangle is divided into smaller triangles. How many triangles of all sizes are there?
Answer: D
There are 9 smallest triangles, 3 triangles made from four small triangles, and 1 whole triangle. The total is 9 + 3 + 1 = 13.
8
digit-algebra
hard
In the following problem, A, B, and C are each a different number from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. In other words, AB × A = CBB. What is the value of A × B × C?
Answer: D
The multiplication is 95 × 9 = 855, so A = 9, B = 5, and C = 8. Therefore A × B × C = 9 × 5 × 8 = 360.
9
geometry-area
medium
In the figure below, a U-shaped region is shaded. All the dots are one unit apart from each other horizontally and vertically. What is the area of the U-shape in square units?
Answer: D
The outer rectangle is 6 units wide and 6 units tall, with area 36. The missing rectangular opening is 2 units wide and 4 units tall, with area 8. The shaded area is 36 − 8 = 28 square units.
10
geometry-area
easy
The width of a rectangle is 2 units. Its length is three times as long as its width. What is the area of the rectangle in square units?
Answer: C
The length is 3 × 2 = 6 units. The area is length × width = 6 × 2 = 12 square units.
11
multiplication
easy
Maria had four pies. She first sliced each pie into five equal slices. Then she found that there were not enough slices for her friends. So she cut each slice into three smaller pieces. How many smaller pieces of pie are there now?
Answer: D
Four pies cut into 5 slices each make 4 × 5 = 20 slices. Cutting every slice into 3 pieces makes 20 × 3 = 60 smaller pieces.
12
data-interpretation
easy
At the end of the school year, all children brought some fruits to their class for a party. A third-grade math teacher counted the types of fruit in her class and made a bar graph. What fraction of the fruits are pears?
Answer: A
The graph shows 14 + 22 + 10 + 8 + 6 = 60 fruits. There are 10 pears, so the fraction is 10/60 = 1/6.
13
symmetry
easy
There are five shapes shown below. Which one of these shapes cannot be folded in half?
Answer: B
Shapes A, C, and D each have a line of symmetry, so they can be folded into matching halves. The irregular shape B has no line of symmetry.
14
averages
medium
Mr. Wilson has 20 students in his math class. On a test, the average score of the whole class is 80. He then removed one highest score and the new average became 79. What is the highest score on the test? Note that the average is the sum of all the scores divided by the number of students.
Answer: C
The original total was 20 × 80 = 1600. The remaining 19 scores total 19 × 79 = 1501. The removed score was 1600 − 1501 = 99.
15
geometry-perimeter
hard
In the following figure, ABCD is a square; the shaded region EFGH is a rectangle. ED = 12 inches, and BG = 10 inches. What is the perimeter of the shaded rectangle EFGH in inches?
Answer: C
Let the shaded width be w, the left margin be x, and the right margin be y. Then BG = x + w = 10 and ED = y + w = 12. The rectangle's perimeter is 2[(x + w + y) + w] = 2[(x + w) + (y + w)] = 2(10 + 12) = 44 inches.
Section B
Open-Ended Questions — 5 Questions
(+5)
16
geometry-perimeter
medium
Part of a garden is marked off and paved with bricks, as shown below. All the corners have right angles. The left edge AB is five meters long. The bottom edge AC is seven meters long. If you walked along the edges of the area all around, what is the distance, in meters, did you walk?
Answer: 24
The stair-step boundary has total downward length 5 meters and total rightward length 7 meters. Including AB and AC, the perimeter is 5 + 7 + 5 + 7 = 24 meters.
17
time
easy
Bob started reading a book at 10:15 in the morning and finished at 12:30 in the afternoon. How many minutes did he spend reading?
Answer: 135
From 10:15 to 12:15 is 2 hours, or 120 minutes. The additional 15 minutes to 12:30 gives 120 + 15 = 135 minutes.
18
circular-arrangement
medium
Twenty tennis balls are numbered from 1 to 20 and placed evenly and in order around a big circle in the school yard. The first ball is next to the 20th ball. What is the number of the ball that is opposite across the circle to the ball numbered 17?
Answer: 7
With 20 evenly spaced balls, the opposite ball is 10 positions away. Moving 10 positions from 17 gives 27, which wraps around to 7.
19
age-problems
medium
In 2024, Beth is seven years old and her father is 35 years old. In how many years will her father be three times as old as Beth?
Answer: 7
After x years, their ages will be 35 + x and 7 + x. Solving 35 + x = 3(7 + x) gives 14 = 2x, so x = 7 years.
20
fractions
medium
A survey was given to 60 students about their favorite class. One student could select only one class. All students replied. A pie chart was made of the data. How many students like science according to the chart?
Answer: 10
The science fraction is 1 − 1/2 − 1/4 − 1/12 = 1/6. Therefore 60 × 1/6 = 10 students like science.
Section C
Open-Ended Questions — 5 Questions
(+6)
21
linear-equations
medium
In the diagram below, three numbers A, B, and C are placed at the corners of the triangle. Each square has the sum of the two numbers in the neighboring circles. For example, B + C = 77. What is the value of C?
Answer: 32
The diagram gives A + C = 86, B + C = 77, and A + B = 99. Adding the first two and subtracting the third gives 2C = 86 + 77 − 99 = 64, so C = 32.
22
series
medium
What is the value of the following expression? 2 + 4 + 8 + 16 + 32 + 64 + · · · + 1024 + 2048
Answer: 4094
This is 2¹ + 2² + ··· + 2¹¹. The sum is 2¹² − 2 = 4096 − 2 = 4094.
23
geometry-area
hard
On the following geoboard, all dots are one unit apart in both horizontal and vertical directions. Two squares are defined within the geoboard. What is the area, in square units, of the shaded region?
Answer: 6
The outer square is 4 units by 4 units, so its area is 16. A side of the tilted inner square moves 3 units horizontally and 1 unit vertically, so its area is 3² + 1² = 10. The shaded area between the squares is 16 − 10 = 6 square units.
24
weight
medium
Your math teacher has a big coffee mug. When it is full of coffee, it weighs 690 grams. When half of the coffee is gone, it weighs 465 grams. What is the weight in grams of the coffee mug when it is empty?
Answer: 240
The difference 690 − 465 = 225 grams is half the coffee. A full mug contains 450 grams of coffee, so the empty mug weighs 690 − 450 = 240 grams.
25
triangles
hard
Todd has a 12-meter-long plastic tube. He wants to cut it into three pieces to form a triangle. Each piece must have a whole-number length. How many different triangles can Todd make?
Answer: 3
For side lengths a ≤ b ≤ c with a + b + c = 12, the triangle condition requires a + b > c. The possible triples are (2, 5, 5), (3, 4, 5), and (4, 4, 4). Therefore Todd can make 3 different triangles.