Combined ISEE Upper Level Math Practice Test – 8
1. (Standard) Probability
A bag contains 5 red marbles, 8 blue marbles, and 7 green marbles. If one marble is drawn at random, what is the probability that it is not blue?
(A) 2/5
(B) 3/5
(C) 8/20
(D) 12/20
1. (Challenge) Probability & Percentages
A box contains 40 marbles: 25% are red, 35% are blue, and the rest are green. If 4 more green marbles are added to the box, what is the new probability of choosing a marble that is not green?
(A) 9/22
(B) 12/22
(C) 6/11
(D) 13/20
2. (Standard) Linear Equations
If 3x – 7 = 14, what is the value of 6x + 2?
(A) 7
(B) 21
(C) 42
(D) 44
2. (Challenge) Algebra & Geometry
The length of a rectangle is (2x + 3) and the width is (x – 1). If the perimeter of the rectangle is 46 cm, what is the area of the rectangle in square centimeters?
(A) 115
(B) 117
(C) 126
(D) 132
3. (Standard) Percent and Tax
A store is offering a 20% discount on a jacket originally priced at $120. After the discount, a 5% sales tax is applied to the sale price. What is the final cost?
(A) $96.00
(B) $100.80
(C) $101.00
(D) $102.00
3. (Challenge) Rates & Logic
Machine A can complete a job in 6 hours, and Machine B can complete the same job in 9 hours. If both machines work together for 2 hours and then Machine A breaks down, how many additional hours will it take Machine B to finish the job alone?
(A) 3
(B) 4
(C) 5
(D) 6
4. (Standard) Rational Numbers
Which of the following numbers is the greatest?
(A) 0.005
(B) 5%
(C) 1/20
(D) 0.51
4. (Challenge) Functions & Sequences
In a sequence where an = 2an-1 + 3, if a1 = 5, what is the value of a4?
(A) 13
(B) 29
(C) 61
(D) 65
5. (Standard) Data Analysis (Mean)
The average (mean) of five numbers is 12. If four of the numbers are 8, 10, 15, and 12, what is the fifth number?
(A) 12
(B) 15
(C) 18
(D) 60
5. (Challenge) Statistics (Weighted Mean)
A student’s average score on 4 tests is 85. If the next two tests are weighted double (each counts as two test grades), what score does the student need to average on those two tests to bring their total weighted average to 90?
(A) 92.5
(B) 94
(C) 95
(D) 97.5
6. (Standard) Rates and Proportions
A car travels 180 miles in 3 hours. At this same constant rate, how many miles will it travel in 5 hours and 30 minutes?
(A) 300
(B) 315
(C) 330
(D) 360
6. (Challenge) Coordinate Geometry & Slopes
Line L passes through (2, 5) and (6, 13). Line K is perpendicular to Line L and passes through the midpoint of the segment connecting (2, 5) and (6, 13). What is the y-intercept of Line K?
(A) 10
(B) 11
(C) 12
(D) 13
7. (Standard) Coordinate Geometry (Slope)
What is the slope of the line passing through the points (2, -3) and (5, 6)?
(A) 1/3
(B) 1
(C) 3
(D) 9
7. (Challenge) Number Theory
If x and y are prime numbers such that x + y = 31, what is the value of xy?
(A) 54
(B) 58
(C) 62
(D) There is no such pair of primes.
8. (Standard) Measurement (Volume/Surface Area)
If a cube has a surface area of 150 square inches, what is its volume in cubic inches?
(A) 25
(B) 125
(C) 150
(D) 625
8. (Challenge) 3D Geometry & Ratios
A cylinder and a cone have the same radius. If the volume of the cylinder is twice the volume of the cone, what is the ratio of the cylinder’s height to the cone’s height?
(A) 2:3
(B) 1:2
(C) 3:2
(D) 1:3
9–12. Quantitative Comparisons
Compare Column A and Column B.
| Pair | Column A | Column B |
| 9 (Standard) | 15% of 80 | 25% of 48 |
| 9 (Challenge) | The value of absolute(-5 + 3) | The value of absolute(3 – 5) |
| 10 (Standard) | x squared | x cubed (where 0 < x < 1) |
| 10 (Challenge) | Slope of 3x – 4y = 12 | Slope of line perp. to y = -4/3x + 5 |
| 11 (Standard) | Area of a circle with radius 4 | 50 |
| 11 (Challenge) | x squared + 1 | 2x |
| 12 (Standard) | 2a + 3 | 3a + 2 |
| 12 (Challenge) | Area of a square with diagonal 10 | 50 |
13. (Standard) Algebraic Expressions
Simplify: 4(2x – 3) – 3(x + 5)
(A) 5x – 27
(B) 5x + 3
(C) 11x – 27
(D) 5x – 7
13. (Challenge) Rational Expressions
Simplify: ((x^2 – 9) / (x^2 + 5x + 6)) divided by ((x – 3) / (x + 2))
(A) 1
(B) (x-3)/(x+3)
(C) (x+2)/(x-3)
(D) 0
14. (Standard) Factoring Quadratics
What are the factors of x^2 – 5x – 14?
(A) (x – 7)(x + 2)
(B) (x + 7)(x – 2)
(C) (x – 14)(x + 1)
(D) (x – 5)(x + 14)
14. (Challenge) Quadratics & Systems
At what point(s) does the line y = 2x + 1 intersect the parabola y = x^2 – 2?
(A) (3, 7) and (-1, -1)
(B) (3, 7) and (1, 3)
(C) (-3, -5) and (1, 3)
(D) No intersection points.
15. (Standard) Geometry (Cylinder)
A cylinder has a radius of 3 cm and a height of 10 cm. Using 3.14 for pi, what is the volume?
(A) 94.2 cubic cm
(B) 188.4 cubic cm
(C) 282.6 cubic cm
(D) 314.0 cubic cm
15. (Challenge) Advanced Exponents
Solve for x: 9 to the power (x-1) = 27 to the power (x+2)
(A) -8
(B) -4
(C) 4
(D) 8
16. (Standard) Exponents
What is the value of 2 to the power -3 multiplied by 16?
(A) 1/2
(B) 2
(C) 4
(D) 8
16. (Challenge) Transformations & Circles
A circle is defined by (x – 3)^2 + (y + 2)^2 = 16. If translated 4 units left and 5 units up, what is the new equation?
(A) (x + 1)^2 + (y – 3)^2 = 16
(B) (x – 7)^2 + (y + 3)^2 = 16
(C) (x + 1)^2 + (y + 7)^2 = 16
(D) (x – 1)^2 + (y – 3)^2 = 16
17. (Standard) Geometric Transformations
Point A is at (3, 4). If point A is reflected across the y-axis and then shifted 2 units down, what are the new coordinates?
(A) (-3, 2)
(B) (3, 2)
(C) (-3, 6)
(D) (-5, 4)
17. (Challenge) Combinatorics
A committee of 3 students is chosen from 5 boys and 4 girls. If it must include at least one girl, how many committees are possible?
(A) 74
(B) 80
(C) 84
(D) 120
18. (Standard) Geometry: Similarity
In triangle ABC, DE is parallel to BC (D is on AB, E is on AC). If AD = 4, DB = 6, and DE = 8, what is BC?
(A) 12
(B) 16
(C) 20
(D) 24
18. (Challenge) Geometry: Similarity & Algebra
In triangle ABC, ADE is a similar inner triangle. If AD = x, DB = x + 2, AE = 6, and EC = 9, what is x?
(A) 2
(B) 3
(C) 4
(D) 6
19. (Standard) Inequality
Solve: -2x + 5 > 13
(A) x > -4
(B) x < -4
(C) x > 4
(D) x < 4
19. (Challenge) Compound Inequalities
How many integer solutions exist for: -3 <= 2x – 5 < 7?
(A) 4
(B) 5
(C) 6
(D) 7
20. (Standard) Algebra: Systems
30 people go to the cinema. Adults cost $10, children $6. Total bill was $220. How many adults?
(A) 10
(B) 15
(C) 20
(D) 25
20. (Challenge) Algebra: Mixture Problems
How many mL of a 50% acid solution must be mixed with a 20% solution to create 600 mL of a 30% acid solution?
(A) 200 mL
(B) 300 mL
(C) 400 mL
(D) 500 mL
21. (Standard) Data Analysis (Median)
What is the median of: 14, 22, 10, 35, 18, 22, 30?
(A) 18
(B) 21.5
(C) 22
(D) 23
21. (Challenge) Logarithms
If log base 2 of (x) + log base 2 of (x – 2) = 3, what is the value of x?
(A) 2
(B) 4
(C) 6
(D) 8
22. (Standard) Regular Polygons
A regular polygon has exactly 11 sides. What are the measures of one exterior angle and one interior angle?
(A) 30 and 150
(B) 32 8/11 and 147 3/11
(C) 36 and 144
(D) 40 and 140
22. (Challenge) Regular Polygons (Exterior/Interior)
Identify the exterior and interior angle of an 11-sided regular polygon.
(A) 32 8/11 and 147 3/11
(B) 30 5/11 and 149 6/11
(C) 32 7/11 and 147 4/11
(D) 33 and 147
23. (Standard) Radicals
Which is equivalent to the square root of 72?
(A) 6 times sqrt(2)
(B) 2 times sqrt(6)
(C) 36
(D) 9 times sqrt(8)
23. (Challenge) Rationalizing Denominators
Rationalize: 4 / (sqrt(5) – 1)
(A) sqrt(5) – 1
(B) sqrt(5) + 1
(C) 2sqrt(5) + 2 (D) 4sqrt(5) + 4
24. (Standard) Trigonometry
In a right triangle, the opposite side is 5 and the hypotenuse is 13. What is tan(theta)?
(A) 5/12
(B) 12/13
(C) 5/13
(D) 12/5
24. (Challenge) Trigonometry & Coordinate Plane
The terminal side of angle theta passes through (-3, 4). What is the value of cos(theta)?
(A) -3/5
(B) 4/5
(C) -4/3
(D) 3/5
25. (Standard) Function Notation
If f(x) = x^2 – 3x + 1, what is f(-2)?
(A) -9
(B) -1
(C) 11
(D) 13
25. (Challenge) Function Composition
If f(x) = 3x – 2 and g(x) = x^2 + 1, what is f(g(-2))?
(A) 5
(B) 13
(C) 17
(D) 25
