Math Practice — Pre-Algebra & Geometry

📐 Pre-Algebra

Problems 1–10 · Topics most students get wrong · Read the example, then try on your own!

📋 Topics Covered

1. Order of Operations (PEMDAS) #1

2. Integer Operations (Negative Numbers) #2

3. Fractions — LCM & Adding #3

4. Distributive Property #4

5. Solving One-Step Equations #5

6. Inequalities #6

7. Ratios & Proportions #7

8. Percent Problems #8

9. Absolute Value #9

10. Exponent Rules #10

1. Order of Operations PEMDAS ⚠ Common Trap

Evaluate: $ 3 + 4 \times 2^2 - (6 \div 3) $

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PEMDAS = Parentheses → Exponents → Multiply/Divide → Add/Subtract
Trap: Don't add before multiplying! Left → Right for ×÷ and +−

✏️ Worked Example

Simplify: $ 5 + 2 \times 3^2 - (4 \div 2) $
Step 1 — Parentheses: $ (4 \div 2) = 2 $
Step 2 — Exponents: $ 3^2 = 9 $
Step 3 — Multiply: $ 2 \times 9 = 18 $
Step 4 — Left to Right: $ 5 + 18 - 2 = \mathbf{21} $

✍ Your Turn

Evaluate: $ 10 - 2 \times (3 + 1)^2 \div 4 $

2. Negative Number Operations Integers ⚠ Sign Flip!

Calculate: $ (-5) \times (-3) + (-8) \div 4 - (-2) $

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SAME signs → POSITIVE | DIFFERENT signs → NEGATIVE
Subtracting a negative = Adding: $ a - (-b) = a + b $

✏️ Worked Example

$ (-4) \times (-2) + (-6) \div 3 $
Step 1: $ (-4)\times(-2) = +8 $ (same signs → positive)
Step 2: $ (-6)\div 3 = -2 $ (different signs → negative)
Step 3: $ 8 + (-2) = 8 - 2 = \mathbf{6} $

✍ Your Turn

Calculate: $ (-9) \div (-3) - (-4) + (-1) \times 5 $

3. Adding Fractions with Different Denominators Fractions / LCM

Simplify: $ \dfrac{2}{3} + \dfrac{5}{6} - \dfrac{1}{4} $

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LCD first, THEN add numerators
LCM of 3, 6, 4 → List multiples or use prime factorization

✏️ Worked Example

$ \dfrac{1}{4} + \dfrac{2}{3} $
LCD of 4 and 3 = 12
$ \dfrac{1}{4} = \dfrac{3}{12}, \quad \dfrac{2}{3} = \dfrac{8}{12} $
Answer: $ \dfrac{3+8}{12} = \dfrac{11}{12} $

✍ Your Turn

Simplify: $ \dfrac{3}{5} - \dfrac{1}{4} + \dfrac{7}{10} $

4. Distributive Property Algebra ⚠ Don't forget the minus!

Expand and simplify: $ 3(2x - 4) - 2(x + 5) $

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a(b + c) = ab + ac & Combine Like Terms LAST
Trap: $-2(x+5) = -2x \color{red}{-10}$, NOT $-2x+5$

✏️ Worked Example

$ 4(x - 2) - 3(2x + 1) $
= $ 4x - 8 - 6x - 3 $
= $ (4x - 6x) + (-8 - 3) $
= $ -2x - 11 $

✍ Your Turn

Expand: $ 5(3x + 2) - 4(x - 7) $

5. Solving Two-Step Equations Equations

Solve for $x$: $ 3x + 7 = 25 $

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UNDO in reverse order: − first, ÷ second
Whatever you do to one side → do the SAME to the other

✏️ Worked Example

$ 4x - 3 = 13 $
Step 1 — Add 3: $ 4x = 16 $
Step 2 — Divide by 4: $ x = 4 $
✅ Check: $ 4(4) - 3 = 13 $ ✓

✍ Your Turn

(a) Solve: $ 2x - 5 = 11 $ (b) Solve: $ \dfrac{x}{3} + 4 = 9 $

6. Solving & Graphing Inequalities Inequalities ⚠ Flip the sign!

Solve and describe the solution: $ -2x + 3 > 11 $

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Multiply or Divide by NEGATIVE → FLIP the sign
$ > $ becomes $ < $ | $ \leq $ becomes $ \geq $

✏️ Worked Example

$ -3x + 6 \leq 15 $
Subtract 6: $ -3x \leq 9 $
Divide by −3 FLIP!: $ x \geq -3 $
Solution: all numbers −3 or greater → $ [-3, \infty) $

✍ Your Turn

Solve: $ -4x - 1 \leq 15 $ then draw a number line.

7. Ratios & Proportions Ratio

A recipe uses 3 cups of flour for every 2 cups of sugar. If you use 9 cups of flour, how many cups of sugar do you need?
Also: If $\dfrac{x}{12} = \dfrac{5}{8}$, find $x$.

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Cross Multiply: $ \dfrac{a}{b} = \dfrac{c}{d} \Rightarrow ad = bc $

✏️ Worked Example

$ \dfrac{4}{x} = \dfrac{6}{15} $
Cross multiply: $ 4 \times 15 = 6x $
$ 60 = 6x \Rightarrow x = 10 $

✍ Your Turn

Solve: $ \dfrac{3}{x} = \dfrac{9}{21} $

8. Percent — Increase & Decrease Percent ⚠ Multiply, don't divide!

A shirt costs $40. It is on sale for 25% off. What is the sale price?
Also: What is 15% of 80?

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Percent of a number: $ \text{part} = \dfrac{\%}{100} \times \text{whole} $
Discount → Multiply by (1 − rate): 25% off = × 0.75

✏️ Worked Example

30% of 60 = $ \dfrac{30}{100} \times 60 = 0.3 \times 60 = 18 $

20% discount on $50:
Discount = $ 0.20 \times 50 = \$10 $
Sale price = $ 50 - 10 = \mathbf{\$40} $ (or $ 50 \times 0.80 = \$40 $)

✍ Your Turn

A price of $120 increases by 15%. What is the new price?

9. Absolute Value Absolute Value ⚠ Two answers possible!

Solve: $ |2x - 3| = 7 $

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|x| = distance from 0 → always ≥ 0
|a| = b means a = b OR a = −b

✏️ Worked Example

$ |x + 1| = 5 $
Case 1: $ x + 1 = 5 \Rightarrow x = 4 $
Case 2: $ x + 1 = -5 \Rightarrow x = -6 $
Solutions: $ x = 4 $ or $ x = -6 $

✍ Your Turn

Solve: $ |3x + 2| = 11 $

10. Exponent Rules Exponents ⚠ Don't multiply the base!

Simplify: $ \dfrac{x^5 \cdot x^{-2}}{x^3} $ and $ (2x^2)^3 $

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Multiply → ADD exponents | Divide → SUBTRACT exponents
Power → MULTIPLY exponents | $ x^0 = 1 $, $ x^{-n} = \dfrac{1}{x^n} $

✏️ Worked Example

$ \dfrac{a^6 \cdot a^{-1}}{a^2} = a^{6 + (-1) - 2} = a^3 $

$ (3x^2)^2 = 3^2 \cdot x^{2\times 2} = 9x^4 $

✍ Your Turn

Simplify: $ (2a^3)(4a^{-1}) \div (a^2) $

📏 Geometry

Problems 11–20 · Area, Angles, Triangles, Circles, and more!

📋 Topics Covered

11. Complementary & Supplementary Angles #11

12. Triangle Angle Sum #12

13. Pythagorean Theorem #13

14. Area of Triangles & Quadrilaterals #14

15. Area & Circumference of Circles #15

16. Parallel Lines & Transversals #16

17. Similar Triangles #17

18. Volume of 3D Shapes #18

19. The Coordinate Plane — Distance & Midpoint #19

20. Perimeter & Area — Mixed Composite Figures #20

11. Complementary & Supplementary Angles Angles ⚠ 90 or 180?

Two angles are supplementary. One angle is $3x + 10$°. The other is $2x - 5$°. Find both angles.

🧠

Complementary = 90° ("C" comes before "S")
Supplementary = 180° ("S" = Straight line)

✏️ Worked Example

Two complementary angles: $ (2x+5)° $ and $ (x+10)° $
$ (2x+5) + (x+10) = 90 $
$ 3x + 15 = 90 \Rightarrow 3x = 75 \Rightarrow x = 25 $
Angles: $ 2(25)+5 = 55°$ and $ 25+10 = 35° $ ✓

✍ Your Turn

Supplementary angles: $ (4x-10)° $ and $ (2x+20)° $. Find $x$ and both angles.

12. Triangle Angle Sum Triangles

In triangle $ABC$, angle $A = 2x°$, angle $B = (x+20)°$, angle $C = (3x-10)°$. Find each angle.

🧠

Sum of angles in ANY triangle = 180°
Exterior angle = sum of the two non-adjacent interior angles

✏️ Worked Example

Angles: $ 3x°, (x+10)°, (2x-4)° $
$ 3x + x + 10 + 2x - 4 = 180 $
$ 6x + 6 = 180 \Rightarrow x = 29 $
Angles: $ 87°, 39°, 54° $ Check: $ 87+39+54=180 $ ✓

✍ Your Turn

A triangle has angles $ x°, (x+30)°, (2x-6)° $. Find all angles.

13. Pythagorean Theorem Right Triangles ⚠ c is the hypotenuse!

$ a^2 + b^2 = c^2 $ (c = hypotenuse, longest side)

A right triangle has legs $ a = 6 $ and $ b = 8 $. Find the hypotenuse $ c $. Also: if $ c = 13 $ and $ a = 5 $, find $ b $.

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a² + b² = c² (always)
Common triples: 3-4-5, 5-12-13, 8-15-17

✏️ Worked Example

Legs: $ a = 9, b = 12 $. Find $ c $.
$ c^2 = 9^2 + 12^2 = 81 + 144 = 225 $
$ c = \sqrt{225} = \mathbf{15} $

✍ Your Turn

(a) Find $c$: $ a=5, b=12 $ (b) Find $b$: $ a=7, c=25 $

14. Area of Triangles & Quadrilaterals Area ⚠ Height ⊥ base!

Find the area of: (a) A triangle with base 10 cm and height 7 cm (b) A trapezoid with parallel sides 5 and 9, height 4

🔷 Triangle: $ A = \dfrac{1}{2} \times b \times h $ 🔷 Trapezoid: $ A = \dfrac{1}{2}(b_1 + b_2) \times h $ 🔷 Parallelogram: $ A = b \times h $

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Height must be PERPENDICULAR to the base
Trapezoid: average the two bases, then × height

✍ Your Turn

A parallelogram has base 13 m and height 6 m. Find its area. Then find the area of a triangle with $b=14, h=9$.

15. Circles — Area & Circumference Circles ⚠ r vs d confusion!

A circle has radius $r = 7$. Find: (a) Circumference (b) Area.
Leave answers in terms of $\pi$.

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C = 2πr | A = πr²
Given diameter? → $r = d \div 2$ FIRST

✍ Your Turn

A circle has diameter 10. Find its circumference and area. Leave answers in terms of $\pi$.

16. Parallel Lines Cut by a Transversal Angles ⚠ Know which pairs!

Lines $\ell_1 \parallel \ell_2$ cut by a transversal. If angle 1 = $ (3x + 5)° $ and angle 5 = $ (5x - 15)° $, find $x$ and both angles.

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Alternate Interior = EQUAL | Co-Interior (Same-side) = 180°
Corresponding = EQUAL | Vertical = EQUAL

✍ Your Turn

Corresponding angles: $ (4x+10)° $ and $ (6x-20)° $. Find $x$.

17. Similar Triangles — Scale Factor Similar Figures

Triangle $ABC \sim$ Triangle $DEF$. The sides of $\triangle ABC$ are 6, 8, 10. If the shortest side of $\triangle DEF$ is 9, find the other two sides.

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Corresponding sides are PROPORTIONAL
Find scale factor first: $ k = \dfrac{\text{new}}{\text{original}} $, then multiply all sides by $k$

✏️ Worked Example

$\triangle ABC$ sides: 4, 6, 8 $\triangle DEF$ shortest side = 6
Scale factor: $ k = \dfrac{6}{4} = 1.5 $
Other sides: $ 6 \times 1.5 = 9 $, $ 8 \times 1.5 = 12 $

✍ Your Turn

$\triangle PQR \sim \triangle XYZ$. Sides of $\triangle PQR$: 5, 7, 9. Largest side of $\triangle XYZ$ = 27. Find the other two sides.

18. Volume of 3D Shapes Volume ⚠ Don't confuse V and SA!

(a) Find the volume of a rectangular prism: $ l = 5, w = 4, h = 3 $
(b) Find the volume of a cylinder: $ r = 6, h = 10 $ (leave in terms of $\pi$)

📦 Rectangular Prism: $ V = l \times w \times h $ 🥫 Cylinder: $ V = \pi r^2 h $ 🔺 Pyramid: $ V = \dfrac{1}{3} \times B \times h $

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Prism/Cylinder: $ V = \text{Base Area} \times h $
Pyramid/Cone: $ V = \dfrac{1}{3} \times \text{Base Area} \times h $

✍ Your Turn

Find the volume of a cone with $ r = 3 $ and $ h = 8 $. Leave in terms of $\pi$.

19. Distance & Midpoint on the Coordinate Plane Coordinate Geometry

Points $ A = (2, 3) $ and $ B = (8, 11) $.
(a) Find the distance $ AB $ (b) Find the midpoint $ M $ of $ AB $

📍 Distance: $ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $ 📍 Midpoint: $ M = \left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right) $

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Distance = Pythagorean Theorem on a grid!
Midpoint = average of x's, average of y's

✍ Your Turn

Find the distance and midpoint between $ P(1, -2) $ and $ Q(7, 6) $.

20. Composite Figures — Area & Perimeter Mixed ⚠ Watch what to ADD or SUBTRACT

circle

Rectangle 14×8 with a semicircle added on one side ($r=4$)

Find the total area of the figure above: a rectangle (14 × 8) with a semicircle (r = 4) attached to one of the short sides.

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Break it into SIMPLE shapes → Add (or subtract) the parts
Semicircle area = $\dfrac{1}{2}\pi r^2$

💡 Strategy tip: For any composite figure:
1) Identify each simple shape
2) Calculate each area separately
3) Add or subtract depending on the figure

✍ Your Turn

A square (side 10) has a triangle on top (base 10, height 6). Find the total area.

Pre-Algebra& Geometry

Pre-Algebra
& Geometry