In this article, we present 25 math problems designed specifically for 8th graders. These problems cover a wide range of topics, including algebra, geometry, statistics, and more. Each question is labeled with its respective math subcategory and difficulty level, ranging from easy to hard. We have also provided detailed step-by-step explanations to help you solve these problems. Let's dive in and sharpen our mathematical abilities!

Problem 1: Solving Equations (Algebra) - Easy

Simplify the expression: 3x + 5 - (2x - 3)

Solution:
Step 1: Distribute the negative sign: 3x + 5 - 2x + 3
Step 2: Combine like terms: (3x - 2x) + (5 + 3)
Step 3: Simplify: x + 8

Answer: The simplified expression is x + 8.

Problem2 : Area and Perimeter (Geometry) - Easy

Find the area and perimeter of a rectangle with length 6 cm and width 4 cm.

Solution:
Step 1: Use the formula for the area of a rectangle: Area = length × width
Step 2: Substitute the given values: Area = 6 cm × 4 cm = 24 cm²
Step 3: Use the formula for the perimeter of a rectangle: Perimeter = 2(length + width)
Step 4: Substitute the given values: Perimeter = 2(6 cm + 4 cm) = 2 × 10 cm = 20 cm

Answer: The area of the rectangle is 24 cm² and the perimeter is 20 cm.

Problem 3: Linear Equations (Algebra) - Medium

Solve the equation: 2x + 3 = 5x - 1

Solution:
Step 1: Subtract 2x from both sides: 3 = 3x - 1
Step 2: Add 1 to both sides: 4 = 3x
Step 3: Divide both sides by 3: x = 4/3

Answer: The solution to the equation is x = 4/3.

Problem 4: Pythagorean Theorem (Geometry) - Medium

Find the length of the hypotenuse in a right triangle with sides measuring 5 cm and 12 cm.

Solution:
Step 1: Apply the Pythagorean theorem: a² + b² = c²
Step 2: Substitute the given values: 5² + 12² = c²
Step 3: Simplify: 25 + 144 = c²
Step 4: Calculate: 169 = c²
Step 5: Take the square root of both sides: c = √169 = 13

Answer: The length of the hypotenuse is 13 cm.

Problem 5: Simplifying Expressions (Algebra) - Easy

Simplify the expression: 2(3x - 4) + 5x - 2(2x + 1)

Solution:
Step 1: Distribute the 2 and -2: 6x - 8 + 5x - 4x - 2
Step 2: Combine like terms: 6x + 5x - 4x - 8 - 2
Step 3: Simplify: 7x - 10

Answer: The simplified expression is 7x - 10.

Problem 6: Volume of a Cylinder (Geometry) - Medium

Find the volume of a cylinder with a radius of 4 cm and a height of 8 cm. (Use π ≈ 3.14)

Solution:
Step 1: Use the formula for the volume of a cylinder: Volume = πr²h
Step 2: Substitute the given values: Volume = 3.14 × (4 cm)² × 8 cm
Step 3: Simplify: Volume = 3.14 × 16 cm² × 8 cm
Step 4: Calculate: Volume = 3.14 × 128 cm³
Step 5: Simplify further: Volume ≈ 401.92 cm³

Answer: The volume of the cylinder is approximately 401.92 cm³.

Problem 7: Solving Inequalities (Algebra) - Medium

Solve the inequality: 2x - 5 ≤ 3x + 2

Solution:
Step 1: Subtract 2x from both sides: -5 ≤ x + 2
Step 2: Subtract 2 from both sides: -7 ≤ x

Answer: The solution to the inequality is x ≥ -7.

Problem 8: Area of a Triangle (Geometry) - Easy

Find the area of a triangle with base length 10 cm and height 6 cm.

Solution:
Step 1: Use the formula for the area of a triangle: Area = (1/2) × base × height
Step 2: Substitute the given values: Area = (1/2) × 10 cm × 6 cm
Step 3: Calculate: Area = 5 cm × 6 cm = 30 cm²

Answer: The area of the triangle is 30 cm².

Problem 9: Exponents and Radicals (Algebra) - Easy

Simplify the expression: √(9) + 4² - √(16)

Solution:
Step 1: Evaluate the square roots: 3 + 4² - 4
Step 2: Simplify: 3 + 16 - 4
Step 3: Calculate: 19

Answer: The simplified expression is 19.

Problem 10: Mean, Median, and Mode (Statistics) - Medium

Find the mean, median, and mode of the following data set: 5, 7, 4, 5, 6, 7, 5, 8

Solution:
Step 1: Mean: Add up all the numbers and divide by the total count.
Mean = (5 + 7 + 4 + 5 + 6 + 7 + 5 + 8) / 8 = 47 / 8 ≈ 5.875

Step 2: Median: Arrange the numbers in ascending order and find the middle value.
4, 5, 5, 5, 6, 7, 7, 8
Median = (5 + 6) / 2 = 11 / 2 = 5.5

Step 3: Mode: Determine the most frequent number(s) in the data set.
Mode = 5 (since it appears most frequently)

Answer: The mean is approximately 5.875, the median is 5.5, and the mode is 5.

Problem 11: Systems of Equations (Algebra) - Hard

Solve the system of equations:
2x + 3y = 10
3x - 4y = 5

Solution:
Step 1: Multiply the first equation by 2 and the second equation by 3 to eliminate x:
4x + 6y = 20
9x - 12y = 15

Step 2: Add the equations: (4x + 6y) + (9x - 12y) = 20 + 15
13x - 6y = 35

Step 3: Multiply the first equation by 3 and the second equation by 2 to eliminate y:
6x + 9y = 30
6x - 8y = 10

Step 4: Subtract the equations: (6x + 9y) - (6x - 8y) = 30 - 10
17y = 20
y = 20/17

Step 5: Substitute the value of y into one of the original equations (e.g., the first equation):
2x + 3(20/17) = 10
2x + 60/17 = 10
2x = 10 - 60/17
2x = 170/17 - 60/17
2x = 110/17
x = 110/17 * 1/2
x = 110/34
x = 55/17

Answer: The solution to the system of equations is x = 55/17 and y = 20/17.

Problem 12: Similar Figures (Geometry) - Medium

In similar figures, the ratio of their corresponding side lengths is 3:5. If the smaller figure has a perimeter of 18 cm, what is the perimeter of the larger figure?

Solution:
Step 1: Set up a proportion using the ratio of the corresponding side lengths:
3/5 = 18 cm / Perimeter
Step 2: Cross-multiply and solve for the perimeter:
3 × Perimeter = 5 × 18 cm
Perimeter = (5 × 18 cm) / 3
Perimeter = 30 cm

Answer: The perimeter of the larger figure is 30 cm.

Problem 13: Quadratic Equations (Algebra) - Medium

Solve the quadratic equation: x² + 4x - 5 = 0

Solution:
Step 1: Factorize or use the quadratic formula to find the roots of the equation.

Factoring: (x - 1)(x + 5) = 0
Setting each factor equal to zero: x - 1 = 0 or x + 5 = 0
Solving for x: x = 1 or x = -5
Answer: The solutions to the quadratic equation are x = 1 and x = -5.

Problem 14: Area of a Circle (Geometry) - Easy

Find the area of a circle with a radius of 6 cm. (Use π ≈ 3.14)

Solution:
Step 1: Use the formula for the area of a circle: Area = πr²
Step 2: Substitute the given value: Area = 3.14 × (6 cm)²
Step 3: Simplify: Area = 3.14 × 36 cm²
Step 4: Calculate: Area ≈ 113.04 cm²

Answer: The area of the circle is approximately 113.04 cm².

Problem 15: Operations with Fractions (Algebra) - Easy

Simplify the expression: (2/3) + (5/6) - (1/4)

Solution:
Step 1: Find a common denominator: The least common multiple of 3, 6, and 4 is 12.
Step 2: Rewrite each fraction with a denominator of 12:
(2/3) = (8/12)
(5/6) = (10/12)
(1/4) = (3/12)
Step 3: Add the fractions: (8/12) + (10/12) - (3/12) = 15/12
Step 4: Simplify the fraction: 15/12 = (5/4) or 1 1/4

Answer: The simplified expression is (5/4) or 1 1/4.

Problem 16: Surface Area of a Rectangular Prism (Geometry) - Medium

Find the surface area of a rectangular prism with length 8 cm, width 5 cm, and height 3 cm.

Solution:
Step 1: Use the formula for the surface area of a rectangular prism: Surface Area = 2lw + 2lh + 2wh
Step 2: Substitute the given values: Surface Area = 2(8 cm × 5 cm) + 2(8 cm × 3 cm) + 2(5 cm × 3 cm)
Step 3: Calculate: Surface Area = 2(40 cm²) + 2(24 cm²) + 2(15 cm²)
Step 4: Simplify: Surface Area = 80 cm² + 48 cm² + 30 cm²
Step 5: Calculate further: Surface Area = 158 cm²

Answer: The surface area of the rectangular prism is 158 cm².

Problem 17: Probability (Statistics) - Easy

A bag contains 4 red balls, 3 blue balls, and 5 green balls. What is the probability of randomly selecting a red ball?

Solution:
Step 1: Determine the total number of balls in the bag: 4 + 3 + 5 = 12
Step 2: Calculate the probability: Number of red balls / Total number of balls = 4/12 = 1/3

Answer: The probability of randomly selecting a red ball is 1/3.

Problem 18: Percentages (Algebra) - Easy

If a shirt originally costs $40 and is on sale for 20% off, what is the sale price?

Solution:
Step 1: Calculate the amount of discount: 20% of $40 = (20/100) × $40 = $8
Step 2: Subtract the discount from the original price: $40 - $8 = $32

Answer : The sale price of the shirt is $32.

Problem 19: Angles in a Triangle (Geometry) - Easy

In a triangle, the measures of two angles are 30° and 75°. What is the measure of the third angle?

Solution:
Step 1: Subtract the sum of the given angles from 180° (since the sum of angles in a triangle is 180°):
180° - 30° - 75° = 75°

Answer: The measure of the third angle is 75°.

Problem 20: Order of Operations (Algebra) - Easy

Evaluate the expression: 2 + 3 × (4 - 1)

Solution:
Step 1: Perform the operation inside the parentheses: 4 - 1 = 3
Step 2: Multiply: 3 × 3 = 9
Step 3: Add: 2 + 9 = 11

Answer: The value of the expression is 11.

Problem 21: Surface Area of a Cylinder (Geometry) - Medium

Find the surface area of a cylinder with a radius of 5 cm and a height of 10 cm. (Use π ≈ 3.14)

Solution:
Step 1: Calculate the lateral surface area: Lateral Surface Area = 2πrh
Step 2: Substitute the given values: Lateral Surface Area = 2 × 3.14 × 5 cm × 10 cm = 314 cm²
Step 3: Calculate the area of the two bases: Base Area = πr²
Base Area = 3.14 × (5 cm)² = 3.14 × 25 cm² = 78.5 cm²
Step 4: Calculate the total surface area: Total Surface Area = Lateral Surface Area + 2(Base Area)
Total Surface Area = 314 cm² + 2(78.5 cm²) = 471 cm²

Answer: The surface area of the cylinder is 471 cm².

Problem 22: Probability of Compound Events (Statistics) - Medium

A bag contains 4 red marbles, 3 blue marbles, and 5 green marbles. If you randomly select two marbles without replacement, what is the probability of selecting a red marble followed by a blue marble?

Solution:
Step 1: Determine the total number of marbles in the bag: 4 + 3 + 5 = 12
Step 2: Calculate the probability of selecting a red marble on the first draw: 4/12
Step 3: After the first marble is drawn, there are 11 marbles left in the bag.
Step 4: Calculate the probability of selecting a blue marble on the second draw, without replacing the first marble: 3/11
Step 5: Multiply the probabilities: (4/12) × (3/11) = 1/11

Answer: The probability of selecting a red marble followed by a blue marble is 1/11.

Problem 23: Simple Interest (Algebra) - Medium

John invests $500 in a savings account with an annual interest rate of 4%. How much interest will he earn after 2 years?

Solution:
Step 1: Calculate the interest earned per year: $500 × 4% = $500 × (4/100) = $20
Step 2: Multiply the interest per year by the number of years: $20 × 2 = $40

Answer: John will earn $40 in interest after 2 years.

Problem 24: Volume of a Cone (Geometry) - Hard

Find the volume of a cone with a radius of 8 cm and a height of 12 cm. (Use π ≈ 3.14)

Solution:
Step 1: Use the formula for the volume of a cone: Volume = (1/3) × πr²h
Step 2: Substitute the given values: Volume = (1/3) × 3.14 × (8 cm)² × 12 cm
Step 3: Simplify: Volume = (1/3) × 3.14 × 64 cm² × 12 cm
Step 4: Calculate: Volume ≈ 803.84 cm³

Answer: The volume of the cone is approximately 803.84 cm³.

Problem 25: Percent Increase (Algebra) - Hard
The population of a city increased from 10,000 to 12,500. What was the percent increase?
Solution:
Step 1: Calculate the difference in population: 12,500 - 10,000 = 2,500
Step 2: Calculate the percent increase: (2,500 / 10,000) × 100% = 25%

Answer: The percent increase in the population was 25%.

Congratulations on completing the 25 math problems for 8th graders! We covered a wide range of topics, including algebra, geometry, statistics, and more. By practicing these problems and understanding the step-by-step solutions, you have honed your mathematical skills. Keep practicing and exploring new math concepts to further strengthen your abilities. Remember, math is all around us, and with practice, you can excel in this fascinating subject! You can find more math resources in out on our math page.