Emma has 3 times as many stickers as Jake. Together they have 48 stickers.
How many stickers does Emma have?
⚠️ Careful — many students find Jake's count instead of Emma's!
🧠
Memory Point
LET the smaller = x E = 3x, J = x → x + 3x = 48 → x = 12 Emma = 3 × 12 = 36 ← answer the RIGHT person!
📖 Step-by-Step Explanation
Let Jake = x stickers. Since Emma has 3 times as many, Emma = 3x.
Together: x + 3x = 48 → 4x = 48 → x = 12 (Jake's count).
Emma = 3 × 12 = 36.
The classic mistake: stopping at x = 12 and picking (A). Always re-read who the question asks about!
02
Rate & TimeModerate⚠ Common Mistake
A car travels at 60 mph for 2 hours, then at 40 mph for 3 hours.
What is the average speed for the entire trip?
⚠️ Do NOT just average 60 and 40!
🧠
Memory Point
Average speed = Total Distance ÷ Total Time NOT (v₁ + v₂) ÷ 2 when times differ!
📖 Step-by-Step Explanation
Total distance = (60 × 2) + (40 × 3) = 120 + 120 = 240 miles.
Total time = 2 + 3 = 5 hours.
Average speed = 240 ÷ 5 = 48 mph.
The trap answer (A) 50 mph comes from averaging speeds directly — only valid when time segments are equal!
03
Percent & DiscountBeginner
A jacket originally costs $120. It is on sale for 25% off, then a further 10% off the sale price.
What is the final price?
🧠
Memory Point
MULTIPLY multipliers: × 0.75 × 0.90 NOT 35% off! Successive % ≠ sum of %
📖 Step-by-Step Explanation
After 25% off: $120 × 0.75 = $90.
After another 10% off: $90 × 0.90 = $81.
Common mistake: 25% + 10% = 35% → $120 × 0.65 = $78 — WRONG! Each percent applies to the already-reduced price.
04
InequalitiesModerate⚠ Sign Flip!
Solve: −3x + 7 > 16
Which value of x satisfies the inequality?
🧠
Memory Point
FLIP the sign when ÷ or × by NEGATIVE! −3x > 9 → x < −3 (sign flips!)
📖 Step-by-Step Explanation
−3x + 7 > 16 → −3x > 9 → x < −3 (flip! dividing by −3).
Test each: Only x = −5 satisfies x < −3.
Check: −3(−5) + 7 = 15 + 7 = 22 > 16 ✓
x = −2: −3(−2)+7 = 13, not > 16 ✗
05
Systems of EquationsModerate
Two friends order food. Alex pays $14 for 2 burgers and 1 drink. Sam pays $20 for 2 burgers and 3 drinks.
How much does one drink cost?
🧠
Memory Point
ELIMINATE: subtract eq1 from eq2 2b + 3d − (2b + 1d) = 20 − 14 → 2d = 6 → d = 3
📖 Step-by-Step Explanation
Eq1: 2b + d = 14 | Eq2: 2b + 3d = 20
Subtract Eq1 from Eq2: 2d = 6 → d = $3.
Back-substitute: 2b + 3 = 14 → b = $5.50. Check Eq2: 2(5.5)+3(3)=11+9=20 ✓
06
Slope & InterceptBeginner⚠ Read carefully
A line passes through (2, 5) and (6, 13).
What is the y-intercept of this line?
🧠
Memory Point
SLOPE first: m = (y₂−y₁)/(x₂−x₁) Then: y = mx + b → plug one point → solve b
📖 Step-by-Step Explanation
Slope m = (13−5)/(6−2) = 8/4 = 2.
Use y = 2x + b. Plug in (2, 5): 5 = 2(2) + b → 5 = 4 + b → b = 1.
Equation: y = 2x + 1. Verify: 2(6)+1 = 13 ✓
07
FactoringModerate⚠ Sign trap
Which are the solutions to: x² − x − 12 = 0?
🧠
Memory Point
Find two nums: MULTIPLY = −12, ADD = −1 +3 and −4 ✓ → (x+3)(x−4) = 0
📖 Step-by-Step Explanation
Factor: need two numbers that multiply to −12 and add to −1.
+3 × (−4) = −12 ✓ and (+3) + (−4) = −1 ✓
(x + 3)(x − 4) = 0 → x = −3 or x = 4.
Common mistake: flipping signs → x = 3 or x = −4 (B). Always set each factor = 0!
08
FunctionsBeginner
If f(x) = 3x² − 2x + 1, what is f(−2)?
🧠
Memory Point
SUBSTITUTE carefully: (−2)² = +4, NOT −4 Negative number squared = POSITIVE!
Set up proportion: 2.5/12 = x/60.
Cross-multiply: 12x = 150 → x = 150/12 = 12.5 cups.
Or notice 60 ÷ 12 = 5 batches, so 2.5 × 5 = 12.5.
10
Word Problem — AgeModerate⚠ Future age
Mom is 4 times as old as her son now. In 6 years, she will be 3 times as old.
How old is the son now?
🧠
Memory Point
NOW vs FUTURE — make a table! Now: M=4s | Future: M+6 = 3(s+6) → solve s
📖 Step-by-Step Explanation
Let son's age now = s. Mom's age now = 4s.
In 6 years: (4s + 6) = 3(s + 6)
4s + 6 = 3s + 18 → s = 12.
Check: Son is 12, Mom is 48. In 6 years: Son 18, Mom 54. 54 = 3×18 ✓
△
Geometry
Triangles · Circles · Area · Angles · Proofs
11
Pythagorean TheoremBeginner⚠ Which is the hypotenuse?
A right triangle has legs of length 7 and 24.
What is the length of the hypotenuse?
Co-interior (same-side interior) angles are supplementary: they add up to 180°.
180° − 65° = 115°.
Don't confuse with alternate interior angles (which are equal at 65°). "Same side" = supplementary; "opposite side" = equal.
14
Triangle — Exterior AngleModerate⚠ Easy to confuse
In a triangle, two interior angles are 47° and 68°.
What is the measure of the exterior angle at the third vertex?
🧠
Memory Point
Exterior angle = SUM of two NON-adjacent interior angles OR: interior = 180−47−68 = 65, exterior = 180−65 = 115
📖 Step-by-Step Explanation
Third interior angle = 180° − 47° − 68° = 65°.
Exterior angle = 180° − 65° = 115°.
Shortcut: exterior = 47° + 68° = 115° (exterior angle theorem). (A) is the interior angle — a very common wrong answer!
15
Area of Composite FiguresModerate
An L-shaped figure has a total width of 10 m, total height of 8 m. A rectangle of 4 m × 3 m is cut from the top-right corner.
What is the area of the L-shape?
🧠
Memory Point
SUBTRACT method: Big rectangle − cut-out (10×8) − (4×3) = 80 − 12 = 68 m²
📖 Step-by-Step Explanation
Big rectangle area = 10 × 8 = 80 m².
Cut-out = 4 × 3 = 12 m².
L-shape area = 80 − 12 = 68 m².
(B) 80 is the full rectangle before cutting — always subtract the removed piece!
16
Similar TrianglesModerate⚠ Scale factor
Two similar triangles have corresponding sides of 6 cm and 9 cm.
The area of the smaller triangle is 24 cm². What is the area of the larger?
🧠
Memory Point
Area ratio = (side ratio)² ← SQUARE IT! Side ratio 6:9 = 2:3 → Area ratio = 4:9
📖 Step-by-Step Explanation
Scale factor = 9/6 = 3/2.
Area ratio = (3/2)² = 9/4.
Larger area = 24 × (9/4) = 54 cm².
Mistake: multiplying by just 3/2 (not squaring) → 24 × 1.5 = 36 (A). Areas scale by the square of the linear ratio!
17
Volume of CylinderBeginner⚠ Radius vs Diameter
A cylindrical can has a diameter of 8 cm and a height of 15 cm.
What is its volume? (Use π ≈ 3.14)
🧠
Memory Point
V = πr²h → r = d/2 = 4 HALVE diameter FIRST, then square!
📖 Step-by-Step Explanation
r = 8 ÷ 2 = 4 cm.
V = π × 4² × 15 = 3.14 × 16 × 15 = 3.14 × 240 = 753.6 cm³.
Mistake (A): using d = 8 as radius → 3.14 × 64 × 15 = 3,014.4 — 4× too large!
18
Coordinate GeometryModerate
What is the distance between points (1, 2) and (7, 10)?
Δx = 7−1 = 6, Δy = 10−2 = 8.
Distance = √(6² + 8²) = √(36 + 64) = √100 = 10.
This is a 6-8-10 right triangle (3-4-5 scaled by 2). Recognizing Pythagorean triples saves computation!
19
Polygons — Interior AnglesModerate
What is the sum of interior angles of a hexagon (6 sides)?
🧠
Memory Point
Sum = (n − 2) × 180° n = number of sides → n=6: (6−2)×180 = 720°
📖 Step-by-Step Explanation
Formula: (n − 2) × 180° = (6 − 2) × 180° = 4 × 180° = 720°.
(A) 540° is for a pentagon (5 sides). (D) 1080° is for an octagon (8 sides). Always apply the formula with the correct n!
20
Surface AreaModerate⚠ Count all faces!
A rectangular box (cuboid) measures 5 cm × 4 cm × 3 cm.
What is the total surface area?
SA = 2(lw + lh + wh) = 2(5×4 + 5×3 + 4×3)
= 2(20 + 15 + 12) = 2 × 47 = 94 cm².
(B) 47 is just half — forgetting the "×2" since each pair has two faces is a classic error!
(A) 60 = volume, not surface area.