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Self-Study Workbook

Pre‑Algebra
& Geometry

20 essential problems. Pick an answer, get instant feedback. ★ tags show quick memory tricks.

Pre-Algebra
10 Problems · Variables · Equations · Fractions · Ratios
01
ORDER OF OPERATIONS
Easy
Evaluate: 3 + 4 × 2 − 1
⚠️ Common mistake: adding 3+4 first!
Memory Point
PEMDAS — Parentheses · Exponents · Multiply · Divide · Add · Subtract
Always multiply/divide before add/subtract.
📘 Explanation
Step 1 — Multiply first: 4 × 2 = 8
Step 2 — Then left to right: 3 + 8 = 11
Step 3: 11 − 1 = 10 ✓
Why A is wrong: 3+4=7, then 7×2=14, then 14−1=13 — but this breaks PEMDAS!
02
SOLVING EQUATIONS
Easy
Solve for x: 2x + 5 = 13
Memory Point
UNDO — work backwards. Subtract first, then divide.
Think: what was done to x? Undo it in reverse order.
📘 Explanation
Step 1 — Subtract 5 from both sides: 2x = 13 − 5 = 8
Step 2 — Divide both sides by 2: x = 8 ÷ 2 = 4 ✓
Check: 2(4) + 5 = 8 + 5 = 13 ✓
03
NEGATIVE NUMBERS
Tricky
What is (−3) × (−4) + (−2)?
⚠️ Tricky: double negative becomes positive!
Memory Point
SAME SIGN → POSITIVE  |  DIFFERENT SIGN → NEGATIVE
(−) × (−) = (+)     (+) × (−) = (−)
📘 Explanation
Step 1 — Multiply: (−3) × (−4) = +12 (same signs = positive)
Step 2 — Add: 12 + (−2) = 12 − 2 = 10 ✓
04
FRACTIONS
Tricky
Simplify: 34 + 16
Memory Point
LCD — Least Common Denominator. Find the smallest number both denominators divide into.
4 and 6 → LCD = 12
📘 Explanation
LCD of 4 and 6 = 12
Convert: 3/4 = 9/12    1/6 = 2/12
Add: 9/12 + 2/12 = 11/12 ✓
Why A is wrong: you can't just add numerators AND denominators (4+10 doesn't work).
05
DISTRIBUTIVE PROPERTY
Tricky
Expand: 3(x − 4) + 2x
Memory Point
DISTRIBUTE — multiply outside number by each term inside.
3(x − 4) = 3·x − 3·4 = 3x − 12
📘 Explanation
Step 1 — Distribute 3: 3(x−4) = 3x − 12
Step 2 — Add 2x: 3x − 12 + 2x = 5x − 12 ✓
A is wrong: 3(x−4) gives −12, not −4. Don't forget to multiply 3×4=12!
06
RATIO & PROPORTION
Easy
If a car travels 120 miles in 2 hours, how far does it travel in 5 hours at the same speed?
Memory Point
UNIT RATE — find miles per 1 hour first, then multiply.
120 ÷ 2 = 60 mph → 60 × 5 = ?
📘 Explanation
Unit rate: 120 ÷ 2 = 60 miles/hour
5 hours: 60 × 5 = 300 miles ✓
07
EXPONENTS
Tricky
What is the value of 23 × 22?
⚠️ Common mistake: multiplying the exponents!
Memory Point
SAME BASE → ADD exponents
am × an = am+n    23 × 22 = 23+2 = 25
📘 Explanation
Add exponents (same base): 23 × 22 = 23+2 = 25
Calculate: 25 = 2×2×2×2×2 = 32 ✓
B is wrong: multiplying exponents (3×2=6) gives 26=64 — that rule is for (23)2!
08
PERCENT
Easy
A jacket costs $80. It is on sale for 25% off. What is the sale price?
Memory Point
PERCENT OFF → multiply by (1 − rate)
25% off = 75% remains → $80 × 0.75 = ?
📘 Explanation
Discount amount: 25% of $80 = 0.25 × 80 = $20
Sale price: $80 − $20 = $60 ✓
D is only the discount amount, not the price you pay!
09
INEQUALITIES
Tricky
Solve: −2x > 8
⚠️ Watch out: dividing by a negative flips the sign!
Memory Point
FLIP — dividing or multiplying by a negative number → flip the inequality sign.
(>) becomes (<)    (<) becomes (>)
📘 Explanation
Divide both sides by −2:
−2x ÷ (−2) > 8 ÷ (−2) → FLIP the sign!
x < −4 ✓
A is wrong: forgot to flip > to < when dividing by negative!
10
SLOPE
Tricky
Find the slope of the line passing through points (2, 3) and (6, 11).
Memory Point
RISE over RUN — slope = y₂ − y₁x₂ − x₁
Change in y ÷ Change in x
📘 Explanation
Rise (Δy): 11 − 3 = 8
Run (Δx): 6 − 2 = 4
Slope: 8 ÷ 4 = 2 ✓
Geometry
10 Problems · Angles · Area · Pythagorean Theorem · Circles
G1
SUPPLEMENTARY ANGLES
Easy
Two angles are supplementary. One angle measures 65°. What is the other angle?
⚠️ Don't confuse supplementary (180°) with complementary (90°)!
Memory Point
SUPPLEMENTARY = 180° (Straight line)   S is for Straight.
COMPLEMENTARY = 90° (Corner/Right angle)   C is for Corner.
📘 Explanation
Supplementary angles add up to 180°
Other angle = 180° − 65° = 115° ✓
A (25°) is the complement of 65° — those add to 90°, not 180°.
G2
PYTHAGOREAN THEOREM
Easy
A right triangle has legs of length 3 and 4. What is the hypotenuse?
Memory Point
a² + b² = c² — legs² + legs² = hypotenuse²
Memorize the 3-4-5 triple! (Also: 5-12-13, 8-15-17)
📘 Explanation
a² + b²: 3² + 4² = 9 + 16 = 25
c = √25 = 5 ✓
This is the famous 3-4-5 Pythagorean triple — worth memorizing!
G3
AREA OF TRIANGLE
Easy
A triangle has a base of 10 cm and a height of 6 cm. What is its area?
Memory Point
Area = ½ × base × height
A triangle is HALF of a rectangle — always divide by 2!
📘 Explanation
Area = ½ × b × h
= ½ × 10 × 6 = ½ × 60 = 30 cm² ✓
A (60) forgot to multiply by ½. A triangle is only half a rectangle!
G4
CIRCLE AREA
Tricky
A circle has a diameter of 10 cm. What is its area? (Use π ≈ 3.14)
⚠️ Diameter ≠ Radius! Divide by 2 first.
Memory Point
Area = πr²   but r = d ÷ 2 !
diameter = 10 → radius = 5 → r² = 25
📘 Explanation
Radius: r = 10 ÷ 2 = 5
Area: π × 5² = 3.14 × 25 = 78.5 cm² ✓
A uses r=10 (didn't divide diameter by 2): 3.14 × 100 = 314 — wrong!
G5
VERTICAL ANGLES
Easy
Two lines intersect. One pair of vertical angles measures 70°. What is the measure of the adjacent angle?
⚠️ Adjacent ≠ vertical! Vertical angles are equal; adjacent are supplementary.
Memory Point
VERTICAL = EQUAL (opposite angles)
ADJACENT = 180° (next to each other on a line)
📘 Explanation
Adjacent angles on a line are supplementary (add to 180°)
Adjacent angle = 180° − 70° = 110° ✓
A (70°) would be correct for the vertical (opposite) angle, not the adjacent one!
G6
TRIANGLE ANGLE SUM
Easy
A triangle has angles of 45° and 85°. What is the third angle?
Memory Point
TRIANGLE SUM = 180° — always, no exceptions.
Third angle = 180° − (first + second)
📘 Explanation
Sum of angles in triangle = 180°
Third angle = 180° − 45° − 85° = 180° − 130° = 50° ✓
G7
PERIMETER VS AREA
Tricky
A rectangle is 8 cm long and 5 cm wide. A student says the perimeter is 40 cm. Is this correct?
⚠️ Most confused formula: Perimeter vs. Area!
Memory Point
PERIMETER = 2(l + w) = total fence around
AREA = l × w = carpet inside the room
📘 Explanation
Perimeter: P = 2(l + w) = 2(8 + 5) = 2 × 13 = 26 cm ✓
Area: A = l × w = 8 × 5 = 40 cm²
The student calculated the Area (40), not the Perimeter! Common mix-up.
G8
SIMILAR TRIANGLES
Tricky
Two similar triangles have sides in ratio 1:3. If the smaller triangle has an area of 4 cm², what is the area of the larger one?
⚠️ Area scales by the square of the ratio!
Memory Point
SIDE RATIO → AREA RATIO²
If sides ratio = 1:3, then area ratio = 1²:3² = 1:9
📘 Explanation
Side ratio = 1:3 → Area ratio = 1²:3² = 1:9
Larger area: 4 × 9 = 36 cm² ✓
A (12 = 4×3) only multiplied by the side ratio, not its square!
G9
VOLUME OF RECTANGULAR PRISM
Easy
A box is 5 cm long, 4 cm wide, and 3 cm tall. What is its volume?
Memory Point
V = l × w × h — length × width × height
Think of stacking layers: one layer = l × w, then stack h layers.
📘 Explanation
V = l × w × h
= 5 × 4 × 3 = 20 × 3 = 60 cm³ ✓
G10
CIRCUMFERENCE
Tricky
A wheel has a radius of 7 cm. How far does it travel in one full rotation? (Use π ≈ 3.14)
⚠️ One full rotation = circumference, not area!
Memory Point
C = 2πr (or C = πd) — the distance around a circle.
"One rotation = one circumference" — the wheel rolls its own edge.
📘 Explanation
C = 2πr = 2 × 3.14 × 7
= 6.28 × 7 = 43.96 cm ✓
A (153.86) is the area πr² = 3.14 × 49. B only multiplied πr (forgot the 2).
Pre-Algebra & Geometry · Self-Study Workbook · 20 Problems Total