Math Practice — Algebra & Geometry

Algebra 1

Word Problems

Translate real-life situations into equations. Common mistake: forgetting to define your variable clearly before setting up the equation.

⚡ Quick Memory Keys

DEFINE name your variable first

TRANSLATE words → math symbols

SOLVE isolate the variable

CHECK plug answer back in

SUM + (plus / more than)

DIFFERENCE − (less than / fewer)

PRODUCT × (times / of)

QUOTIENT ÷ (divided by / per)

⭐ Very Easy · One-Step Equation

Sarah has some apples. After eating 3 apples, she has 11 apples left. How many apples did she start with?

💡 Let x = starting number. "After eating 3" means subtract.

Solution

Let x = starting apples.

x − 3 = 11 x = 11 + 3 = 14

Sarah started with 14 apples. ✓ Check: 14 − 3 = 11 ✔

⭐ Very Easy · Two Numbers Sum

Two numbers add up to 20. One number is 6 more than the other. What are the two numbers?

💡 If one number is x, the other is x + 6. Both must add to 20.

Solution

Let x = smaller number, so x + 6 = larger number.

x + (x + 6) = 20 2x + 6 = 20 → 2x = 14 → x = 7

Numbers are 7 and 13. Check: 7 + 13 = 20 ✔, 13 − 7 = 6 ✔

⭐⭐ Easy · Rate / Unit Price

A store sells pens at $2 each. Tom buys some pens and pays $14 in total. How many pens did he buy?

💡 Rate × quantity = total. This is a division problem.

Solution

Let n = number of pens.

2n = 14 n = 14 ÷ 2 = 7

Tom bought 7 pens. ✓

⭐⭐ Easy · Age Problem (TRICKY — "more than" direction!)

Emma is 3 years older than her brother. The sum of their ages is 19. How old is Emma?

⚠️ Common mistake: forgetting who is older. Emma = brother + 3.

Solution

Let b = brother's age. Emma = b + 3.

b + (b + 3) = 19 2b + 3 = 19 → 2b = 16 → b = 8 Emma = 8 + 3 = 11

Emma is 11 years old. ✓ Check: 8 + 11 = 19 ✔

⭐⭐ Easy · Perimeter Problem

A rectangle's length is twice its width. The perimeter is 36 cm. What is the width?

💡 Perimeter = 2l + 2w. Replace l with 2w.

Solution

Let w = width. Length = 2w.

2(2w) + 2(w) = 36 4w + 2w = 36 → 6w = 36 → w = 6

Width = 6 cm, Length = 12 cm. ✓ Check: 2(12) + 2(6) = 24 + 12 = 36 ✔

⭐⭐ Easy · Money / Coins (TRICKY — value vs. count!)

Jake has dimes and nickels totaling $1.20. He has twice as many nickels as dimes. How many dimes does he have?

⚠️ Remember: dime = 10¢, nickel = 5¢. Value ≠ count!

Solution

Let d = dimes. Nickels = 2d.

10d + 5(2d) = 120 (in cents) 10d + 10d = 120 → 20d = 120 → d = 6

Jake has 6 dimes and 12 nickels. ✓ Check: 60¢ + 60¢ = $1.20 ✔

⭐⭐⭐ Medium · Distance = Rate × Time

A car travels at 60 km/h. How long does it take to travel 210 km?

💡 FORMULA: d = r × t → t = d ÷ r

Solution

Using d = r × t:

210 = 60 × t t = 210 ÷ 60 = 3.5 hours

The trip takes 3.5 hours (3 hours 30 minutes). ✓

⭐⭐⭐ Medium · Consecutive Integers (CONFUSING setup!)

The sum of three consecutive integers is 48. What is the largest of the three integers?

⚠️ Consecutive means n, n+1, n+2. The question asks for the LARGEST, not n!

Solution

Let n = smallest integer. The three are n, n+1, n+2.

n + (n+1) + (n+2) = 48 3n + 3 = 48 → 3n = 45 → n = 15 Largest = n + 2 = 15 + 2 = 17

⚠️ Many students answer 15 — that's the SMALLEST. The largest is 17. ✓

⭐⭐⭐ Medium · Percent / Discount

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

💡 Sale price = Original − Discount. Discount = % × Original.

Solution

Discount = 25% × $80:

0.25 × 80 = $20 discount Sale price = $80 − $20 = $60

⚠️ Option A ($20) is the discount amount, not the sale price! Answer: $60. ✓

⭐⭐⭐ Medium · Work / Mixture (TRICKY wording!)

Maria earns $12 per hour at her job. She also earned a one-time bonus of $30. If she earned $114 in total, how many hours did she work?

⚠️ The bonus is NOT per hour — it's added once, not multiplied!

Solution

Let h = hours worked.

12h + 30 = 114 12h = 114 − 30 = 84 h = 84 ÷ 12 = 7

Maria worked 7 hours. ✓ Check: 12(7) + 30 = 84 + 30 = 114 ✔

Geometry · Circles

Circle Problems

From basics to arc length — master the vocabulary and formulas first, then the problems become straightforward.

⚡ Circle Formula Memory Keys

RADIUS r = center to edge

DIAMETER d = 2r (across)

CIRCUMFERENCE C = 2πr = πd

AREA A = πr²

π ≈ 3.14 or use 22/7

CHORD line inside circle

ARC part of circumference

TANGENT touches at 1 point

⭐ Very Easy · Radius vs Diameter

A circle has a diameter of 10 cm. What is its radius?

💡 DIAMETER = 2 × RADIUS. So RADIUS = DIAMETER ÷ 2.

Solution

r = d ÷ 2 = 10 ÷ 2 = 5 cm

The radius is 5 cm. ✓ (Diameter is always twice the radius.)

⭐ Very Easy · Circumference (given radius)

Find the circumference of a circle with radius = 7 cm. Use π ≈ 3.14.

💡 C = 2πr. Don't confuse with Area = πr²!

Solution

C = 2 × π × r = 2 × 3.14 × 7 C = 6.28 × 7 = 43.96 cm

⚠️ Option C (153.86) is the Area πr². Don't mix up the formulas! Answer: 43.96 cm. ✓

⭐⭐ Easy · Area of a Circle

What is the area of a circle with diameter = 12 cm? Use π ≈ 3.14.

⚠️ The problem gives DIAMETER. You must find radius FIRST before using A = πr²!

Solution

Step 1: Find radius from diameter.

r = d ÷ 2 = 12 ÷ 2 = 6 cm A = πr² = 3.14 × 6² = 3.14 × 36 = 113.04 cm²

⚠️ Option B used r = 12 (forgot to halve). Always convert diameter → radius first! Answer: 113.04 cm². ✓

⭐⭐ Easy · Find Radius from Circumference

A circle has a circumference of 62.8 cm. What is the radius? Use π ≈ 3.14.

💡 Work backwards: C = 2πr → r = C ÷ (2π). Rearrange the formula!

Solution

C = 2πr → 62.8 = 2 × 3.14 × r 62.8 = 6.28 × r → r = 62.8 ÷ 6.28 = 10 cm

The radius is 10 cm. ✓

⭐⭐ Easy · Central Angle

A circle is divided into 4 equal sectors. What is the central angle of each sector?

💡 A full circle = 360°. Divide equally among sectors.

Solution

Central angle = 360° ÷ 4 = 90°

Each sector has a central angle of 90°. ✓ (These form a "plus" shape — right angles!)

⭐⭐⭐ Medium · Arc Length (TRICKY fraction!)

A circle has radius 9 cm. What is the arc length of a sector with a central angle of 120°? Use π ≈ 3.14.

💡 Arc = (angle/360) × 2πr. The fraction (120/360) = 1/3 of the circle.

Solution

Arc Length = (θ/360) × 2πr = (120/360) × 2 × 3.14 × 9 = (1/3) × 56.52 = 18.84 cm

The arc length is 18.84 cm. ✓

⭐⭐⭐ Medium · Area of Sector

A pizza has a radius of 10 cm. It is cut into 8 equal slices. What is the area of one slice? Use π ≈ 3.14.

💡 Area of sector = (angle/360) × πr². One slice = 360/8 = 45° sector.

Solution

Each slice = 360° ÷ 8 = 45°

Total area = πr² = 3.14 × 100 = 314 cm² One slice = 314 ÷ 8 = 39.25 cm²

One slice has an area of 39.25 cm². ✓

⭐⭐⭐ Medium · Tangent Line (TRICKY concept!)

A tangent line touches a circle at point P. The radius to point P is 5 cm. What is the angle between the radius and the tangent line?

⚠️ KEY THEOREM: A tangent line is ALWAYS perpendicular (90°) to the radius at the point of tangency!

Solution

Tangent-Radius Theorem: A tangent to a circle is always perpendicular to the radius drawn to the point of tangency.

Angle = 90° (always, regardless of radius length)

The angle is always 90°. ✓ The length of the radius (5 cm) is irrelevant to the angle!

⭐⭐⭐ Medium · Inscribed Angle Theorem

An inscribed angle in a circle intercepts an arc of 80°. What is the measure of the inscribed angle?

💡 INSCRIBED ANGLE THEOREM: Inscribed angle = ½ × intercepted arc. Central angle = arc directly.

Solution

Inscribed Angle Theorem:

Inscribed Angle = (1/2) × Intercepted Arc = (1/2) × 80° = 40°

⚠️ Many students answer 80° (confusing inscribed angle with central angle). Inscribed angles are always half the arc! Answer: 40°. ✓

⭐⭐⭐ Medium · Semicircle — Thales' Theorem

Triangle ABC is inscribed in a circle. AB is the diameter. What is the measure of angle ACB?

💡 THALES' THEOREM: Any angle inscribed in a semicircle (where the hypotenuse = diameter) is always 90°!

Solution

Thales' Theorem: An angle inscribed in a semicircle is always a right angle.

Since AB is a diameter → arc AB = 180° Angle ACB = (1/2) × 180° = 90°

Angle ACB is always 90°, no matter where C is on the circle! ✓