Math Practice — Pre-Algebra & Geometry

Pre-Algebra

Part 1 · Q1–10

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Quick Memory Points — Pre-Algebra

ORDER OF OPERATIONS → PEMDAS: Parentheses · Exponents · Multiply/Divide · Add/Subtract
VARIABLE → a letter that holds an unknown number (solve for it!)
INVERSE OPERATION → undo × with ÷, undo + with −
PROPORTION → cross-multiply to solve: a/b = c/d → a×d = b×c
PERCENT → Part = Percent × Whole ÷ 100
NEGATIVE RULE → (−) × (−) = (+), but (−) × (+) = (−)

Q 01 Pre-Algebra ★★☆ PEMDAS

Evaluate: 3 + 4 × 2 − (6 ÷ 3)

⚠ Trap: Do NOT add 3 + 4 first. Multiplication comes before addition!

PEMDAS → Parentheses first, then ×÷, then +−

📖 Explanation

Step 1: Parentheses → (6 ÷ 3) = 2
Step 2: Multiply → 4 × 2 = 8
Step 3: Left to right → 3 + 8 − 2 = 9
Common mistake: Adding 3 + 4 = 7 first, then 7 × 2 = 14. This ignores PEMDAS!

Q 02 Pre-Algebra ★★☆ Equations

Solve for x: 2x + 5 = 17

⚠ Trap: Subtract 5 first, THEN divide. Do not divide 17 by 2 first!

Isolate x → subtract 5 from both sides → divide both sides by 2

📖 Explanation

2x + 5 = 17
→ Subtract 5: 2x = 12
→ Divide by 2: x = 6
Check: 2(6) + 5 = 12 + 5 = 17 ✓

Q 03 Pre-Algebra ★★★ Negative Numbers

What is −3 × (−4) + (−2)?

⚠ Trap: (−) × (−) is POSITIVE, not negative! Then add −2 at the end.

(−) × (−) = (+) | (+) + (−) = subtract

📖 Explanation

Step 1: −3 × (−4) = +12 (negative × negative = positive)
Step 2: 12 + (−2) = 12 − 2 = 10
Most common error: treating (−)×(−) as negative, getting −12 + (−2) = −14.

Q 04 Pre-Algebra ★★★ Proportions

A recipe needs 3 cups of flour for 12 cookies. How many cups are needed for 20 cookies?

⚠ Trap: Set up the proportion carefully — keep the same units on each side.

3 cups12 cookies = x cups20 cookies → Cross-multiply

📖 Explanation

Cross-multiply: 3 × 20 = 12 × x
60 = 12x
x = 60 ÷ 12 = 5 cups

Q 05 Pre-Algebra ★★★ Percent

A jacket costs $80. It is on sale for 25% off. What is the sale price?

⚠ Trap: 25% off means you pay 75% of the price — don't forget to subtract the discount!

Discount = 80 × 0.25 | Sale Price = 80 − Discount

📖 Explanation

Discount = 80 × 0.25 = $20
Sale Price = 80 − 20 = $60
Or: 80 × 0.75 = $60 (since you keep 75%)

Q 06 Pre-Algebra ★★★ Inequalities

Which value of x makes this TRUE? 3x − 2 > 10

⚠ Trap: Solve like an equation, but remember — if you multiply/divide by a NEGATIVE, flip the sign!

3x − 2 > 10 → 3x > 12 → x > 4

📖 Explanation

3x − 2 > 10 → 3x > 12 → x > 4
Only x = 5 satisfies x > 4. Test: 3(5) − 2 = 13 > 10 ✓
x = 4: 3(4) − 2 = 10, which is NOT greater than 10.

Q 07 Pre-Algebra ★★★ Exponents

Simplify: 2³ × 2²

⚠ Trap: Do NOT multiply the exponents! When multiplying same bases, you ADD the exponents.

a^m × aⁿ = a^m+n → 2³⁺² = 2⁵

📖 Explanation

Same base → ADD exponents: 2³ × 2² = 2³⁺² = 2⁵ = 32
Common error: 2^3×2 = 2⁶ = 64 — this is the POWER rule (a^m)ⁿ, not product rule!

Q 08 Pre-Algebra ★★☆ Expressions

Simplify by combining like terms: 5x + 3 − 2x + 7

⚠ Trap: You can only combine terms with the same variable. 5x and 3 are NOT like terms!

Like terms: 5x and −2x → combine. Constants: 3 and 7 → combine.

📖 Explanation

x-terms: 5x − 2x = 3x
Constants: 3 + 7 = 10
Answer: 3x + 10

Q 09 Pre-Algebra ★★★ Fractions in Equations

Solve for x: x3 + 2 = 5

⚠ Trap: Subtract 2 first, then multiply both sides by 3. Don't multiply by 3 first on a mixed expression!

x/3 + 2 = 5 → x/3 = 3 → x = 9

📖 Explanation

x/3 + 2 = 5
Subtract 2: x/3 = 3
Multiply by 3: x = 9
Check: 9/3 + 2 = 3 + 2 = 5 ✓

Q 10 Pre-Algebra ★★★ Word Problem · Rate

Maria walks 3 miles per hour. She needs to reach a store that is 7.5 miles away. How long will it take her?

⚠ Trap: Use Time = Distance ÷ Rate. Don't multiply distance × rate!

Rate × Time = Distance → Time = Distance ÷ Rate

📖 Explanation

Time = Distance ÷ Rate = 7.5 ÷ 3 = 2.5 hours
Check: 3 mph × 2.5 h = 7.5 miles ✓

Geometry

Part 2 · Q11–20

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Quick Memory Points — Geometry

TRIANGLE SUM → 3 angles always add up to 180°
AREA of RECTANGLE → A = length × width
AREA of TRIANGLE → A = ½ × base × height
CIRCLE → Area = πr², Circumference = 2πr (r = radius)
PYTHAGOREAN THEOREM → a² + b² = c² (right triangle only!)
SUPPLEMENTARY → two angles add to 180° | COMPLEMENTARY → add to 90°
VOLUME of RECTANGULAR PRISM → V = l × w × h

Q 11 Geometry ★★☆ Triangle Angles

A triangle has angles of 45° and 70°. What is the measure of the third angle?

⚠ Trap: All 3 angles must sum to 180° — not 90° or 360°!

∠A + ∠B + ∠C = 180° → ∠C = 180° − 45° − 70°

📖 Explanation

Third angle = 180° − 45° − 70° = 65°
Check: 45 + 70 + 65 = 180 ✓

Q 12 Geometry ★★☆ Area of Triangle

A triangle has a base of 10 cm and a height of 6 cm. What is its area?

⚠ Trap: Don't forget the ½! Many students use base × height without dividing by 2.

Area = ½ × base × height = ½ × 10 × 6

📖 Explanation

A = ½ × 10 × 6 = ½ × 60 = 30 cm²
The ½ is essential — a triangle is exactly half of a rectangle with the same base and height.

Q 13 Geometry ★★★ Circle — Circumference

A circular pizza has a diameter of 14 inches. What is the circumference? (Use π ≈ 3.14)

⚠ Trap: The formula uses RADIUS, but you are given DIAMETER. Radius = Diameter ÷ 2.

C = 2πr → r = 14 ÷ 2 = 7 → C = 2 × 3.14 × 7

📖 Explanation

Radius r = 14 ÷ 2 = 7 in
C = 2 × 3.14 × 7 = 43.96 inches
Option A (21.98) is C = π × r (missing the 2). Option B is the area (πr²).

Q 14 Geometry ★★★ Pythagorean Theorem

A right triangle has legs of 3 ft and 4 ft. What is the length of the hypotenuse?

⚠ Trap: The hypotenuse is NOT 3 + 4 = 7. You must use squares!

a² + b² = c² → 3² + 4² = c² → 9 + 16 = c² → c = √25

📖 Explanation

a² + b² = c²
3² + 4² = 9 + 16 = 25
c = √25 = 5 ft
(3, 4, 5) is the most famous Pythagorean triple — memorize it!

Q 15 Geometry ★★★ Supplementary Angles

Two angles are supplementary. One angle measures 112°. What is the other angle?

⚠ Trap: Supplementary = 180° total. Don't use 90° (that's complementary)!

Supplementary: ∠A + ∠B = 180° | Complementary: ∠A + ∠B = 90°

📖 Explanation

Supplementary → total = 180°
Other angle = 180° − 112° = 68°
If complementary: 90° − 112° is impossible (negative). Always identify supplementary vs. complementary!

Q 16 Geometry ★★★ Area of Circle

A circle has a radius of 5 m. What is its area? (Use π ≈ 3.14)

⚠ Trap: Area uses r² (radius SQUARED). Don't confuse Area = πr² with Circumference = 2πr.

Area = π × r² = 3.14 × 5² = 3.14 × 25

📖 Explanation

Area = π × r² = 3.14 × 5² = 3.14 × 25 = 78.5 m²
Option A (31.4) = 2πr = circumference, not area!

Q 17 Geometry ★★☆ Perimeter

A rectangle has a length of 9 cm and a width of 4 cm. What is the perimeter?

⚠ Trap: Perimeter = all 4 sides added. Don't just add 9 + 4 = 13. There are TWO of each side!

P = 2 × (length + width) = 2 × (9 + 4)

📖 Explanation

P = 2(9 + 4) = 2 × 13 = 26 cm
A rectangle has 2 lengths and 2 widths: 9 + 9 + 4 + 4 = 26 ✓

Q 18 Geometry ★★★ Volume

A box has a length of 5 cm, width of 3 cm, and height of 4 cm. What is its volume?

⚠ Trap: Volume uses all THREE dimensions multiplied. Area only uses two!

V = l × w × h = 5 × 3 × 4

📖 Explanation

V = 5 × 3 × 4 = 15 × 4 = 60 cm³
Note: volume is in cubic units (cm³), not square units (cm²).

Q 19 Geometry ★★★★ Missing Side — Pythagorean

A right triangle has a hypotenuse of 13 m and one leg of 5 m. What is the length of the other leg?

⚠ Trap: The hypotenuse is c (the LONGEST side). Solve for a missing leg: a² = c² − b²

a² = c² − b² = 13² − 5² = 169 − 25 = 144 → a = √144

📖 Explanation

a² = 13² − 5² = 169 − 25 = 144
a = √144 = 12 m
(5, 12, 13) is another famous Pythagorean triple — memorize it!

Q 20 Geometry ★★★★ Composite Shapes

A figure is made of a rectangle (6 m × 4 m) with a triangle on top (base 6 m, height 3 m). What is the total area of the figure?

⚠ Trap: Add the AREAS — not perimeters. Use correct formula for each shape separately!

Total Area = Rectangle + Triangle = (l×w) + (½×b×h)

📖 Explanation

Rectangle: 6 × 4 = 24 m²
Triangle: ½ × 6 × 3 = 9 m²
Total = 24 + 9 = 33 m²