Math Mastery — Algebra & Geometry

KEY: let x = unknown → write equation → solve

Emma earns $12 per hour babysitting. She wants to buy a jacket that costs $78. She already has $18 saved. How many hours does she need to work?

Quick Example

If she earns $10/hr and needs $50, already has $10 → $10x + 10 = 50$ → $x = 4$ hours.

Step-by-Step Solution

Set up the equation: money earned + savings = jacket price

$12x + 18 = 78$

Subtract 18 from both sides:

$12x = 60 \Rightarrow x = 5$

Emma needs to work 5 hours. ✓ The tricky part: don't forget to subtract the savings she already has before dividing!

KEY: rate × time = distance → set equal

Two cars leave the same point traveling in opposite directions. Car A goes 55 mph and Car B goes 45 mph. After how many hours will they be 250 miles apart?

Quick Example

60 mph + 40 mph = 100 mph combined rate. To be 200 miles apart: $\frac{200}{100} = 2$ hours.

Step-by-Step Solution

Opposite directions → add the speeds. Combined rate = 55 + 45 = 100 mph

$100t = 250 \Rightarrow t = 2.5 \text{ hours}$

Common mistake: Using only one car's speed. When going opposite directions, you ALWAYS add the rates!

KEY: inequality → flip sign when ÷ or × by negative

A roller coaster requires riders to be at least 48 inches tall. Marcus is currently 41 inches tall and grows about 2.5 inches per year. What is the minimum number of whole years he must wait?

Quick Example

If 40 in. now, grows 3 in/yr, needs 50 in: $40 + 3y \geq 50$ → $y \geq 3.33$ → wait 4 years (whole number!).

Step-by-Step Solution

$41 + 2.5y \geq 48$

$2.5y \geq 7 \Rightarrow y \geq 2.8$

Since we need whole years, round up: 3 years. ✓ Trap: choosing 2 years because 2.8 ≈ 2. But 2 years gives only 46 inches — not enough!

KEY: system of equations → eliminate one variable

A theater sells adult tickets for $9 and child tickets for $5. One evening, 200 tickets were sold for a total of $1,220. How many adult tickets were sold?

Quick Example

$a + c = 10$ and $8a + 4c = 56$ → multiply first eq by 4 → subtract → find $a$.

Step-by-Step Solution

Let $a$ = adult tickets, $c$ = child tickets

$a + c = 200 \quad (1)$

$9a + 5c = 1220 \quad (2)$

Multiply (1) by 5: $5a + 5c = 1000$. Subtract from (2):

$4a = 220 \Rightarrow a = 55$

KEY: slope = rise/run = (y₂−y₁)/(x₂−x₁)

A candle is 10 inches tall when lit. After 3 hours, it is 7 inches tall. If the candle burns at a constant rate, what is its height after 8 hours?

Quick Example

Start 12 in., after 2 hrs = 8 in. → burns 2 in/hr → after 5 hrs: $12 - 2(5) = 2$ in.

Step-by-Step Solution

Rate = $\frac{10-7}{3} = 1$ inch per hour. Linear model:

$h(t) = 10 - 1 \cdot t$

$h(8) = 10 - 8 = 2 \text{ inches}$

Trap answer D (0) tempts you if you miscalculate the burn rate as more than 1 in/hr.

KEY: percent → decimal: 15% = 0.15 → multiply

A store marks up all items by 40%. A customer then uses a coupon for 25% off the marked-up price. If the original cost is $80, what is the final price the customer pays?

Quick Example

$100 → 40% markup → $140 → 25% off → $140 \times 0.75 = \$105$. Not the original price!

Step-by-Step Solution

Marked-up price: $80 \times 1.40 = \$112$

After 25% off: $112 \times 0.75 = \$84$

The big trap: Many students think +40% then −25% = +15%, arriving at $92. Wrong! Percent changes multiply, they don't add. Also, B and C are the same — always read all choices carefully!

KEY: mixture → amt × concentration + amt × concentration = total

How many liters of a 60% acid solution must be mixed with 10 liters of a 30% acid solution to produce a 50% acid solution?

Quick Example

Mixing $x$ liters of 80% with 5 liters of 20% to get 60%: $0.8x + 0.2(5) = 0.6(x+5)$

Step-by-Step Solution

$0.60x + 0.30(10) = 0.50(x + 10)$

$0.60x + 3 = 0.50x + 5$

$0.10x = 2 \Rightarrow x = 20 \text{ liters}$

KEY: consecutive integers → n, n+1, n+2

The sum of three consecutive odd integers is 81. What is the largest of the three integers?

Quick Example

Consecutive odd integers differ by 2: $n, n+2, n+4$. Sum = $3n + 6$. Solve for n.

Step-by-Step Solution

Let the integers be $n, n+2, n+4$:

$n + (n+2) + (n+4) = 81$

$3n + 6 = 81 \Rightarrow n = 25$

Largest = $25 + 4 = \mathbf{29}$. Trap: choosing 27 = the middle, or 25 = the smallest!

KEY: direct variation → y = kx; find k first, then solve

The number of pages Sofia reads varies directly with the number of hours she reads. She reads 84 pages in 3 hours. How many pages will she read in 5.5 hours?

Quick Example

60 pages in 2 hrs → $k = 30$ pages/hr → in 7 hrs: $30 \times 7 = 210$ pages.

Step-by-Step Solution

$k = \frac{84}{3} = 28 \text{ pages/hour}$

$y = 28 \times 5.5 = 154 \text{ pages}$

Trap: Some students divide 5.5 by 3 and multiply by 84 without simplifying the unit rate first — same answer, but easier to make errors.

KEY: compound inequality → solve each part separately

A theme park offers a "Family Deal" for groups between 4 and 8 people inclusive. Tickets cost $24 each. What is the range of total costs for the Family Deal?

Quick Example

If 3–6 people at $10 each: min = $3 \times 10 = \$30$, max = $6 \times 10 = \$60$. Range: $30–$60.

Step-by-Step Solution

Min: $4 \times 24 = \$96$

Max: $8 \times 24 = \$192$

"Inclusive" means 4 and 8 ARE included. Watch for "between" (exclusive) vs "between... inclusive" — completely different answers!

KEY: Pythagorean theorem → a² + b² = c² (c = hypotenuse)

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

Quick Example

Base = 3, height = 4 → $3^2 + 4^2 = 9 + 16 = 25$ → ladder = $\sqrt{25} = 5$ ft. (The famous 3-4-5 triangle!)

Step-by-Step Solution

$c^2 = 6^2 + 8^2 = 36 + 64 = 100$

$c = \sqrt{100} = 10 \text{ feet}$

This is a 6-8-10 triangle (multiply 3-4-5 by 2). Recognizing Pythagorean triples saves time on tests!

KEY: area of circle = πr²; circumference = 2πr

A circular garden has a diameter of 14 meters. What is the area of the garden? (Use $\pi \approx 3.14$)

Quick Example

Diameter = 10 m → radius = 5 m → $A = \pi(5)^2 = 25\pi \approx 78.5$ m².
Trap: Don't use diameter in the formula — always halve it first!

Step-by-Step Solution

Radius = $14 \div 2 = 7$ m

$A = \pi r^2 = 3.14 \times 7^2 = 3.14 \times 49 = 153.86 \text{ m}^2$

Choice A (615.44) is what you get if you accidentally use diameter (14) instead of radius (7) in the formula — the #1 most common mistake on circle problems!

KEY: triangle angles sum = 180°; exterior angle = sum of 2 non-adjacent interior

In triangle ABC, angle A = 35° and angle B = 78°. An exterior angle at vertex C is formed. What is the measure of this exterior angle?

Quick Example

Interior angles 50° and 60° → exterior at 3rd vertex = $50 + 60 = 110°$. (Exterior angle theorem — no need to find the interior angle first!)

Step-by-Step Solution

Method 1 (Exterior Angle Theorem):

Exterior angle = 35° + 78° = 113°

Method 2 (verify): Interior C = 180° − 35° − 78° = 67°. Exterior = 180° − 67° = 113° ✓

Choice A (67°) is the interior angle at C — a classic misdirection!

KEY: similar triangles → matching sides are proportional

A 6-foot person casts a 4-foot shadow. At the same time, a nearby flagpole casts a 20-foot shadow. How tall is the flagpole?

Quick Example

Person height / shadow = Pole height / shadow → $\frac{6}{4} = \frac{h}{20}$

Step-by-Step Solution

$\frac{6}{4} = \frac{h}{20}$

$4h = 120 \Rightarrow h = 30 \text{ feet}$

Key: set up the proportion with matching parts — height to height, shadow to shadow. Don't mix them up!

KEY: area of trapezoid = ½(b₁ + b₂) × h

A trapezoidal field has two parallel sides of 30 m and 50 m, and a height (perpendicular distance) of 20 m. What is the area of the field?

Quick Example

Bases 10 and 20, height 5: $A = \frac{1}{2}(10 + 20) \times 5 = \frac{1}{2}(30)(5) = 75$ m².

Step-by-Step Solution

$A = \frac{1}{2}(30 + 50) \times 20 = \frac{1}{2}(80)(20) = 800 \text{ m}^2$

Choice A (1,600) = forgetting the ½. Choice B (600) = using only one base. Always use BOTH bases and divide by 2!

KEY: volume of cylinder = πr²h

A cylindrical water tank has a radius of 5 feet and a height of 12 feet. How much water can it hold? (Use $\pi \approx 3.14$)

Quick Example

r = 3, h = 10 → $V = 3.14 \times 9 \times 10 = 282.6$ ft³. Volume = base area × height.

Step-by-Step Solution

$V = \pi r^2 h = 3.14 \times 5^2 \times 12 = 3.14 \times 25 \times 12 = 942 \text{ ft}^3$

Choice C (3,768) comes from using diameter (10) instead of radius (5). Choice D (1,884) = forgetting to square the radius. Square the radius FIRST, then multiply by height!

KEY: complementary = 90° total; supplementary = 180° total

Two angles are supplementary. One angle is three times the other. What is the measure of the larger angle?

Quick Example

Complementary version: $x + 3x = 90$ → $x = 22.5°$. Larger = $67.5°$.
For supplementary, replace 90 with 180.

Step-by-Step Solution

$x + 3x = 180 \Rightarrow 4x = 180 \Rightarrow x = 45°$

Larger angle = $3 \times 45° = 135°$

Choice D (45°) is the smaller angle. The question asks for the LARGER. Read carefully!

KEY: perimeter = add ALL sides; don't confuse with area

A rectangular swimming pool is 25 meters long and 10 meters wide. A fence is to be built 3 meters away from each side of the pool. What is the perimeter of the fenced area?

Quick Example

Pool 10×4, fence 2m away → new dimensions: $(10+4) \times (4+4) = 14 \times 8$ → perimeter = $2(14+8) = 44$ m.

Step-by-Step Solution

Fence is 3 m away on EACH side → add 6 m to each dimension:

New length = $25 + 6 = 31$ m; New width = $10 + 6 = 16$ m

Perimeter = $2(31 + 16) = 2(47) = 94$ m

Wait — let's recheck B and D: Perimeter = 94 m. The trap (A = 70 m) comes from using the original pool dimensions. Always add the buffer to ALL four sides (which means +6 to length AND +6 to width).

KEY: arc length = (θ/360) × 2πr; sector area = (θ/360) × πr²

A slice of pizza is shaped like a sector of a circle. The pizza has a diameter of 16 inches and the slice has a central angle of 45°. What is the area of the slice? (Use $\pi \approx 3.14$)

Quick Example

r = 6, angle = 60°: $A = \frac{60}{360} \times 3.14 \times 36 = \frac{1}{6} \times 113.04 = 18.84$ in².

Step-by-Step Solution

Radius = $16 \div 2 = 8$ inches.

$A = \frac{45}{360} \times 3.14 \times 8^2 = \frac{1}{8} \times 3.14 \times 64 = \frac{1}{8} \times 200.96 = 25.12 \text{ in}^2$

Choice A uses diameter (16) instead of radius (8) in the formula. Once again, convert diameter to radius before substituting!

KEY: scale factor k → area scales by k²; volume scales by k³

Two similar rectangles have a scale factor of 1:3 (small to large). The area of the small rectangle is 8 cm². What is the area of the large rectangle?

Quick Example

Scale factor 1:2 → area multiplied by $2^2 = 4$. Small area 5 cm² → large area = 20 cm².
Never just multiply by the scale factor — always SQUARE it for area!

Step-by-Step Solution

Area scale factor = $3^2 = 9$

Large area = $8 \times 9 = 72 \text{ cm}^2$

Choice A (24) = multiplying by 3 (the scale factor, not squared). Choice D (27) = multiplying by $3^3$ (that's for volume). For area, always square the linear scale factor!

Word Problems
that actually teach

Algebra 1 — Word Problems

Geometry — Word Problems