Algebra & Linear Equations
Q 01
Linear
Trap ⚠
Easy
⚡ Memory Point: "TOTAL = RATE × TIME" — always assign one variable, express the other in terms of it.
A shop sells pens for $3 each and notebooks for $5 each. Maya bought a total of 12 items and paid $44. How many pens did she buy?
📐 Worked Example (Similar)
Sam buys apples ($1) and oranges ($2), 10 items, pays $14.Let p = apples → p + o = 10 and p + 2o = 14
Subtract: o = 4, so p = 6. ✓
Trap: Many students add coefficients wrong or confuse which variable is which after substitution.
💡 Explanation (disappears in 5s)
Let p = pens, n = notebooks.System: p + n = 12 and 3p + 5n = 44.
From eq 1: n = 12 − p. Substitute: 3p + 5(12 − p) = 44 → 3p + 60 − 5p = 44 → −2p = −16 → p = 8.
Q 02
Percent Change
Trap ⚠
Easy
⚡ Memory Point: "PERCENT CHANGE = (New − Old) / Old × 100" — never divide by the new value!
A jacket originally costs $80. After a discount, it sells for $60. What is the percent decrease in price?
📐 Worked Example (Similar)
Price drops from $50 → $40.% decrease = (50−40)/50 × 100 = 20%
% change = (|New − Old|) / Old × 100
Trap: Students divide by $60 (the new price) instead of $80 (the original). Always divide by the original.
💡 Explanation (disappears in 5s)
(80 − 60) / 80 × 100 = 20/80 × 100 = 25%. The trap is dividing by 60, which gives ≈33.3%.
Q 03
Rate · Distance
Trap ⚠
Medium
⚡ Memory Point: "AVERAGE SPEED ≠ average of speeds" — use Total Distance / Total Time.
Carlos drives 60 miles at 30 mph, then 60 miles at 60 mph. What is his average speed for the entire trip, in mph?
📐 Worked Example (Similar)
Drive 20 mi at 20 mph → time = 1 hr.Drive 20 mi at 40 mph → time = 0.5 hr.
Avg speed = 40 mi / 1.5 hr ≈ 26.67 mph (NOT 30 mph!)
Trap: The "obvious" answer of 45 mph is wrong. Equal distances (not times) means harmonic mean applies.
💡 Explanation (disappears in 5s)
Time₁ = 60/30 = 2 hr. Time₂ = 60/60 = 1 hr. Total time = 3 hr. Total distance = 120 mi.Avg speed = 120/3 = 40 mph. (Trap answer 45 = simple average of 30 and 60.)
Q 04
Proportional Reasoning
Easy
⚡ Memory Point: "SCALE UP/DOWN consistently" — if 3 workers take 6 days, 1 worker takes 18 days (inverse proportion for work).
A recipe for 4 people requires 3 cups of flour. If you want to make the recipe for 10 people, how many cups of flour do you need? (Enter as a fraction or decimal.)
Trap: Students add 3 extra cups rather than scaling by the ratio. Always form the proportion correctly.
💡 Explanation (disappears in 5s)
Ratio: 3 cups / 4 people = x / 10 people → x = 30/4 = 7.5 cups.
Q 05
Inequalities
Trap ⚠
Medium
⚡ Memory Point: "FLIP the inequality sign when multiplying or dividing by a NEGATIVE number."
A store policy says the number of items n a customer can return satisfies: −2n + 10 ≥ 4. What is the maximum number of items a customer can return?
📐 Worked Example (Similar)
Solve −3x + 9 ≥ 3:−3x ≥ −6 → divide by −3, flip sign → x ≤ 2. Max = 2.
Trap: Forgetting to flip the inequality sign gives x ≥ 3 instead of x ≤ 3, reversing the entire solution set.
💡 Explanation (disappears in 5s)
−2n + 10 ≥ 4 → −2n ≥ −6 → divide by −2, FLIP sign → n ≤ 3. Maximum = 3.
Quadratics & Functions
Q 06
Vertex Form
Trap ⚠
Medium
⚡ Memory Point: "VERTEX = (h, k) in f(x) = a(x − h)² + k" — watch the sign of h!
A ball is thrown upward. Its height in feet is given by h(t) = −16(t − 3)² + 144, where t is seconds. What is the maximum height, in feet?
📐 Worked Example (Similar)
f(x) = −2(x − 1)² + 8 → vertex at (1, 8) → max = 8.
Trap: Students confuse the vertex time (t = 3) with the max height. Read what the question actually asks.
💡 Explanation (disappears in 5s)
Vertex form: max height is the k-value = 144 feet at t = 3 sec. The −16 coefficient confirms it's a maximum (opens downward).
Q 07
Quadratic Roots
Trap ⚠
Medium
⚡ Memory Point: "ROOTS from factored form (x − r₁)(x − r₂) = 0 → x = r₁ or x = r₂. Sign flips!"
The profit (in hundreds of dollars) of a business is modeled by P(x) = (x − 4)(x − 10), where x is months since opening. For how many months is the profit negative? (i.e., how many integer values of x between the roots?)
Trap: Students include the endpoints (x = 4 and x = 10) where P = 0. "Negative" means strictly between the roots.
💡 Explanation (disappears in 5s)
Roots at x = 4 and x = 10. Between them (exclusive): x = 5, 6, 7, 8, 9 → 5 integer months of negative profit. (The parabola opens upward, so P < 0 between roots.)
Q 08
Function Notation
Trap ⚠
Medium
⚡ Memory Point: "f(a + b) ≠ f(a) + f(b) in general — substitute the WHOLE expression into the function."
If f(x) = x² − 3x + 2, what is f(x + 1) − f(x)?
📐 Worked Example (Similar)
g(x) = 2x + 1. Find g(x + 1) − g(x).g(x + 1) = 2(x + 1) + 1 = 2x + 3.
g(x + 1) − g(x) = (2x + 3) − (2x + 1) = 2.
Trap: Students try f(x+1) = f(x) + f(1) — this is wrong for non-linear functions. Always fully expand.
💡 Explanation (disappears in 5s)
f(x+1) = (x+1)² − 3(x+1) + 2 = x²+2x+1 − 3x−3+2 = x²−x.f(x+1) − f(x) = (x²−x) − (x²−3x+2) = 2x − 2... wait: = x²−x − x²+3x−2 = 2x − 2. Also acceptable: 2(x−1).
Ratios, Rates & Classic Traps
Q 09
Work Rate
Trap ⚠
Medium
⚡ Memory Point: "COMBINED RATE = Rate₁ + Rate₂. Work rate = 1/time to complete job."
Pipe A fills a tank in 6 hours. Pipe B fills the same tank in 3 hours. How many hours will it take to fill the tank if both pipes are open? (Express as a fraction.)
📐 Worked Example (Similar)
A: 1/4 tank/hr. B: 1/6 tank/hr.Combined: 1/4 + 1/6 = 3/12 + 2/12 = 5/12 tank/hr.
Time = 12/5 hours.
Trap: Adding hours directly (6 + 3 = 9) or averaging them (4.5). Always add RATES, not times.
💡 Explanation (disappears in 5s)
Rate A = 1/6, Rate B = 1/3 = 2/6. Combined = 3/6 = 1/2 tank/hr. Time = 1 ÷ (1/2) = 2 hours.
Q 10
Ratio Parts
Trap ⚠
Easy
⚡ Memory Point: "RATIO a:b → total parts = a + b. Each share = Total × (part / total parts)."
Three friends split a $240 prize in the ratio 1:2:3. How much does the person with the largest share receive?
Trap: Students divide $240 by 3 (the number of people) rather than by 6 (the total ratio parts). Always sum the ratio first.
💡 Explanation (disappears in 5s)
Total parts = 1+2+3 = 6. Each part = 240/6 = $40. Largest share = 3 × $40 = $120.
Statistics & Data Interpretation
Q 11
Mean / Median
Trap ⚠
Medium
⚡ Memory Point: "MEAN uses all values. If one value is unknown, use: unknown = (target mean × n) − (sum of known values)."
A student scored 72, 85, 90, and 78 on four tests. What score does she need on her fifth test to have a mean (average) of exactly 82?
📐 Worked Example (Similar)
Known scores: 70, 80. Want mean = 78 over 3 tests.Needed = 78 × 3 − (70 + 80) = 234 − 150 = 84.
Trap: Finding what score is needed to bring each test up to 82, instead of using the target total.
💡 Explanation (disappears in 5s)
Target total = 82 × 5 = 410. Known sum = 72+85+90+78 = 325. Needed = 410 − 325 = 85.
Q 12
Median Shift
Trap ⚠
Hard
⚡ Memory Point: "MEDIAN is position-based, not value-based — adding an extreme outlier does NOT change the median much."
A dataset has 7 values: 2, 5, 7, 9, 11, 14, 20. An 8th value of 100 is added. By how many does the median increase?
Trap: Students think the outlier (100) raises the median significantly. Remember: median is the middle value, unaffected by extremes.
💡 Explanation (disappears in 5s)
Original (7 values): median = 4th value = 9. New (8 values): median = avg of 4th & 5th = (9+11)/2 = 10. Increase = 10 − 9 = 1.
Q 13
Scatterplot / Correlation
Easy
⚡ Memory Point: "LINE OF BEST FIT: substitute x into the equation to predict y. Don't confuse slope with correlation."
A line of best fit is given by y = 2.5x + 10, where x is hours studied and y is test score. What score does the model predict for a student who studies 8 hours?
Trap: Multiplying 2.5 by 10 (confusing the constant with the coefficient). Substitute carefully.
💡 Explanation (disappears in 5s)
y = 2.5(8) + 10 = 20 + 10 = 30.
Geometry & Measurement
Q 14
Circle Area
Trap ⚠
Medium
⚡ Memory Point: "RADIUS = diameter / 2. Area = πr² (not πd²!). When diameter is given, always halve it first."
A circular garden has a diameter of 10 meters. A path of width 1 meter is built around the outside of the garden. What is the area of the path alone, in square meters? (Leave answer in terms of π.)
📐 Worked Example (Similar)
Inner circle r = 4 m. Outer r = 5 m.Area of ring = π(5²) − π(4²) = 25π − 16π = 9π m².
Trap: Using diameter instead of radius in the formula, or forgetting to compute the difference (ring) vs. just the outer circle.
💡 Explanation (disappears in 5s)
Garden radius = 5 m. Outer radius (garden + path) = 6 m.Path area = π(6²) − π(5²) = 36π − 25π = 11π m².
Q 15
Similar Triangles
Trap ⚠
Medium
⚡ Memory Point: "SIMILAR FIGURES → sides are proportional, areas scale by the SQUARE of the ratio."
Triangle A is similar to Triangle B. The ratio of their corresponding sides is 3:5. If the area of Triangle A is 27 cm², what is the area of Triangle B, in cm²?
📐 Worked Example (Similar)
Side ratio 1:2 → Area ratio = 1²:2² = 1:4.If A₁ = 9, then A₂ = 36.
Trap: Multiplying area by 5/3 instead of (5/3)². Area scales by the SQUARE of the linear scale factor.
💡 Explanation (disappears in 5s)
Area ratio = (3/5)² = 9/25. So 27/A_B = 9/25 → A_B = 27 × 25/9 = 75 cm².
Exponents, Growth & Advanced
Q 16
Exponential Growth
Trap ⚠
Medium
⚡ Memory Point: "EXPONENTIAL GROWTH: A = P(1 + r)ᵗ — percentage growth compounds each period, not adds linearly."
A bacteria population doubles every 3 hours. If there are 500 bacteria now, how many will there be after 9 hours?
📐 Worked Example (Similar)
Doubles every 2 hours. Start: 100. After 6 hours = 3 periods:100 × 2³ = 100 × 8 = 800.
Trap: Multiplying by 2 × 3 (= 6) instead of 2³. After 9 hours there are 9/3 = 3 doubling periods.
💡 Explanation (disappears in 5s)
9 hours / 3 hours per period = 3 doublings. 500 × 2³ = 500 × 8 = 4000.
Q 17
Systems of Equations
Trap ⚠
Hard
⚡ Memory Point: "NO SOLUTION when lines are parallel (same slope, different y-intercept). INFINITE solutions when identical lines."
For what value of k does the system below have no solution?
2x + 3y = 7
4x + ky = 5
4x + ky = 5
📐 Worked Example (Similar)
For no solution: lines must be parallel → same slope, different intercept.x + 2y = 3 and 2x + 4y = 8 → multiply eq 1 by 2: identical → infinite solutions.
x + 2y = 3 and 2x + 4y = 7 → parallel → no solution.
Trap: Setting the equations equal to each other rather than comparing slope ratios. Parallel means a₁/a₂ = b₁/b₂ ≠ c₁/c₂.
💡 Explanation (disappears in 5s)
For parallel lines: 2/4 = 3/k → k = 6. Check intercepts: 7/5 ≠ 2/4, so they are indeed parallel (not identical) → k = 6.
Q 18
Absolute Value
Trap ⚠
Hard
⚡ Memory Point: "|expression| = k → expression = k OR expression = −k. Always check BOTH solutions."
A factory's quality control requires that the diameter d of a part satisfy |d − 50| ≤ 2 mm. What is the largest acceptable diameter?
📐 Worked Example (Similar)
|x − 10| ≤ 3 → −3 ≤ x − 10 ≤ 3 → 7 ≤ x ≤ 13.Largest = 13.
Trap: Solving only one branch (d − 50 = 2 → d = 52) and ignoring the lower bound, or setting up the inequality direction wrong.
💡 Explanation (disappears in 5s)
|d − 50| ≤ 2 → −2 ≤ d − 50 ≤ 2 → 48 ≤ d ≤ 52. Largest acceptable = 52 mm.
Q 19
Compound Interest
Trap ⚠
Hard
⚡ Memory Point: "COMPOUND vs SIMPLE: A = P(1+r)ᵗ vs A = P(1+rt). Compound grows faster — never confuse the two."
$1,000 is invested at an annual interest rate of 10%, compounded annually. After 2 years, how much interest has been earned (not total amount)?
📐 Worked Example (Similar)
$500 at 20% compounded, 2 years.A = 500(1.2)² = 500 × 1.44 = $720. Interest = $220.
Trap: Simple interest gives $200 (10% × $1000 × 2). But compounded = $210 because year 2 earns interest on interest.
💡 Explanation (disappears in 5s)
A = 1000(1.1)² = 1000 × 1.21 = $1,210. Interest = 1210 − 1000 = $210. (Simple interest would give only $200.)
Q 20
Unit Conversion
Boss Level 👑
Hard
⚡ Memory Point: "DIMENSIONAL ANALYSIS — multiply by conversion fractions so unwanted units cancel. Write units out explicitly!"
A car travels at 90 kilometers per hour. How many meters per second is this? (1 km = 1000 m; 1 hour = 3600 seconds.)
📐 Worked Example (Similar)
60 km/hr × (1000 m / 1 km) × (1 hr / 3600 s)
= 60,000 / 3600 m/s
= 16.67 m/s
= 60,000 / 3600 m/s
= 16.67 m/s
Trap: Dividing by 1000 instead of multiplying, or using 60 seconds instead of 3600. Units guide every step — write them out.
💡 Explanation (disappears in 5s)
90 km/hr × (1000 m/km) × (1 hr/3600 s) = 90,000 / 3600 = 25 m/s.