Math Problems For 8th Graders With Answers

Mathematics in the 8th grade is a pivotal stage, bridging the gap between basic arithmetic and more advanced algebraic concepts. Mastering these mathematical challenges is crucial for success in higher-level math courses. This article provides a comprehensive collection of math problems tailored for 8th graders, complete with step-by-step solutions, covering a wide range of topics including algebra, geometry, and data analysis.

Algebra Problems

Solving Linear Equations

Problem 1: Solve for x: 5x + 3 = 2x + 12

Solution:

1. Subtract 2x from both sides: 5x - 2x + 3 = 2x - 2x + 12, which simplifies to 3x + 3 = 12.
2. Subtract 3 from both sides: 3x + 3 - 3 = 12 - 3, which simplifies to 3x = 9.
3. Divide both sides by 3: (3x)/3 = 9/3, which gives x = 3.

Problem 2: Solve for y: -2(y - 4) = 3y - 2

Solution:

1. Distribute the -2 on the left side: -2y + 8 = 3y - 2.
2. Add 2y to both sides: -2y + 2y + 8 = 3y + 2y - 2, which simplifies to 8 = 5y - 2.
3. Add 2 to both sides: 8 + 2 = 5y - 2 + 2, which simplifies to 10 = 5y.
4. Divide both sides by 5: 10/5 = (5y)/5, which gives y = 2.

Problem 3: Solve for z: (2/3)z + 5 = 11

Solution:

1. Subtract 5 from both sides: (2/3)z + 5 - 5 = 11 - 5, which simplifies to (2/3)z = 6.
2. Multiply both sides by 3/2: (3/2) * (2/3)z = 6 * (3/2), which simplifies to z = 9.

Systems of Linear Equations

Problem 4: Solve the system of equations:

• x + y = 5
• x - y = 1

Solution:

1. Add the two equations: (x + y) + (x - y) = 5 + 1, which simplifies to 2x = 6.
2. Divide both sides by 2: (2x)/2 = 6/2, which gives x = 3.
3. Substitute x = 3 into the first equation: 3 + y = 5.
4. Subtract 3 from both sides: 3 - 3 + y = 5 - 3, which gives y = 2. Therefore, the solution is x = 3 and y = 2.

Problem 5: Solve the system of equations:

• 2x + y = 7
• x - y = -1

Solution:

1. Add the two equations: (2x + y) + (x - y) = 7 + (-1), which simplifies to 3x = 6.
2. Divide both sides by 3: (3x)/3 = 6/3, which gives x = 2.
3. Substitute x = 2 into the second equation: 2 - y = -1.
4. Add y and 1 to both sides: 2 + 1 = y, which gives y = 3. Therefore, the solution is x = 2 and y = 3.

Problem 6: Solve the system of equations:

• 3x + 2y = 8
• x - y = 1

Solution:

1. Multiply the second equation by 2: 2(x - y) = 2(1), which simplifies to 2x - 2y = 2.
2. Add the modified second equation to the first equation: (3x + 2y) + (2x - 2y) = 8 + 2, which simplifies to 5x = 10.
3. Divide both sides by 5: (5x)/5 = 10/5, which gives x = 2.
4. Substitute x = 2 into the second equation: 2 - y = 1.
5. Subtract 2 from both sides: 2 - 2 - y = 1 - 2, which simplifies to -y = -1.
6. Multiply both sides by -1: y = 1. Therefore, the solution is x = 2 and y = 1.

Exponents and Scientific Notation

Problem 7: Simplify: (3x²y³)²

Solution:

1. Apply the power to each term inside the parentheses: 3² * (x²)² * (y³)².
2. Simplify each term: 9 * x⁴ * y⁶. Therefore, the simplified expression is 9x⁴y⁶.

Problem 8: Simplify: (a^-2b³) / (a³b^-1)

Solution:

1. Use the quotient rule for exponents: a^-2-3 * b^3-(-1).
2. Simplify the exponents: a^-5 * b⁴.
3. Rewrite with positive exponents: b⁴ / a⁵.

Problem 9: Express 0.000045 in scientific notation.

Solution:

1. Move the decimal point 5 places to the right to get 4.5.
2. Since we moved the decimal point 5 places to the right, the exponent is -5. Therefore, 0.000045 in scientific notation is 4.5 × 10^-5.

Problem 10: Express 3.2 × 10⁴ in standard notation.

Solution:

1. Move the decimal point 4 places to the right: 32000. Therefore, 3.2 × 10⁴ in standard notation is 32,000.

Factoring

Problem 11: Factor: x² + 5x + 6

Solution:

1. Find two numbers that multiply to 6 and add to 5. These numbers are 2 and 3.
2. Rewrite the expression as (x + 2)(x + 3). Therefore, the factored form of x² + 5x + 6 is (x + 2)(x + 3).

Problem 12: Factor: y² - 4y - 21

Solution:

1. Find two numbers that multiply to -21 and add to -4. These numbers are -7 and 3.
2. Rewrite the expression as (y - 7)(y + 3). Therefore, the factored form of y² - 4y - 21 is (y - 7)(y + 3).

Problem 13: Factor: 4a² - 9

Solution:

1. Recognize this as a difference of squares: (2a)² - 3².
2. Rewrite the expression as (2a - 3)(2a + 3). Therefore, the factored form of 4a² - 9 is (2a - 3)(2a + 3).

Geometry Problems

Pythagorean Theorem

Problem 14: A right triangle has legs of lengths 6 cm and 8 cm. Find the length of the hypotenuse.

Solution:

1. Use the Pythagorean Theorem: a² + b² = c², where a and b are the legs and c is the hypotenuse.
2. Substitute the given values: 6² + 8² = c².
3. Simplify: 36 + 64 = c², which gives 100 = c².
4. Take the square root of both sides: √100 = √c², which gives c = 10. Therefore, the length of the hypotenuse is 10 cm.

Problem 15: A right triangle has a leg of length 5 inches and a hypotenuse of length 13 inches. Find the length of the other leg.

Solution:

1. Use the Pythagorean Theorem: a² + b² = c².
2. Substitute the given values: 5² + b² = 13².
3. Simplify: 25 + b² = 169.
4. Subtract 25 from both sides: b² = 169 - 25, which gives b² = 144.
5. Take the square root of both sides: √b² = √144, which gives b = 12. Therefore, the length of the other leg is 12 inches.

Problem 16: A ladder is leaning against a wall. The base of the ladder is 3 meters away from the wall, and the ladder reaches a height of 4 meters on the wall. How long is the ladder?

Solution:

1. Use the Pythagorean Theorem: a² + b² = c².
2. Substitute the given values: 3² + 4² = c².
3. Simplify: 9 + 16 = c², which gives 25 = c².
4. Take the square root of both sides: √25 = √c², which gives c = 5. Therefore, the length of the ladder is 5 meters.

Area and Perimeter

Problem 17: Find the area and perimeter of a rectangle with length 12 cm and width 5 cm.

Solution:

1. Area of a rectangle: A = length × width = 12 cm × 5 cm = 60 cm².
2. Perimeter of a rectangle: P = 2(length + width) = 2(12 cm + 5 cm) = 2(17 cm) = 34 cm. Therefore, the area is 60 cm² and the perimeter is 34 cm.

Problem 18: Find the area and circumference of a circle with radius 7 inches.

Solution:

1. Area of a circle: A = πr² = π(7 inches)² = 49π inches².
2. Circumference of a circle: C = 2πr = 2π(7 inches) = 14π inches. Therefore, the area is 49π inches² and the circumference is 14π inches.

Problem 19: Find the area of a triangle with base 10 m and height 8 m.

Solution:

1. Area of a triangle: A = (1/2) × base × height = (1/2) × 10 m × 8 m = 40 m². Therefore, the area of the triangle is 40 m².

Angles

Problem 20: Two angles are supplementary. One angle is 65 degrees. Find the measure of the other angle.

Solution:

1. Supplementary angles add up to 180 degrees.
2. Let the other angle be x. Then, 65 + x = 180.
3. Subtract 65 from both sides: x = 180 - 65, which gives x = 115. Therefore, the measure of the other angle is 115 degrees.

Problem 21: Two angles are complementary. One angle is 30 degrees. Find the measure of the other angle.

Solution:

1. Complementary angles add up to 90 degrees.
2. Let the other angle be x. Then, 30 + x = 90.
3. Subtract 30 from both sides: x = 90 - 30, which gives x = 60. Therefore, the measure of the other angle is 60 degrees.

Problem 22: In a triangle, two angles measure 45 degrees and 75 degrees. Find the measure of the third angle.

Solution:

1. The sum of the angles in a triangle is 180 degrees.
2. Let the third angle be x. Then, 45 + 75 + x = 180.
3. Simplify: 120 + x = 180.
4. Subtract 120 from both sides: x = 180 - 120, which gives x = 60. Therefore, the measure of the third angle is 60 degrees.

Data Analysis Problems

Mean, Median, Mode, and Range

Problem 23: Find the mean, median, mode, and range of the following data set: 4, 7, 10, 12, 15, 4, 6.

Solution:

1. Mean: (4 + 7 + 10 + 12 + 15 + 4 + 6) / 7 = 58 / 7 ≈ 8.29.
2. Median: First, sort the data set: 4, 4, 6, 7, 10, 12, 15. The median is the middle value, which is 7.
3. Mode: The mode is the value that appears most frequently, which is 4.
4. Range: The range is the difference between the largest and smallest values: 15 - 4 = 11. Therefore, the mean is approximately 8.29, the median is 7, the mode is 4, and the range is 11.

Problem 24: Find the mean, median, mode, and range of the following data set: 22, 25, 25, 30, 32, 35, 40.

Solution:

1. Mean: (22 + 25 + 25 + 30 + 32 + 35 + 40) / 7 = 209 / 7 = 29.86.
2. Median: The data set is already sorted. The median is the middle value, which is 30.
3. Mode: The mode is the value that appears most frequently, which is 25.
4. Range: The range is the difference between the largest and smallest values: 40 - 22 = 18. Therefore, the mean is approximately 29.86, the median is 30, the mode is 25, and the range is 18.

Problem 25: Find the mean, median, mode, and range of the following data set: 1, 3, 5, 5, 7, 9, 11, 11, 11.

Solution:

1. Mean: (1 + 3 + 5 + 5 + 7 + 9 + 11 + 11 + 11) / 9 = 63 / 9 = 7.
2. Median: The data set is already sorted. The median is the middle value, which is 7.
3. Mode: The mode is the value that appears most frequently, which is 11.
4. Range: The range is the difference between the largest and smallest values: 11 - 1 = 10. Therefore, the mean is 7, the median is 7, the mode is 11, and the range is 10.

Probability

Problem 26: A bag contains 5 red balls, 3 blue balls, and 2 green balls. What is the probability of drawing a red ball?

Solution:

1. Total number of balls: 5 (red) + 3 (blue) + 2 (green) = 10.
2. Probability of drawing a red ball: (Number of red balls) / (Total number of balls) = 5 / 10 = 1/2. Therefore, the probability of drawing a red ball is 1/2 or 50%.

Problem 27: A spinner has 8 equal sections, numbered 1 through 8. What is the probability of spinning an odd number?

Solution:

1. Total number of sections: 8.
2. Odd numbers: 1, 3, 5, 7. There are 4 odd numbers.
3. Probability of spinning an odd number: (Number of odd numbers) / (Total number of sections) = 4 / 8 = 1/2. Therefore, the probability of spinning an odd number is 1/2 or 50%.

Problem 28: A fair six-sided die is rolled. What is the probability of rolling a number greater than 4?

Solution:

1. Total number of outcomes: 6 (numbers 1 through 6).
2. Numbers greater than 4: 5, 6. There are 2 numbers greater than 4.
3. Probability of rolling a number greater than 4: (Number of outcomes greater than 4) / (Total number of outcomes) = 2 / 6 = 1/3. Therefore, the probability of rolling a number greater than 4 is 1/3.

Ratios and Proportions

Problem 29: If 3 apples cost $1.50, how much will 7 apples cost?

Solution:

1. Find the cost of one apple: $1.50 / 3 = $0.50.
2. Find the cost of 7 apples: 7 × $0.50 = $3.50. Therefore, 7 apples will cost $3.50.

Problem 30: A recipe calls for 2 cups of flour and 3 cups of sugar. If you want to make a larger batch using 8 cups of flour, how many cups of sugar do you need?

Solution:

1. Find the ratio of flour to sugar: 2 cups flour / 3 cups sugar.
2. Set up a proportion: 2/3 = 8/x, where x is the amount of sugar needed.
3. Cross-multiply: 2x = 3 × 8, which gives 2x = 24.
4. Divide both sides by 2: x = 24 / 2, which gives x = 12. Therefore, you need 12 cups of sugar.

Problem 31: A map has a scale of 1 inch = 25 miles. If two cities are 4 inches apart on the map, what is the actual distance between them?

Solution:

1. Multiply the distance on the map by the scale factor: 4 inches × 25 miles/inch = 100 miles. Therefore, the actual distance between the two cities is 100 miles.

Word Problems

Problem 32: John is 5 years older than Mary. The sum of their ages is 31. How old are John and Mary?

Solution:

1. Let Mary's age be m. Then John's age is m + 5.
2. The sum of their ages is m + (m + 5) = 31.
3. Simplify: 2m + 5 = 31.
4. Subtract 5 from both sides: 2m = 26.
5. Divide both sides by 2: m = 13. So, Mary is 13 years old.
6. John's age is m + 5 = 13 + 5 = 18. Therefore, Mary is 13 years old, and John is 18 years old.

Problem 33: A store sells shirts for $15 each and pants for $25 each. If a customer buys 3 shirts and 2 pants, how much does the customer spend in total?

Solution:

1. Cost of 3 shirts: 3 × $15 = $45.
2. Cost of 2 pants: 2 × $25 = $50.
3. Total cost: $45 + $50 = $95. Therefore, the customer spends $95 in total.

Problem 34: A train travels at a speed of 80 miles per hour. How far will the train travel in 2.5 hours?

Solution:

1. Use the formula: distance = speed × time.
2. Substitute the given values: distance = 80 miles/hour × 2.5 hours = 200 miles. Therefore, the train will travel 200 miles.

Mastering these 8th-grade math problems is essential for building a strong foundation in mathematics. By understanding the concepts and practicing regularly, students can enhance their problem-solving skills and prepare for future mathematical challenges. The problems provided here cover a broad range of topics and offer detailed solutions to aid in comprehension and practice.