Continental Math League — Grade 7 to 9 Practice
A compiled set of CML-style questions for Grades 7–9. Use for printable practice, quizzes, or study reference.
Grades 7, 8 & 9
Grade 7 Questions (1–70)
1) Find the sum of 19 hundreds and 11 tens.
2) If a ⊕ b = (a + b)/(a – b) and a ⊕ b = 3/5. If a = 8, then, in simplest form, b = _____.
3) On a number line, the point labeled number 10 is twice as far from the point labeled number 6 as it is from the point labeled number 12. There is another point labeled number _____ that is also twice as far from the point labeled number 6 as it is from the point labeled number 12.
4) A one-foot cube is to be divided completely into 1″ cubes. If all the 1″ cubes were to be placed one on top of one another, the stack of 1″ cubes would be _____ feet high.
12″×12″×12″ cube divided into 1″ cubes
5) Find the perimeter of the figure shown.
+-----12-----+
| |
| +----7---+
| | |
+---+---+----+
| 5 |
+---+
6) In the multiplication problem, A, B, C, D and E represent different digits. Each time the same letter appears it represents the same digit. Find the number represented by ABCD.
ABCD
× E
____
EBEA
7) What 3-digit palindrome has digits that add to 7 and whose product of the digits is as large as possible?
8) If k · 3/4 = 1/2 · 1/4, find k in simplest form.
9) Ishmael made a long jump that was 4 ft. 9 in. further than Juwan’s long jump. Together, they jumped a total distance of 30 ft. 6 in. In simplest form, Ishmael’s long jump was _____ ft. _____ in.
10) The area of △SRP equals ½ the area of △RST. The area of △RUQ equals ⅓ the area of △RST. The area of △RST = 9 sq. units. The area of quadrilateral PQUT = _____ sq. units.
11) Find the sum of 23 hundreds and 17 tens.
12) What is the largest 4-digit number that uses each of the digits 2, 5, 7, and 9 exactly once?
13) A rectangular garden has a perimeter of 48 feet. If the length is 3 times the width, what is the width?
14) If x + 15 = 3x – 9, find the value of x.
15) How many different routes are there from Point A to Point B if you can only move right or up?
B o --- o
| |
o --- o
| |
A o --- o
16) Find the area of a triangle with base 12 cm and height 8 cm.
17) What is (3/8) + (5/12) in simplest form?
18) A number increased by 25% becomes 150. What is the original number?
19) How many prime numbers are there between 20 and 40?
20) If a cube has a volume of 64 cubic inches, what is the length of each edge?
21) Find the value of (2^5 – 3^3).
22) A bag contains 3 red marbles and 7 blue marbles. What is the probability of drawing a red marble?
23) What is the sum of the interior angles of a hexagon?
24) If (3x – 7 = 14), what is the value of (x + 5)?
25) How many factors does 36 have?
26) A circle has a diameter of 14 cm. What is its circumference? (Use π ≈ 3.14)
27) What is (2/3) × (9/8) in simplest form?
28) If a rectangle has an area of 72 square units and a width of 8 units, what is its length?
29) Find the median of the numbers: 15, 23, 18, 31, 27, 19, 25.
30) What is the greatest common factor of 48 and 72?
31) A train travels 240 miles in 4 hours. What is its average speed?
32) If (y = 3x + 2) and (x = 4), what is the value of (y)?
33) How many sides does a polygon have if the sum of its interior angles is 1440°?
34) What is 15% of 80?
35) Find the least common multiple of 12 and 18.
36) A cylinder has a radius of 5 cm and height of 10 cm. What is its volume? (Use π ≈ 3.14)
37) What is (5/6) – (2/9) in simplest form?
38) If (2x + 3y = 19) and (x = 5), what is the value of (y)?
39) How many different ways can the letters in MATH be arranged?
40) What is the surface area of a cube with edge length 6 units?
41) Find the value of (√144 + √81).
42) A store offers a 20% discount. If an item costs $45 after the discount, what was the original price?
43) What is the next term in the sequence: 2, 6, 18, 54, ___?
44) How many diagonals does a pentagon have?
45) What is (4^3 × 4^2)?
46) A jar contains 24 coins. If ⅓ are quarters and the rest are dimes, how many dimes are there?
47) If the radius of a circle is tripled, by what factor does the area increase?
48) What is the value of x if (x/4) + 7 = 12?
49) How many perfect squares are there between 1 and 100 (inclusive)?
50) Find the area of a parallelogram with base 15 cm and height 8 cm.
51) What is (7/8) ÷ (3/4)?
52) A bookshelf has 5 shelves with 8 books on each shelf. If 12 books are removed, how many books remain?
53) What is the slope of a line passing through points (2, 5) and (6, 13)?
54) How many edges does a rectangular prism have?
55) Find the value of (3! + 4!).
56) If a triangle has angles measuring 45°, 45°, and 90°, what type of triangle is it?
57) What is 125% expressed as a decimal?
58) Find the value of x if 5x – 3 = 2x + 12.
59) How many prime factors does 60 have?
60) What is the area of a trapezoid with bases 8 and 12 units and height 5 units?
61) Find the value of (√169 – √36).
62) A recipe calls for 2½ cups of flour. If you want to make 3 batches, how much flour do you need?
63) What is the y-intercept of the line y = 3x – 7?
64) How many vertices does an octagon have?
65) What is (2^8 ÷ 2^3)?
66) If a coin is flipped 3 times, what is the probability of getting at least one head?
67) Find the perimeter of a regular pentagon with side length 6 cm.
68) What is the value of x if (2x/3) = 8?
69) How many two-digit numbers are divisible by 7?
70) What is the volume of a sphere with radius 3 units? (Use π ≈ 3.14)
Grade 8 Questions (71–140)
71) 55 × 888 ? 88 × 555. a) < b) > c) = (Hint: look at the factors.)
72) a ⊕ b = (a + b)/(a – b) and a △ b = (a × b)/(a + b). The value of a ⊕ b is between 1/2 and 2. The value of a △ b is between
a) 0 and 1 b) 1 and 2 c) 2 and 3 d) 1/2 and 2 e) 10 and 20
73) Elena is 5″ short of being 5′ tall. Her height is ⅝ of her brother Rinaldo’s height. Rinaldo is _____ ft. _____ in. tall.
74) Jeremy purchased a large 60 lb. bag of jelly beans for $96. From these he packaged ½ lb. bags of jelly beans and all were sold for $1.25 each. How much profit did Jeremy make in that business venture?
75) If (n × 4^2 × 5^5 = 10^6), find (n).
76) A container is ⅔ full with red and yellow dye in the ratio of 5:3 (red to yellow). Blue dye is then poured in until the container is full. In simplest form, what is the ratio of blue dye to red dye?
77) For a 2×2 determinant, det([a c; b d]) = ad − bc. Find det([x 4; 3 7]) in simplest form if det([x 4; 3 7]) = x + 24.
78) A potato weighs 6 ounces plus 1/6 of its weight. The potato weighs _____ lbs.
79) The average of n numbers is 32. The average of 9 of those numbers is 26. The average of the remaining numbers is _____.
80) Circle O is inscribed in square ABCD. E, F, G and H are midpoints of the sides of square ABCD. Square EFGH is thus formed. The area of the shaded region is 16(π – 2) sq.in. The perimeter of square ABCD is _____ in.
Square with inscribed circle
and inner square EFGH
81) 27n^2 · 9a^2 = 3^k. k = _____.
82) The probability that it will rain tomorrow is x. The probability it will not rain tomorrow is y. The value of the expression 7x^2 + 14xy + 7y^2 – 1 is _____.
83) Find the value of √(64 + 36).
84) If (3x + 2y = 14) and (x – y = 1), find the value of (x + y).
85) What is the area of a rhombus with diagonals of length 12 and 16?
86) Find the sum of the first 20 positive integers.
87) If f(x) = 2x^2 – 3x + 1, find f(3).
88) How many different ways can 5 people sit in a row?
89) What is the distance between points (2, 3) and (6, 6)?
90) If log_2 x = 5, find the value of (x).
91) What is the coefficient of x^2 in the expansion of (x + 3)^3?
92) A regular hexagon has a perimeter of 42 cm. What is the area? (Use √3 ≈ 1.732)
93) If 2^x = 8^3, find the value of (x).
94) What is the median of the numbers: 12, 18, 15, 22, 19, 17, 14, 21?
95) Find the area of a sector with central angle 60° in a circle with radius 6.
96) If x^2 – 5x – 14 = 0, find the sum of the solutions.
97) How many different 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5 without repetition?
98) What is the surface area of a cone with radius 4 and slant height 5? (Use π ≈ 3.14)
99) If sin θ = 3/5 and θ is in the first quadrant, find cos θ.
100) What is the value of Σ(i=1 to 10) i^2?
101) Find the equation of a line passing through (2, 5) with slope 3.
102) What is the volume of a pyramid with base area 36 and height 15?
103) If (x+3)/2 = (x-1)/3, find the value of (x).
104) How many integers are there between -10 and 15 (exclusive)?
105) What is the area of an equilateral triangle with side length 8?
106) If a = 2 and b = 3, find the value of a^b + b^a.
107) What is the probability of rolling a sum of 7 with two dice?
108) Find the length of the hypotenuse of a right triangle with legs 9 and 12.
109) If 3x^2 – 12x + 12 = 0, find the discriminant.
110) What is the sum of the interior angles of a dodecagon?
111) Find the value of (d/dx)(x^3 + 2x^2 – 5x + 7) at x = 2.
112) How many ways can you choose 3 books from 8 books?
113) What is the circumference of an ellipse with semi-major axis 5 and semi-minor axis 3? (Use approximation formula)
114) If log_3 (2x – 1) = 2, find the value of (x).
115) What is the range of the function f(x) = x^2 – 4x + 5?
116) Find the area bounded by y = x^2 and y = 4 from x = 0 to x = 2.
117) What is the period of sin(3x)?
118) If matrix A = [ [2, 3], [1, 4] ], find det(A).
119) How many positive divisors does 180 have?
120) What is the limit of (x^2 – 4)/(x – 2) as x approaches 2?
121) Find the area of a regular octagon with side length 4.
122) If e^x = 10, find the value of (x) (use ln(10) ≈ 2.3).
123) What is the variance of the dataset: 2, 4, 6, 8, 10?
124) Find the equation of the circle with center (3, -2) and radius 5.
125) If tan θ = 4/3, find sec θ (θ in first quadrant).
126) What is the sum of an infinite geometric series with first term 3 and ratio 1/2?
127) How many edges does a cube have?
128) Find the value of ∫_0^2 x^2 dx.
129) What is the amplitude of y = 5sin(2x) + 3?
130) If f(x) = e^{2x}, find f'(x).
131) How many ways can 6 people be arranged in a circle?
132) What is the volume of a torus with major radius 5 and minor radius 2?
133) If z = 3 + 4i, find |z|.
134) What is the probability of getting exactly 2 heads in 5 coin flips?
135) Find the focus of the parabola y = x^2.
136) If ⃗a = (2, 3) and ⃗b = (4, -1), find ⃗a · ⃗b.
137) What is the sum of the first n positive odd integers (in terms of n)?
138) Find the surface area of a sphere with radius 6.
139) If log_a b = 2 and log_a c = 3, find log_a(bc).
140) What is the standard deviation of the numbers: 1, 2, 3, 4, 5?
Grade 9 Questions (141–200)
Grade 9 Questions (141–200)
141) The harmonic mean, m, of x and y is given by 2/(1/x + 1/y). Find the harmonic mean of 5 and 10.
142) Numbers are said to be relatively prime if 1 is their only common factor. For example, 4 and 9 are relatively prime numbers. Two numbers, both relatively prime to 36 are _____.
143) Mr. Felding traveled 4 hours averaging 60 m.p.h. to reach his destination. Had he averaged 40 m.p.h. it would have taken him an additional _____ hours traveling time.
144) The average of n numbers is 32. The average of 9 of those numbers is 26. The average of the remaining numbers is _____.
145) The probability that it will rain tomorrow is x. The probability it will not rain tomorrow is y. The value of the expression 7x^2 + 14xy + 7y^2 – 1 is _____.
146) A printer which can print 80 symbols across a page is printing out the alphabet as follows: A B C D . . . X Y Z A B C . . . What is the 46th letter on the 5th line?
147) If (a,b,c) means (a – b) · (a – c) · (b + c), then (x, 8, y) = _____ in simplest form.
148) Find the sum of the first 8 digits to the right of the decimal point in the decimal representation of 7/99.
149) If 3^x = a, express 3^{2x+1} in terms of a.
150) The positive number _____ is equal to four-fifths of the sum of the number and its reciprocal.
151) △A ~ △B. The area of △A = 20 sq. ft. Find x and y in simplest radical form.
Triangle A: sides x, y, 8
Triangle B: sides 10′, y’, 8′
152) A stack of 2″ squares are placed inside an isosceles triangle (AB = AC), as shown. The perimeter of △ABC is _____ “. (Leave your answer in radical form.)
Isosceles triangle with
squares stacked inside
Base = 8″
153) Find the value of lim(x → 0) sin x/x.
154) If f(x) = x^3 – 2x^2 + x – 1, find the remainder when f(x) is divided by (x – 2).
155) What is the equation of the tangent line to y = x^2 at the point (3, 9)?
156) Find the area enclosed by y = x^2 and y = 2x.
157) If sin A + cos A = √3/2, find sin A cos A.
158) What is the coefficient of x^3 in the expansion of (1 + 2x)^5?
159) Find the sum of the infinite series Σ(n=1 to ∞) 1/n(n+1).
160) If matrix A has eigenvalues 2 and 3, what are the eigenvalues of A^2?
161) What is the radius of convergence of Σ(n=0 to ∞) x^n/n!?
162) Find the area of the region bounded by y = e^x, y = 0, x = 0, and x = 1.
163) If f(x) = ln(x^2 + 1), find f”(0).
164) What is the period of tan(2x + π/4)?
165) Find the volume generated by rotating y = √x about the x-axis from x = 0 to x = 4.
166) If z_1 = 1 + i and z_2 = 1 – i, find z_1^8.
167) What is the Laplace transform of t^2 e^{3t}?
168) Find the critical points of f(x) = x^4 – 4x^3 + 4x^2.
169) What is the curl of the vector field ⃗F = (xy, x^2, z)?
170) Find the Fourier coefficient a_1 for f(x) = x on [-π, π].
171) If u = x^2 + y^2 and v = xy, find ∂(u,v)/∂(x,y).
172) What is the general solution to y” – 4y’ + 4y = 0?
173) Find the residue of 1/(z(z-1)^2) at z = 1.
174) What is the chromatic number of the complete graph K_5?
175) Find the minimum value of f(x,y) = x^2 + y^2 – 2x + 4y + 5.
176) If A is a 3×3 matrix with det(A) = 5, find det(2A).
177) What is the order of the group S_4 (symmetric group on 4 elements)?
178) Find the Taylor series expansion of sin x around x = 0 up to the x^5 term.
179) What is the rank of the matrix (1 2 3) in the symmetric group S_5?
180) Find the surface integral ∬_S ⃗F · d⃗S where ⃗F = (x,y,z) and S is the unit sphere.
181) What is the generating function for the Fibonacci sequence?
182) Find the number of ways to color a cube with 6 different colors (one per face).
183) What is the Galois group of x^4 – 2 over ℚ?
184) Find the dimension of the vector space of 2×2 symmetric matrices over ℝ.
185) What is the genus of a torus?
186) Find the probability that a random 3×3 matrix over ℱ_2 is invertible.
187) What is the Euler characteristic of a sphere?
188) Find the commutator [x^2 d/dx, x d/dx] acting on polynomials.
189) What is the minimal polynomial of sqrt(2) + sqrt(3) over Q?
190) Find the number of spanning trees of the complete graph K_4.
191) What is the discriminant of the elliptic curve y^2 = x^3 + x + 1?
192) Find the index of the subgroup 2ℤ in ℤ.
193) What is the Catalan number C_4?
194) Find the primitive 8th root of unity in ℂ.
195) What is the chromatic polynomial of the path graph P_4?
196) Find the Möbius function μ(30).
197) What is the maximum number of edges in a planar graph with 10 vertices?
198) Find the order of the element (1 2 3) in the symmetric group S_5.
199) What is the dimension of the quotient space ℝ^4 / span{(1,1,0,0), (0,1,1,0)}?
200) Find the value of the Riemann zeta function ζ(2).
© CML Practice — Grades 7–9 question compilation. Use for personal study and classroom practice.