Math Practice — Pre-Algebra & Geometry

Q 01

Integers · Number Line

The temperature at midnight was −8°F. By noon, it had risen 15°F. Then it dropped 3°F by evening. What was the temperature at evening?

RISE = ADD, DROP = SUBTRACT

Solution

Start at −8. Rise 15: −8 + 15 = 7. Drop 3: 7 − 3 = 4°F. Trap: students often subtract the rise or forget the drop, landing on 7 or −26.

Q 02

Fractions · Operations

A recipe uses 2/3 cup of sugar. If you make 1½ batches, how much sugar do you need in total?

MIXED NUMBER → IMPROPER FIRST, THEN MULTIPLY

Solution

Convert 1½ = 3/2. Multiply: (2/3) × (3/2) = 6/6 = 1 cup. Common error: adding instead of multiplying, or forgetting to convert the mixed number.

Q 03

Ratios & Proportions

A car travels 150 miles in 3 hours. At the same speed, how many miles will it travel in 5 hours?

UNIT RATE × TIME = DISTANCE

Solution

Unit rate: 150 ÷ 3 = 50 mph. Distance in 5 hrs: 50 × 5 = 250 miles. Trap: students pick 200 by multiplying 150 by (5/3) incorrectly.

Q 04

Percents · Discount

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

SALE PRICE = ORIGINAL × (1 − DISCOUNT%)

Solution

Discount: 80 × 0.25 = $20. Sale price: 80 − 20 = $60. Trap: many answer $20 (the discount amount, not the final price).

Q 05

One-Step Equations

Sarah has $42 after spending some money on lunch. She originally had $57. How much did lunch cost?

x + REMAINING = ORIGINAL → x = ORIGINAL − REMAINING

Solution

Equation: 57 − x = 42, so x = 57 − 42 = $15. Trap: students add both numbers, getting $99.

Q 06

Order of Operations · PEMDAS

Evaluate: 3 + 2 × (8 − 5)² ÷ 6

PEMDAS: PARENTHESES → EXPONENT → × ÷ → + −

Solution

Step by step: (8−5) = 3 → 3² = 9 → 2 × 9 = 18 → 18 ÷ 6 = 3 → 3 + 3 = 6. Most errors come from skipping the exponent step.

Q 07

Inequalities

Jake needs at least $100 to buy a game. He has $67 saved. He earns $11 per hour mowing lawns. What is the minimum number of whole hours he must work?

MINIMUM → CEILING (ROUND UP, NEVER DOWN)

Solution

Need: 100 − 67 = $33 more. Hours: 33 ÷ 11 = 3 exactly. Answer: 3 hours. Trap: if the division weren't exact, you must round UP (not down), because he needs at least $100.

Q 08

Variables & Expressions

A store charges $5 per notebook and $2 per pen. Which expression represents the total cost of n notebooks and p pens?

RATE × QUANTITY, ONE TERM PER ITEM

Solution

5n (notebooks) + 2p (pens) = 5n + 2p. Trap A flips which price goes with which item. Traps C and D treat all items as the same price.

Q 09

Two-Step Equations

Five times a number, decreased by 7, equals 28. What is the number?

UNDO ADDITION/SUBTRACTION FIRST, THEN MULTIPLY/DIVIDE

Solution

5x − 7 = 28 → 5x = 35 → x = 7. Trap: divide before adding, getting 28 ÷ 5 − 7 — wrong order of inverse operations.

Q 10

Unit Conversion

A marathon is approximately 26.2 miles. If 1 mile ≈ 1.61 km, about how many kilometers is a marathon? Round to the nearest whole number.

MULTIPLY TO CONVERT miles → km (bigger unit = bigger number)

Solution

26.2 × 1.61 ≈ 42.18 ≈ 42 km. Trap: dividing (miles ÷ 1.61 ≈ 16) — students divide when they should multiply because km is larger than miles.

Q 11

Perimeter · Rectangle

A rectangular garden is 12 m long and 7 m wide. How much fencing is needed to go all the way around it?

PERIMETER = 2(length + width) — BOTH sides TWICE

Solution

P = 2(12 + 7) = 2 × 19 = 38 m. Trap A adds only once (19). Trap B multiplies both dimensions (12 × 7 = 84 — that's the area!).

Q 12

Area · Triangle

A triangular park has a base of 20 m and a height of 9 m. What is its area?

TRIANGLE AREA = (1/2) × base × height — DON'T FORGET THE HALF!

Solution

A = ½ × 20 × 9 = ½ × 180 = 90 m². Trap A skips the ½ (most common mistake). Trap C just adds.

Q 13

Pythagorean Theorem

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

a² + b² = c² — c IS ALWAYS THE HYPOTENUSE (longest side)

Solution

6² + 8² = 36 + 64 = 100. c = √100 = 10 ft. This is the classic 3-4-5 triple scaled by 2 (6-8-10). Trap A adds the legs directly.

Q 14

Circles · Circumference

A circular fountain has a diameter of 10 m. What is its circumference? Use π ≈ 3.14.

C = πd = 2πr — DIAMETER given? Use πd directly!

Solution

C = πd = 3.14 × 10 = 31.4 m. Trap A computes area (π × 5²). Trap C uses diameter as radius (2πd = 2 × 3.14 × 10 — doubled it).

Q 15

Angles · Supplementary & Complementary

Two angles are supplementary. One angle measures 3x + 10° and the other measures x + 30°. What is the value of x?

SUPPLEMENTARY = 180° | COMPLEMENTARY = 90°

Solution

(3x+10) + (x+30) = 180 → 4x + 40 = 180 → 4x = 140 → x = 35. Check: 3(35)+10 = 115°, 35+30 = 65°, 115+65 = 180° ✓

Q 16

Volume · Rectangular Prism

A fish tank is 60 cm long, 30 cm wide, and 40 cm tall. How many liters of water can it hold? (1 L = 1,000 cm³)

VOLUME = l × w × h, THEN CONVERT cm³ → liters ÷ 1000

Solution

V = 60 × 30 × 40 = 72,000 cm³. Convert: 72,000 ÷ 1,000 = 72 L. Trap C forgets to divide (leaves it as cm³). Trap D only adds the three dimensions.

Q 17

Similar Triangles · Scale

A tree casts a shadow 15 ft long. At the same time, a 5 ft tall person casts a 3 ft shadow. How tall is the tree?

SIMILAR TRIANGLES: height / shadow = CONSTANT ratio

Solution

Proportion: h / 15 = 5 / 3 → h = 15 × (5/3) = 25 ft. Trap B divides wrong (15 × 3 = 45). Always set up the ratio as same-unit / same-unit.

Q 18

Coordinate Geometry · Distance

Point A is at (1, 2) and Point B is at (4, 6). What is the distance between them?

DISTANCE = √[(x₂−x₁)² + (y₂−y₁)²] — PYTHAGOREAN on a grid!

Solution

d = √[(4−1)² + (6−2)²] = √[9 + 16] = √25 = 5. Trap A adds differences directly (3+4=7). Trap D gives the value under the radical (25) without taking the square root.

Q 19

Area · Circle

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

A = πr² — RADIUS = diameter ÷ 2, SQUARE the radius!

Solution

Radius: r = 14 ÷ 2 = 7 in. Area: A = 3.14 × 7² = 3.14 × 49 = 153.86 in². Trap A is the circumference. Trap B uses diameter as radius (3.14 × 14²). Trap D skips π entirely (7² × 4 ≈ 196).

Q 20

Interior Angles · Polygon

What is the sum of the interior angles of a hexagon (6-sided polygon)?

SUM = (n − 2) × 180° — subtract 2 triangles from corners

Solution

(6 − 2) × 180° = 4 × 180° = 720°. Trap A is a pentagon (5 sides). Trap C is an octagon. Trap D is always the sum of exterior angles for any convex polygon.