MathSphere Logo

Practice Problems

Pick a topic, choose your difficulty level, and test your knowledge!

Algebra

Equations, expressions, inequalities, and unknowns.

🟢 Easy — Straightforward one-step and two-step problems. Great for building confidence!
0 / 10 answered

1. Solve for x:   2x + 5 = 15

✔ Correct Answer: x = 5
1Start: 2x + 5 = 15
2Subtract 5: 2x = 10
3Divide by 2: x = 5

2. What is the value of 3y when y = 4?

✔ Correct Answer: 12
1Substitute y = 4 into 3y.
23 × 4 = 12

3. Simplify:   5x + 3x

✔ Correct Answer: 8x
1Like terms share the same variable.
25x + 3x = (5 + 3)x = 8x

4. Solve:   x − 7 = 10

✔ Correct Answer: x = 17
1Add 7 to both sides.
2x = 10 + 7 = 17

5. If f(x) = 2x + 1, what is f(3)?

✔ Correct Answer: 7
1Substitute x = 3: f(3) = 2(3) + 1
2= 6 + 1 = 7

6. What is the y-intercept of y = 4x + 7?

✔ Correct Answer: 7
1In y = mx + b, b is the y-intercept.
2Here b = 7.

7. Solve for n:   n/4 = 9

✔ Correct Answer: n = 36
1Multiply both sides by 4.
2n = 9 × 4 = 36

8. Which expression is equivalent to 4(x + 2)?

✔ Correct Answer: 4x + 8
1Distribute: 4 × x = 4x and 4 × 2 = 8.
2Result: 4x + 8

9. If a = 3 and b = 5, what is 2a − b?

✔ Correct Answer: 1
12a = 2 × 3 = 6
22a − b = 6 − 5 = 1

10. Solve:   3x = 21

✔ Correct Answer: x = 7
1Divide both sides by 3.
2x = 21 ÷ 3 = 7
🟡 Medium — Multi-step equations, factoring, and function analysis.
0 / 10 answered

1. Solve:   3x − 7 = 2x + 4

✔ Correct Answer: x = 11
1Move x terms left: 3x − 2x = 4 + 7
2x = 11

2. Expand:   (x + 3)²

✔ Correct Answer: x² + 6x + 9
1Use (a+b)² = a² + 2ab + b²
2x² + 2(x)(3) + 3² = x² + 6x + 9

3. Factor:   x² − 5x + 6

✔ Correct Answer: (x − 2)(x − 3)
1Find two numbers: multiply to +6, add to −5.
2−2 × −3 = 6 and −2 + (−3) = −5 ✓

4. What is the slope of the line   y = 3x − 7?

✔ Correct Answer: 3
1In y = mx + b, m is the slope.
2Coefficient of x is 3, so slope = 3.

5. Solve the system: x + y = 7 and x − y = 3

✔ Correct Answer: x = 5, y = 2
1Add the equations: 2x = 10 → x = 5
2Substitute: 5 + y = 7 → y = 2

6. Solve the inequality:   2x − 3 < 7

✔ Correct Answer: x < 5
1Add 3: 2x < 10
2Divide by 2: x < 5

7. What are the roots of x² − 4 = 0?

✔ Correct Answer: x = ±2
1x² = 4 → x = ±√4
2x = 2 or x = −2

8. Simplify:   (3x²)(4x³)

✔ Correct Answer: 12x⁵
1Multiply coefficients: 3 × 4 = 12
2Add exponents: x² × x³ = x⁵ → 12x⁵

9. What is the inverse function of f(x) = 2x + 6?

✔ Correct Answer: f⁻¹(x) = (x − 6)/2
1Replace f(x) with y: y = 2x + 6
2Swap x and y, solve for y: x = 2y + 6 → y = (x−6)/2

10. If 5x + 2y = 20 and y = 5, what is x?

✔ Correct Answer: x = 2
1Substitute y = 5: 5x + 10 = 20
25x = 10 → x = 2
🔴 Hard — Quadratic formula, complex inequalities, and advanced functions.
0 / 10 answered

1. Solve using the quadratic formula:   x² − 5x + 6 = 0

✔ Correct Answer: x = 2 or x = 3
1x = [5 ± √(25−24)] / 2 = [5 ± 1] / 2
2x = (5+1)/2 = 3 or x = (5−1)/2 = 2

2. Which inequality represents the solution to   −3x > 12?

✔ Correct Answer: x < −4
1Divide both sides by −3 (flip the inequality sign).
2x < −4

3. If g(x) = x² − 4 and h(x) = 2x, find g(h(3)).

✔ Correct Answer: 32
1First find h(3) = 2(3) = 6
2Then g(6) = 6² − 4 = 36 − 4 = 32

4. What is the discriminant of   2x² + 3x − 5 = 0?

✔ Correct Answer: 49
1Discriminant = b² − 4ac = 3² − 4(2)(−5)
2= 9 + 40 = 49

5. Simplify:   (x² − 9) / (x − 3)

✔ Correct Answer: x + 3
1Factor the numerator: x² − 9 = (x+3)(x−3)
2Cancel (x−3): result is (x + 3), x ≠ 3

6. Solve:   |2x − 4| = 10

✔ Correct Answer: x = 7 or x = −3
1Case 1: 2x − 4 = 10 → x = 7
2Case 2: 2x − 4 = −10 → x = −3

7. The roots of x² + bx + 12 = 0 are x = 3 and x = 4. What is b?

✔ Correct Answer: b = −7
1By Vieta's formulas, sum of roots = −b → 3 + 4 = −b
27 = −b → b = −7

8. Solve for x:   log₂(x) = 5

✔ Correct Answer: x = 32
1log₂(x) = 5 means 2⁵ = x
2x = 32

9. What is the vertex of the parabola y = x² − 6x + 11?

✔ Correct Answer: (3, 2)
1Vertex x = −b/2a = 6/2 = 3
2y = 9 − 18 + 11 = 2 → vertex (3, 2)

10. For what value of k does kx² − 4x + 1 = 0 have exactly one solution?

✔ Correct Answer: k = 4
1One solution means discriminant = 0: b² − 4ac = 0
216 − 4k = 0 → k = 4

Geometry

Shapes, areas, volumes, and angles.

🟢 Easy — Basic shapes, angles, and simple formulas.
0 / 10 answered

1. What is the sum of interior angles of a triangle?

✔ Correct Answer: 180°
1Fundamental theorem: all triangle angles sum to 180°.

2. What is the area of a rectangle with length 8 and width 5?

✔ Correct Answer: 40
1Area = length × width = 8 × 5 = 40

3. How many sides does a hexagon have?

✔ Correct Answer: 6
1"Hex" comes from Greek meaning six. A hexagon has 6 sides.

4. What is the perimeter of a square with side 7?

✔ Correct Answer: 28
1Perimeter = 4 × side = 4 × 7 = 28

5. What type of angle is exactly 90°?

✔ Correct Answer: Right
1Right angle = exactly 90°. Acute < 90°. Obtuse > 90°.

6. What is the circumference of a circle with radius 7? (Use π ≈ 3.14)

✔ Correct Answer: 43.96
1C = 2πr = 2 × 3.14 × 7
2= 43.96

7. What is the area of a triangle with base 10 and height 6?

✔ Correct Answer: 30
1Area = ½ × base × height
2= ½ × 10 × 6 = 30

8. Two angles are supplementary. One is 65°. What is the other?

✔ Correct Answer: 115°
1Supplementary angles add up to 180°.
2180° − 65° = 115°

9. How many degrees are in a full circle?

✔ Correct Answer: 360°
1A full rotation is always 360°.

10. What is the volume of a cube with side length 4?

✔ Correct Answer: 64
1V = side³ = 4³ = 64
🟡 Medium — Circles, Pythagorean theorem, and 3D shapes.
0 / 10 answered

1. What is the area of a circle with radius 3?

✔ Correct Answer: 9π
1Area = πr² = π × 3² = 9π

2. What is the hypotenuse of a right triangle with legs 3 and 4?

✔ Correct Answer: 5
1c² = 3² + 4² = 9 + 16 = 25
2c = √25 = 5

3. What is the volume of a cube with side length 4?

✔ Correct Answer: 64
1V = side³ = 4³ = 64

4. The perimeter of a rectangle is 36. Width is 6. What is the length?

✔ Correct Answer: 12
1P = 2(l+w) → 36 = 2(l+6)
2l + 6 = 18 → l = 12

5. What is the circumference of a circle with diameter 10?

✔ Correct Answer: 10π
1C = π × diameter = π × 10 = 10π

6. What is the area of a circle with radius 5?

✔ Correct Answer: 25π
1Area = πr² = π(5²) = 25π

7. In a right triangle, if one leg is 8 and hypotenuse is 10, what is the other leg?

✔ Correct Answer: 6
1Pythagoras: a² + b² = c² → 64 + b² = 100
2b² = 36 → b = 6

8. What is the sum of interior angles of a pentagon?

✔ Correct Answer: 540°
1Sum = (n − 2) × 180° = (5 − 2) × 180°
2= 3 × 180° = 540°

9. What is the volume of a cylinder with radius 3 and height 5?

✔ Correct Answer: 45π
1V = πr²h = π(9)(5) = 45π

10. Two parallel lines are cut by a transversal. Alternate interior angles are:

✔ Correct Answer: Equal
1Alternate interior angles formed by parallel lines cut by a transversal are always equal (congruent).
🔴 Hard — Coordinate geometry, arc length, sector area, and complex solids.
0 / 10 answered

1. What is the area of a triangle with vertices A(0,0), B(4,0), C(0,3)?

✔ Correct Answer: 6
1Base = 4, Height = 3 (right triangle at origin)
2Area = ½ × base × height = ½ × 4 × 3 = 6

2. What is the arc length of a 60° sector in a circle of radius 6?

✔ Correct Answer: 2π
1Arc length = (θ/360) × 2πr
2= (60/360) × 2π(6) = (1/6) × 12π = 2π

3. What is the volume of a sphere with radius 3?

✔ Correct Answer: 36π
1V = (4/3)πr³ = (4/3)π(27) = 36π

4. What is the distance between points (1, 2) and (5, 5)?

✔ Correct Answer: 5
1d = √[(5−1)² + (5−2)²] = √[16 + 9]
2= √25 = 5

5. What is the sum of interior angles of a decagon (10 sides)?

✔ Correct Answer: 1440°
1Sum = (n − 2) × 180° where n = number of sides
2= (10 − 2) × 180° = 8 × 180° = 1440°

6. What is the area of a sector with radius 6 and central angle 90°?

✔ Correct Answer: 9π
1Area = (θ/360) × πr² = (90/360) × π(36)
2= (1/4) × 36π = 9π

7. What is the midpoint of (2, 4) and (8, 10)?

✔ Correct Answer: (5, 7)
1Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
2= ((2+8)/2, (4+10)/2) = (5, 7)

8. What is the surface area of a cube with side 5?

✔ Correct Answer: 150
1Surface area = 6 × side² = 6 × 25 = 150

9. A cone has radius 3 and height 4. What is its slant height?

✔ Correct Answer: 5
1Slant height l = √(r² + h²) = √(9 + 16)
2= √25 = 5

10. What is the equation of a circle with center (2, −3) and radius 4?

✔ Correct Answer: (x−2)² + (y+3)² = 16
1(x−h)² + (y−k)² = r² where center is (h, k)
2(x−2)² + (y−(−3))² = 4² → (x−2)² + (y+3)² = 16

Calculus

Derivatives, integrals, and limits.

🟢 Easy — Basic derivatives and simple integrals using the power rule.
0 / 10 answered

1. What is the derivative of x²?

✔ Correct Answer: 2x
1Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
2d/dx[x²] = 2x

2. What is the derivative of a constant, e.g. d/dx[7]?

✔ Correct Answer: 0
1Constants don't change, so their rate of change is 0.

3. What is ∫ 2x dx?

✔ Correct Answer: x² + C
1∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
2∫ 2x dx = 2 · x²/2 + C = x² + C

4. What is the derivative of sin(x)?

✔ Correct Answer: cos(x)
1d/dx[sin(x)] = cos(x) — a standard result to memorize.

5. What is lim(x→0) sin(x)/x?

✔ Correct Answer: 1
1As x → 0, sin(x) ≈ x (small angle approximation).
2Therefore sin(x)/x → 1.

6. What is the derivative of 5x³?

✔ Correct Answer: 15x²
1Power rule: d/dx[axⁿ] = n·axⁿ⁻¹
2d/dx[5x³] = 3 × 5x² = 15x²

7. What is the derivative of cos(x)?

✔ Correct Answer: −sin(x)
1d/dx[cos(x)] = −sin(x) — a standard result to memorize.

8. What is ∫ 5 dx?

✔ Correct Answer: 5x + C
1The integral of a constant k is kx + C.
2∫ 5 dx = 5x + C

9. What does f'(x) > 0 on an interval tell us about f(x)?

✔ Correct Answer: f is increasing
1A positive derivative means the function is going up (increasing).

10. What is the derivative of eˣ?

✔ Correct Answer: eˣ
1eˣ is its own derivative — a unique and important property.
🟡 Medium — Chain rule, definite integrals, and product/quotient rules.
0 / 10 answered

1. What is ∫₀² x dx?

✔ Correct Answer: 2
1∫ x dx = x²/2 + C
2[x²/2]₀² = (4/2) − (0/2) = 2

2. What is d/dx [x³ + 5x]?

✔ Correct Answer: 3x² + 5
1Differentiate term by term using the power rule.
2d/dx[x³] = 3x²; d/dx[5x] = 5 → 3x² + 5

3. What is ∫ 3x² dx?

✔ Correct Answer: x³ + C
1∫ 3x² dx = 3 · x³/3 + C = x³ + C

4. Using the chain rule, find d/dx [sin(3x)].

✔ Correct Answer: 3cos(3x)
1Chain rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)
2f = sin, g = 3x → cos(3x) · 3 = 3cos(3x)

5. Find the local minimum of f(x) = x² − 4x + 5.

✔ Correct Answer: x = 2
1f'(x) = 2x − 4; set to zero: 2x − 4 = 0
2x = 2; f''(x) = 2 > 0 → confirmed minimum.

6. Find d/dx [ln(x)].

✔ Correct Answer: 1/x
1Standard result: d/dx[ln(x)] = 1/x (for x > 0).

7. What is ∫ (1/x) dx?

✔ Correct Answer: ln|x| + C
1∫ (1/x) dx = ln|x| + C — a standard integral result.

8. Using the quotient rule, find d/dx [x / eˣ].

✔ Correct Answer: (1 − x) / eˣ
1Quotient rule: (u'v − uv') / v² where u=x, v=eˣ
2= (1·eˣ − x·eˣ) / e²ˣ = eˣ(1−x) / e²ˣ = (1−x)/eˣ

9. What is ∫₁³ 2 dx?

✔ Correct Answer: 4
1∫₁³ 2 dx = [2x]₁³ = 2(3) − 2(1)
2= 6 − 2 = 4

10. What is the second derivative of f(x) = x⁴?

✔ Correct Answer: 12x²
1First derivative: f'(x) = 4x³
2Second derivative: f''(x) = 12x²
🔴 Hard — Integration by substitution, related rates, and advanced limits.
0 / 10 answered

1. What is ∫ e^(2x) dx?

✔ Correct Answer: e^(2x)/2 + C
1Let u = 2x, du = 2 dx → dx = du/2
2∫ eᵘ (du/2) = eᵘ/2 + C = e^(2x)/2 + C

2. Find d/dx [x · eˣ] using the product rule.

✔ Correct Answer: eˣ(x + 1)
1Product rule: (uv)' = u'v + uv'
2u=x, v=eˣ → 1·eˣ + x·eˣ = eˣ(1+x)

3. What is lim(x→∞) (3x² + 1)/(x² − 2)?

✔ Correct Answer: 3
1Divide numerator and denominator by x².
2(3 + 1/x²) / (1 − 2/x²) → 3/1 = 3 as x→∞

4. What is ∫₀^π sin(x) dx?

✔ Correct Answer: 2
1∫ sin(x) dx = −cos(x) + C
2[−cos(x)]₀^π = (−cos π) − (−cos 0) = 1 + 1 = 2

5. A ball's position is s(t) = t³ − 3t. At what time t > 0 is the velocity zero?

✔ Correct Answer: t = 1
1Velocity = s'(t) = 3t² − 3
2Set to zero: 3t² − 3 = 0 → t² = 1 → t = 1

6. What is d/dx [tan(x)]?

✔ Correct Answer: sec²(x)
1Standard result: d/dx[tan(x)] = sec²(x).

7. Evaluate ∫ x·ln(x) dx using integration by parts.

✔ Correct Answer: x²ln(x)/2 − x²/4 + C
1Let u = ln(x), dv = x dx → du = 1/x dx, v = x²/2
2∫ x·ln(x) dx = x²ln(x)/2 − ∫ x²/(2x) dx = x²ln(x)/2 − x²/4 + C

8. What is lim(x→0) (1 − cos x) / x²?

✔ Correct Answer: 1/2
1Apply L'Hôpital twice (0/0 form): differentiate top and bottom.
2lim sin(x)/2x → lim cos(x)/2 = 1/2 as x→0

9. Find the area between y = x² and y = x on [0, 1].

✔ Correct Answer: 1/6
1Area = ∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹
2= 1/2 − 1/3 = 1/6

10. What is ∫ cos²(x) dx?

✔ Correct Answer: x/2 + sin(2x)/4 + C
1Use identity: cos²(x) = (1 + cos(2x)) / 2
2∫ (1 + cos2x)/2 dx = x/2 + sin(2x)/4 + C

Statistics

Data, probability, and distributions.

🟢 Easy — Mean, median, mode, and basic probability.
0 / 10 answered

1. What is the mean of: 4, 8, 6, 5, 7?

✔ Correct Answer: 6
1Sum = 4+8+6+5+7 = 30
2Mean = 30 ÷ 5 = 6

2. What is the median of: 3, 7, 2, 9, 5?

✔ Correct Answer: 5
1Sort: 2, 3, 5, 7, 9 — middle value is 5.

3. A fair die is rolled. What is the probability of getting a 4?

✔ Correct Answer: 1/6
16 equally likely outcomes; one is 4 → P = 1/6.

4. What is the mode of: 2, 4, 4, 5, 7?

✔ Correct Answer: 4
1The mode is the value that appears most often.
24 appears twice — all others appear once.

5. A bag has 3 red and 7 blue balls. What is the probability of drawing a red ball?

✔ Correct Answer: 3/10
1Total = 10. Favourable = 3.
2P(red) = 3/10

6. What is the mode of: 3, 7, 3, 9, 3, 7?

✔ Correct Answer: 3
1The mode is the most frequently occurring value.
23 appears 3 times — more than any other value.

7. A bag has 3 red and 7 blue marbles. What is the probability of picking red?

✔ Correct Answer: 3/10
1P(red) = favourable outcomes / total outcomes
2= 3 / (3 + 7) = 3/10

8. What is the range of: 12, 4, 9, 17, 6?

✔ Correct Answer: 13
1Range = Max − Min = 17 − 4 = 13

9. If you flip a fair coin, what is the probability of getting tails?

✔ Correct Answer: 1/2
1A fair coin has 2 equally likely outcomes: heads or tails.
2P(tails) = 1/2

10. What is the mean of: 10, 20, 30?

✔ Correct Answer: 20
1Sum = 10 + 20 + 30 = 60
2Mean = 60 / 3 = 20
🟡 Medium — Standard deviation, variance, and the empirical rule.
0 / 10 answered

1. What does the standard deviation measure?

✔ Correct Answer: Spread/dispersion of the data
1Standard deviation (σ) tells how spread out values are from the mean.

2. In a normal distribution, ~what % of data falls within 1 standard deviation?

✔ Correct Answer: 68%
1The 68–95–99.7 (empirical) rule: within 1σ → 68%.

3. What is the variance of the data set: 2, 4, 4, 4, 5, 5, 7, 9?

✔ Correct Answer: 4
1Mean = (2+4+4+4+5+5+7+9)/8 = 40/8 = 5
2Variance = [(9+1+1+1+0+0+4+16)/8] = 32/8 = 4

4. Two events A and B are mutually exclusive. P(A)=0.3, P(B)=0.4. What is P(A or B)?

✔ Correct Answer: 0.7
1Mutually exclusive → P(A or B) = P(A) + P(B)
2= 0.3 + 0.4 = 0.7

5. What is the range of the data set: 5, 12, 3, 8, 20, 1?

✔ Correct Answer: 19
1Range = Maximum − Minimum = 20 − 1 = 19

6. Two dice are rolled. What is P(sum = 7)?

✔ Correct Answer: 1/6
1Pairs summing to 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6 ways.
2P = 6/36 = 1/6

7. What is the variance of: 2, 4, 4, 4, 5, 5, 7, 9?

✔ Correct Answer: 4
1Mean = 40/8 = 5. Deviations²: 9,1,1,1,0,0,4,16 → sum = 32
2Variance = 32/8 = 4

8. In a normal distribution, what percentage of data falls within 1 standard deviation of the mean?

✔ Correct Answer: 68%
1The 68-95-99.7 rule: 1σ → 68%, 2σ → 95%, 3σ → 99.7%.

9. What is the interquartile range (IQR) of: 1, 3, 5, 7, 9, 11, 13?

✔ Correct Answer: 8
1Q1 = 3, Q3 = 11 (lower and upper quartiles)
2IQR = Q3 − Q1 = 11 − 3 = 8

10. P(A) = 0.4, P(B) = 0.5, and A and B are independent. What is P(A and B)?

✔ Correct Answer: 0.2
1For independent events: P(A and B) = P(A) × P(B)
2= 0.4 × 0.5 = 0.2
🔴 Hard — Conditional probability, Bayes' theorem, and regression.
0 / 10 answered

1. P(A)=0.5, P(B|A)=0.4. What is P(A and B)?

✔ Correct Answer: 0.2
1P(A and B) = P(A) × P(B|A)
2= 0.5 × 0.4 = 0.2

2. In a binomial distribution with n=5 and p=0.5, what is P(X=3)?

✔ Correct Answer: 5/16
1P(X=k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ
2C(5,3) · (0.5)³ · (0.5)² = 10 · (1/8) · (1/4) = 10/32 = 5/16

3. A z-score of 2 means the value is how many standard deviations above the mean?

✔ Correct Answer: 2
1z = (x − μ) / σ → z = 2 means the value is 2σ above the mean.

4. Which correlation coefficient indicates the strongest linear relationship?

✔ Correct Answer: r = −0.95
1Strength = |r|. The closer to ±1, the stronger the relationship.
2|−0.95| = 0.95 is closest to 1 → strongest.

5. Using Bayes' theorem: P(D)=0.01, P(+|D)=0.99, P(+|¬D)=0.05. What is P(D|+)?

✔ Correct Answer: ~0.17
1P(D|+) = P(+|D)·P(D) / [P(+|D)·P(D) + P(+|¬D)·P(¬D)]
2= (0.99×0.01) / (0.99×0.01 + 0.05×0.99) = 0.0099 / 0.0594 ≈ 0.167

6. A z-score of 2.0 corresponds to what percentile (approx)?

✔ Correct Answer: 97.7th
1z = 2.0 means 2 standard deviations above the mean.
2From the 68-95-99.7 rule: ~95% within ±2σ → ~97.7% below z = +2.

7. A sample has mean 50 and standard deviation 10, n = 25. What is the standard error?

✔ Correct Answer: 2
1Standard error = σ / √n = 10 / √25
2= 10 / 5 = 2

8. Which value of r indicates the strongest linear correlation?

✔ Correct Answer: r = −0.95
1Strength of correlation is measured by |r|, not sign.
2|−0.95| = 0.95 is larger than |0.85| = 0.85 or |0.50|.

9. In hypothesis testing, what does a p-value of 0.03 mean at α = 0.05?

✔ Correct Answer: Reject H₀
1If p-value < α, we reject the null hypothesis.
20.03 < 0.05 → reject H₀; result is statistically significant.

10. How many ways can 5 students be arranged in a line?

✔ Correct Answer: 120
1Number of permutations = 5! = 5 × 4 × 3 × 2 × 1
2= 120

Arithmetic

Numbers, operations, fractions, and percentages.

🟢 Easy — Basic operations, fractions, and simple percentages.
0 / 10 answered

1. What is 15% of 200?

✔ Correct Answer: 30
10.15 × 200 = 30

2. What is 2⁵?

✔ Correct Answer: 32
12×2×2×2×2 = 32

3. Simplify:   3/4 + 1/2

✔ Correct Answer: 5/4
1Convert: 1/2 = 2/4 → 3/4 + 2/4 = 5/4

4. What is the LCM of 4 and 6?

✔ Correct Answer: 12
1Multiples of 4: 4, 8, 12... Multiples of 6: 6, 12... → LCM = 12

5. A shirt costs $80 and is on sale for 25% off. What is the sale price?

✔ Correct Answer: $60
1Discount = 25% × 80 = $20
2Sale price = 80 − 20 = $60

6. What is 15% of 200?

✔ Correct Answer: 30
115% = 15/100 = 0.15
20.15 × 200 = 30

7. What is the LCM of 4 and 6?

✔ Correct Answer: 12
1Multiples of 4: 4, 8, 12… Multiples of 6: 6, 12…
2LCM = 12 (smallest common multiple)

8. What is 3/4 + 1/4?

✔ Correct Answer: 1
1Same denominator: 3/4 + 1/4 = 4/4 = 1

9. What is √49?

✔ Correct Answer: 7
17 × 7 = 49, so √49 = 7

10. What is 2³?

✔ Correct Answer: 8
12³ = 2 × 2 × 2 = 8
🟡 Medium — Ratios, proportions, GCD, and multi-step percentage problems.
0 / 10 answered

1. What is the GCD (HCF) of 36 and 48?

✔ Correct Answer: 12
1Factors of 36: 1,2,3,4,6,9,12,18,36
2Factors of 48: 1,2,3,4,6,8,12,16,24,48 → GCD = 12

2. A car travels 150 km in 3 hours. What is its average speed?

✔ Correct Answer: 50 km/h
1Speed = Distance / Time = 150 / 3 = 50 km/h

3. If 3 : 5 = x : 20, what is x?

✔ Correct Answer: 12
1Cross multiply: 5x = 3 × 20 = 60
2x = 60 / 5 = 12

4. A price increased from $50 to $65. What is the percentage increase?

✔ Correct Answer: 30%
1Increase = 65 − 50 = 15
2% increase = (15/50) × 100 = 30%

5. Simplify:   (2/3) ÷ (4/9)

✔ Correct Answer: 3/2
1Dividing by a fraction = multiplying by its reciprocal.
2(2/3) × (9/4) = 18/12 = 3/2

6. A jacket costs $80 after a 20% discount. What was the original price?

✔ Correct Answer: $100
1$80 = 80% of original price → original = 80 / 0.80
2= $100

7. Evaluate:   3 + 4 × 2 − 1

✔ Correct Answer: 10
1Order of operations (BODMAS): multiplication first → 4 × 2 = 8
23 + 8 − 1 = 10

8. What is the GCD of 48 and 36?

✔ Correct Answer: 12
1Factors of 48: 1,2,3,4,6,8,12,16,24,48
2Factors of 36: 1,2,3,4,6,9,12,18,36 → GCD = 12

9. Express 0.375 as a fraction in simplest form.

✔ Correct Answer: 3/8
10.375 = 375/1000. Simplify by dividing by 125.
2375/1000 = 3/8

10. If 5 workers can finish a job in 8 days, how many days for 10 workers?

✔ Correct Answer: 4
1Inverse proportion: more workers → fewer days.
2Days = (5 × 8) / 10 = 40 / 10 = 4 days
🔴 Hard — Number theory, compound interest, and complex fraction operations.
0 / 10 answered

1. What is compound interest on $1000 at 10% p.a. for 2 years (compounded annually)?

✔ Correct Answer: $210
1A = P(1 + r)ⁿ = 1000(1.1)² = 1000 × 1.21 = $1210
2Interest = 1210 − 1000 = $210

2. What is the smallest prime number greater than 50?

✔ Correct Answer: 53
151 = 3×17 (not prime). 52 = 4×13 (not prime). 53 is prime!

3. A number is divisible by both 4 and 6. It must be divisible by:

✔ Correct Answer: 12
1Must be divisible by LCM(4, 6) = 12.
2Not necessarily 24 (e.g. 12 works but not 24).

4. Evaluate:   √(144/25)

✔ Correct Answer: 12/5
1√(a/b) = √a / √b
2= √144 / √25 = 12 / 5

5. If a train travels at 60 km/h for 2.5 hours, then at 80 km/h for 1.5 hours, what is its average speed?

✔ Correct Answer: 67.5 km/h
1Total distance = 60×2.5 + 80×1.5 = 150 + 120 = 270 km
2Total time = 2.5 + 1.5 = 4 hours → Avg speed = 270/4 = 67.5 km/h

6. What is 2⁻³ expressed as a fraction?

✔ Correct Answer: 1/8
1Negative exponent: a⁻ⁿ = 1/aⁿ
22⁻³ = 1/2³ = 1/8

7. A $5000 investment grows at 6% p.a. compounded monthly for 1 year. What is the amount?

✔ Correct Answer: $5308.39
1A = P(1 + r/n)^(nt) = 5000(1 + 0.06/12)^12
2= 5000 × (1.005)^12 ≈ 5000 × 1.06168 ≈ $5308.39

8. Simplify:   (3/4) × (8/9) ÷ (2/3)

✔ Correct Answer: 1
1(3/4) × (8/9) = 24/36 = 2/3
2(2/3) ÷ (2/3) = 1

9. What is 7! (7 factorial)?

✔ Correct Answer: 5040
17! = 7 × 6 × 5 × 4 × 3 × 2 × 1
2= 5040

10. Which is larger: 2⁸ or 3⁵?

✔ Correct Answer: 2⁸
12⁸ = 256
23⁵ = 243 → 256 > 243, so 2⁸ is larger.