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Part A — Pre-Algebra
Variables, expressions, equations, and number sense
Evaluate: 3 + 4 × (2² − 1)
📌 Quick Concept
Always follow PEMDAS — Parentheses → Exponents → Multiply/Divide → Add/Subtract.
Example: 2 + 3 × (1² + 1) = 2 + 3 × 2 = 2 + 6 = 8
Example: 2 + 3 × (1² + 1) = 2 + 3 × 2 = 2 + 6 = 8
✓ Explanation
Step 1 — Parentheses & exponent: 2² = 4, then 4 − 1 = 3Step 2 — Multiply: 4 × 3 = 12
Step 3 — Add: 3 + 12 = 15
Solve for x: 5x − 7 = 18
📌 Quick Concept
To isolate the variable, do the opposite operation on both sides.
Example: 3x − 4 = 11 → 3x = 15 → x = 5
Example: 3x − 4 = 11 → 3x = 15 → x = 5
✓ Explanation
Add 7 to both sides: 5x = 25Divide by 5: x = 5
If 3 notebooks cost $7.50, how much do 8 notebooks cost?
📌 Quick Concept
Set up a proportion: 3/7.50 = 8/x, then cross-multiply.
OR: find the unit rate first → cost per item × quantity.
OR: find the unit rate first → cost per item × quantity.
✓ Explanation
Unit rate: $7.50 ÷ 3 = $2.50 per notebook8 × $2.50 = $20.00
Which expression has the greatest value?
📌 Quick Concept
(−) × (−) = positive | (−) × (+) = negative
Same signs → positive. Different signs → negative.
Same signs → positive. Different signs → negative.
✓ Explanation
A = +12, B = −12, C = +10, D = +1515 is the greatest value.
A jacket originally priced at $80 is on sale for 35% off. What is the sale price?
📌 Quick Concept
Discount = Original × (% ÷ 100)
Sale price = Original − Discount
Shortcut: Sale price = Original × (1 − 0.35) = Original × 0.65
Sale price = Original − Discount
Shortcut: Sale price = Original × (1 − 0.35) = Original × 0.65
✓ Explanation
Discount: $80 × 0.35 = $28Sale price: $80 − $28 = $52
Simplify: 4x + 3y − 2x + 5y − 1
📌 Quick Concept
Group terms with the same variable together. Constants are like terms too.
Example: 5a + 2b − 3a = 2a + 2b
Example: 5a + 2b − 3a = 2a + 2b
✓ Explanation
x-terms: 4x − 2x = 2xy-terms: 3y + 5y = 8y
Constant: −1
Result: 2x + 8y − 1
Solve: −3x > 12. Which value of x satisfies this?
📌 Quick Concept
When you multiply or divide by a negative number, flip the inequality sign!
−3x > 12 → x < −4
−3x > 12 → x < −4
✓ Explanation
Divide both sides by −3 and flip the sign: x < −4Only −5 is less than −4. Answer: x = −5
Calculate: 2/3 + 3/4 − 1/6
📌 Quick Concept
Find the LCM of the denominators first.
LCM(3, 4, 6) = 12
Convert: 8/12 + 9/12 − 2/12
LCM(3, 4, 6) = 12
Convert: 8/12 + 9/12 − 2/12
✓ Explanation
LCM = 12: 8/12 + 9/12 − 2/12 = 15/12 = 5/4 (or 1¼)
Jake is 3 years older than twice his sister's age. His sister is 7. How old is Jake?
📌 Quick Concept
Translate the sentence into an equation. Let s = sister's age.
Jake = 2s + 3
Jake = 2s + 3
✓ Explanation
Jake = 2 × 7 + 3 = 14 + 3 = 17
Which number is correctly written in scientific notation?
📌 Quick Concept
Scientific notation: a × 10n where 1 ≤ a < 10
The digit before the decimal must be between 1 and 9 (inclusive).
The digit before the decimal must be between 1 and 9 (inclusive).
✓ Explanation
Only 4.23 × 10³ has a coefficient between 1 and 10. All others are out of standard form.
Part B — Geometry
Shapes, angles, area, perimeter, and coordinate geometry
In a triangle, two angles measure 47° and 68°. What is the third angle?
📌 Quick Concept
The sum of all angles in any triangle = 180°
Third angle = 180° − (angle1 + angle2)
Third angle = 180° − (angle1 + angle2)
✓ Explanation
47 + 68 = 115180 − 115 = 65°
A triangle has a base of 14 cm and a height of 9 cm. What is its area?
📌 Quick Concept
A = ½ × b × h
The height must be perpendicular to the base — watch out for slanted sides!
The height must be perpendicular to the base — watch out for slanted sides!
✓ Explanation
A = ½ × 14 × 9 = ½ × 126 = 63 cm²Common mistake: forgetting the ½ (that gives 126).
A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?
📌 Quick Concept
a² + b² = c² — only works for the right angle's opposite side (hypotenuse).
Famous triples to memorize: 3-4-5, 5-12-13, 8-15-17
Famous triples to memorize: 3-4-5, 5-12-13, 8-15-17
✓ Explanation
5² + 12² = 25 + 144 = 169√169 = 13 (a classic 5-12-13 Pythagorean triple)
A circle has a diameter of 10 cm. What is its area? (Use π ≈ 3.14)
📌 Quick Concept
Radius = diameter ÷ 2 — the most common mistake is using diameter in the formula!
Area = π × r² | Circumference = 2πr = πd
Area = π × r² | Circumference = 2πr = πd
✓ Explanation
r = 10 ÷ 2 = 5A = 3.14 × 5² = 3.14 × 25 = 78.5 cm²
Choice B uses diameter (10²) — a very common trap!
Two parallel lines are cut by a transversal. One angle measures 115°. What is the measure of its co-interior (same-side interior) angle?
📌 Quick Concept
With parallel lines + transversal:
• Alternate interior angles → equal
• Co-interior (same-side) angles → supplementary (add to 180°)
• Corresponding angles → equal
• Alternate interior angles → equal
• Co-interior (same-side) angles → supplementary (add to 180°)
• Corresponding angles → equal
✓ Explanation
Co-interior (same-side interior) angles are supplementary.180° − 115° = 65°
An L-shaped figure is made by removing a 3 × 4 rectangle from a 7 × 8 rectangle. What is the perimeter of the L-shape?
📌 Quick Concept
For composite shapes, trace only the outer boundary. Do not add interior lines.
Tip: When a notch is cut, two sides are replaced by two new sides — total outer edges stay countable.
Tip: When a notch is cut, two sides are replaced by two new sides — total outer edges stay countable.
✓ Explanation
Original rectangle perimeter = 2(7+8) = 30.Cutting a notch removes two sides (3 and 4) but adds two new sides (3 and 4) on the inside — so the perimeter = 30 + 4 + 4 = 38 units.
Each cut notch adds its two inner edges to the perimeter.
A box is 6 cm long, 4 cm wide, and 5 cm tall. How many 1 cm³ cubes fit exactly inside?
📌 Quick Concept
Volume = length × width × height
Volume counts 3-dimensional space — make sure all units match.
Volume counts 3-dimensional space — make sure all units match.
✓ Explanation
V = 6 × 4 × 5 = 120 cm³Each 1 cm³ cube fills exactly 1 cm³, so 120 cubes fit.
What is the midpoint of the segment connecting (2, −3) and (8, 7)?
📌 Quick Concept
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
Just average the x-coordinates and average the y-coordinates.
Just average the x-coordinates and average the y-coordinates.
✓ Explanation
x: (2 + 8) ÷ 2 = 10 ÷ 2 = 5y: (−3 + 7) ÷ 2 = 4 ÷ 2 = 2
Midpoint = (5, 2)
Two similar triangles have corresponding sides in the ratio 3 : 5. The smaller triangle has a base of 9 cm. What is the base of the larger triangle?
📌 Quick Concept
Similar triangles have equal angles and proportional sides.
If ratio is 3:5, multiply any side of the smaller triangle by 5/3 to get the larger.
If ratio is 3:5, multiply any side of the smaller triangle by 5/3 to get the larger.
✓ Explanation
Set up proportion: 3/5 = 9/xCross-multiply: 3x = 45 → x = 15 cm
A cube has a side length of 4 cm. What is its total surface area?
📌 Quick Concept
A cube has 6 identical square faces.
Surface Area = 6 × s²
Don't confuse with Volume = s³!
Surface Area = 6 × s²
Don't confuse with Volume = s³!
✓ Explanation
SA = 6 × 4² = 6 × 16 = 96 cm²Choice A (64) is the volume (4³). Don't mix them up!