Math Practice — Algebra 1 & Geometry

Algebra 1

Word Problems

01 Linear Equations Easy

⚡ MEMORY KEY: "EQUAL-BOTH-SIDES" Whatever you do to one side, do the same to the other. Balance = equality.

Emma has 3 times as many stickers as Jake. Together they have 48 stickers. How many stickers does Emma have?

💡 Trap: Don't forget — the question asks for Emma's count, not Jake's.

Step-by-Step Solution

Let Jake = $x$. Then Emma = $3x$.
Equation: $x + 3x = 48 \Rightarrow 4x = 48 \Rightarrow x = 12$
Emma = $3 \times 12 = \mathbf{36}$

Common Mistake: Many students answer 12 (Jake's count) instead of 36 (Emma's). Always re-read what the question asks!

02 Inequalities Medium

⚡ MEMORY KEY: "FLIP-WHEN-NEGATIVE" Multiply / divide both sides by a negative → FLIP the inequality sign!

A store sells notebooks for $4 each. You have at most $30 to spend and need at least 5 notebooks. Which inequality shows the maximum number of notebooks $n$ you can buy?

Step-by-Step Solution

Budget constraint: $4n \leq 30 \Rightarrow n \leq 7.5$, so $n \leq 7$ (whole notebooks).
Minimum constraint: $n \geq 5$
Combined: $\mathbf{5 \leq n \leq 7}$

Trap: Option C forgets the minimum. Option A forgets the maximum. Both constraints must be included!

03 Systems of Equations Medium

⚡ MEMORY KEY: "ELIMINATE-SUBSTITUTE" Two equations, two unknowns: eliminate one variable, then substitute back.

Two friends ordered lunch. Together they spent $19. One meal cost $3 more than the other. How much did the cheaper meal cost?

Step-by-Step Solution

Let cheaper meal = $x$, pricier = $x + 3$.
$x + (x+3) = 19 \Rightarrow 2x + 3 = 19 \Rightarrow 2x = 16 \Rightarrow x = \mathbf{8}$

Trap: Many pick $9 by splitting 19÷2 without accounting for the $3 difference. Always set up equations!

04 Proportions & Rates Easy

⚡ MEMORY KEY: "CROSS-MULTIPLY" Proportion: $\frac{a}{b} = \frac{c}{d}$ → $ad = bc$. Cross the diagonal!

A car travels 150 miles in 3 hours. At the same speed, how many hours will it take to travel 250 miles?

💡 Trap: Don't add 100 miles ÷ 50 mph randomly — set up a proper proportion.

Step-by-Step Solution

Speed = $\frac{150}{3} = 50$ mph.
Time = $\frac{250}{50} = \mathbf{5}$ hours.

Or by proportion: $\frac{150}{3} = \frac{250}{t} \Rightarrow 150t = 750 \Rightarrow t = 5$

05 Percentages Medium

⚡ MEMORY KEY: "IS-OVER-OF × 100" Percent = (part ÷ whole) × 100. "Is" = part, "of" = whole.

A shirt originally costs $80. It goes on sale for 25% off, then an additional 10% off the sale price. What is the final price?

💡 Trap: 25% + 10% ≠ 35% off. The second discount applies to the already-discounted price!

Step-by-Step Solution

After 25% off: $80 \times 0.75 = 60$
After additional 10% off: $60 \times 0.90 = \mathbf{54}$

Trap: $80 \times 0.65 = 52$ — this is wrong! Sequential discounts multiply, they don't add.

06 Slope & Linear Functions Medium

⚡ MEMORY KEY: "RISE-OVER-RUN" Slope $m = \frac{y_2 - y_1}{x_2 - x_1}$ = rise ÷ run. Change in y divided by change in x.

A taxi charges a flat fee of $3 plus $2 per mile. If another taxi charges $5 per mile but no flat fee, after how many miles do both cost the same?

Step-by-Step Solution

Taxi A: $3 + 2m$ Taxi B: $5m$
Set equal: $3 + 2m = 5m \Rightarrow 3 = 3m \Rightarrow m = \mathbf{1}$ mile

Trap: Don't forget to include both the flat fee and per-mile rates. The intersection point is at exactly 1 mile.

07 Exponents Tricky

⚡ MEMORY KEY: "POWER-MEANS-MULTIPLY" $a^m \cdot a^n = a^{m+n}$ — same base, add exponents. Never multiply the bases!

A bacteria colony doubles every hour. If there are 50 bacteria now, how many will there be after 4 hours?

Step-by-Step Solution

Exponential growth: $50 \times 2^4 = 50 \times 16 = \mathbf{800}$

Common Mistake: Multiplying 50 × 4 × 2 = 400 (linear thinking). Doubling is exponential: ×2, ×2, ×2, ×2.

08 Factoring Medium

⚡ MEMORY KEY: "PRODUCT-SUM" For $x^2 + bx + c$: find two numbers that multiply to $c$ and add to $b$.

A rectangular garden has an area of $x^2 + 7x + 12$ square feet. If one side is $(x + 3)$ feet, what is the other side?

Step-by-Step Solution

Factor: $x^2 + 7x + 12$ → need two numbers: product = 12, sum = 7.
Answer: $3 \times 4 = 12$, $3 + 4 = 7$ ✓
So: $(x+3)(x+4)$ → the other side = $\mathbf{(x+4)}$

09 Functions Medium

⚡ MEMORY KEY: "ONE-INPUT-ONE-OUTPUT" A function maps each input (x) to exactly ONE output f(x). Vertical line test!

The function $f(x) = 2x^2 - 3$ models a rollercoaster's height (feet) at position $x$. What is the height at $x = 3$?

Step-by-Step Solution

$f(3) = 2(3)^2 - 3 = 2(9) - 3 = 18 - 3 = \mathbf{15}$

Trap: Answer C = 33 comes from $(2 \times 3)^2 - 3$. Remember: exponent applies only to $x$, not to $2x$!

10 Quadratic Equations Tricky

⚡ MEMORY KEY: "ZERO-PRODUCT-PROPERTY" If $ab = 0$, then $a = 0$ or $b = 0$. Factor first, then set each factor to zero!

A ball is thrown upward. Its height in feet is $h = -16t^2 + 48t$, where $t$ is time in seconds. When does the ball hit the ground?

💡 Ground level means $h = 0$. Factor and use Zero Product Property.

Step-by-Step Solution

Set $h = 0$: $-16t^2 + 48t = 0$
Factor: $-16t(t - 3) = 0$
Solutions: $t = 0$ (launch) or $t = \mathbf{3}$ (landing)

Trap: $t = 0$ is when it was thrown, not when it lands. The answer is $t = 3$.

Geometry

Core Problems

01 Pythagorean Theorem Easy

⚡ MEMORY KEY: "LEG² + LEG² = HYPOTENUSE²" $a^2 + b^2 = c^2$ — the longest side (hypotenuse) is always $c$, opposite the right angle.

A ladder leans against a wall. The base is 6 ft from the wall and the ladder reaches 8 ft up the wall. How long is the ladder?

Step-by-Step Solution

$6^2 + 8^2 = c^2$
$36 + 64 = 100$
$c = \sqrt{100} = \mathbf{10}$ ft

This is a classic 3-4-5 triple scaled by 2: (6, 8, 10). Memorize Pythagorean triples: 3-4-5, 5-12-13, 8-15-17!

02 Area of Triangles Medium

⚡ MEMORY KEY: "HALF-BASE-HEIGHT" $A = \frac{1}{2} \times b \times h$ — height must be PERPENDICULAR to the base, not the slant side!

A triangular park has a base of 14 m and a perpendicular height of 9 m. What is its area?

💡 Trap: Don't multiply base × height without the ½ factor!

Step-by-Step Solution

$A = \frac{1}{2} \times 14 \times 9 = \frac{126}{2} = \mathbf{63} \text{ m}^2$

Trap: B (126) = forgetting the ½. Always halve the base × height product for triangles.

03 Circles — Area & Circumference Medium

⚡ MEMORY KEY: "AROUND-πd, INSIDE-πr²" Circumference uses diameter ($\pi d$), Area uses radius squared ($\pi r^2$). Easy to mix up!

A circular pizza has a diameter of 14 inches. What is its area? (Use $\pi \approx 3.14$)

💡 Trap: 14 is the DIAMETER, not the radius. Always halve it first!

Step-by-Step Solution

Radius = $14 \div 2 = 7$ in
$A = \pi r^2 = 3.14 \times 7^2 = 3.14 \times 49 = \mathbf{153.86}$ in²

Traps: A = circumference. B = used diameter as radius ($14^2$). C = also circumference-related. Always find the radius first!

04 Angle Relationships Easy

⚡ MEMORY KEY: "SUPPLEMENTARY=180, COMPLEMENTARY=90" Supplement → straight line (180°). Complement → corner (90°). S comes before C, 180 > 90.

Two angles are supplementary. One angle is 47°. What is the other angle?

Step-by-Step Solution

Supplementary angles sum to 180°.
$180° - 47° = \mathbf{133°}$

Trap: C and D come from mixing up supplementary (180°) with complementary (90°). Remember: "Supplementary" has more letters → bigger angle sum!

05 Volume of Rectangular Prisms Easy

⚡ MEMORY KEY: "LENGTH × WIDTH × HEIGHT" Volume = $l \times w \times h$. Three dimensions = three factors. Units become cubic (³).

A fish tank is 40 cm long, 20 cm wide, and 25 cm tall. How many cubic centimeters of water can it hold?

Step-by-Step Solution

$V = 40 \times 20 \times 25 = \mathbf{20{,}000}$ cm³

Tip: Multiply step by step: 40 × 20 = 800, then 800 × 25 = 20,000. Don't skip steps!

06 Similar Triangles Medium

⚡ MEMORY KEY: "SAME-SHAPE-SAME-RATIO" Similar triangles have equal angles AND proportional sides. Set up a proportion, then cross-multiply!

A tree casts a 15 ft shadow at the same time a 6 ft person casts a 4 ft shadow. How tall is the tree?

Step-by-Step Solution

Set up proportion: $\frac{\text{tree height}}{\text{tree shadow}} = \frac{\text{person height}}{\text{person shadow}}$
$\frac{h}{15} = \frac{6}{4} \Rightarrow 4h = 90 \Rightarrow h = \mathbf{22.5}$ ft

Trap: Don't mix up the ratio. Keep height-to-shadow consistent on both sides!

07 Interior Angles of Polygons Tricky

⚡ MEMORY KEY: "(n−2)×180" Sum of interior angles of any polygon = $(n-2) \times 180°$ where $n$ = number of sides.

What is the measure of each interior angle of a regular hexagon?

💡 Regular = all sides and angles equal. Find the sum first, then divide.

Step-by-Step Solution

Hexagon has $n = 6$ sides.
Sum = $(6-2) \times 180 = 4 \times 180 = 720°$
Each angle = $720 \div 6 = \mathbf{120°}$

Reference: Pentagon = 108°, Hexagon = 120°, Octagon = 135°, Decagon = 144°.

08 Coordinate Geometry Medium

⚡ MEMORY KEY: "DISTANCE-FORMULA = PYTHAGOREAN" $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ — it's just the Pythagorean theorem on a grid!

What is the distance between points $A(1, 2)$ and $B(4, 6)$?

Step-by-Step Solution

$d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = \mathbf{5}$

This is another 3-4-5 Pythagorean triple! Differences: $\Delta x = 3$, $\Delta y = 4$, so $d = 5$.

09 Surface Area Tricky

⚡ MEMORY KEY: "COUNT-ALL-FACES" Surface area = sum of ALL faces. A rectangular box has 6 faces in 3 matching pairs!

A gift box is 10 cm × 6 cm × 4 cm. How much wrapping paper is needed to cover all sides? (No overlap)

Step-by-Step Solution

SA = $2(lw + lh + wh) = 2(10 \times 6 + 10 \times 4 + 6 \times 4)$
$= 2(60 + 40 + 24) = 2(124) = \mathbf{248}$ cm²

Trap: C = forgot to multiply by 2 (only counted 3 faces). D = multiplied everything together (that's volume-thinking).

10 Parallel Lines & Transversals Tricky

⚡ MEMORY KEY: "Z=ALTERNATE, F=CORRESPONDING, C=CO-INTERIOR" Z-angles (alternate interior) = equal. F-angles (corresponding) = equal. C-angles (co-interior) = 180°.

Two parallel lines are cut by a transversal. One angle formed is 65°. What is the measure of its co-interior (same-side interior) angle?

💡 Trap: Co-interior angles are supplementary (add to 180°), NOT equal!

Step-by-Step Solution

Co-interior angles (also called "consecutive interior" or "same-side interior") are supplementary.
$180° - 65° = \mathbf{115°}$

Trap: A (65°) would be the answer for alternate interior or corresponding angles (those are equal). Don't confuse the angle pairs!