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Grade 8 Mathematics study materials
Grade 8 Mathematics study resources

Grade 8 Mathematics practice questions

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Question 1The Number System

Which statement correctly describes every irrational number?

  1. Its decimal expansion always terminates after a finite number of digits.
  2. It can always be written as a fraction of two integers.
  3. Its decimal expansion is non-terminating and non-repeating. Correct answer.
  4. Its decimal expansion repeats a single digit block forever.

Explanation

By definition, a rational number has a decimal that either terminates or eventually repeats, so an irrational number has a decimal that neither terminates nor settles into a repeating block. Numbers such as the square root of 2 and pi fit this description.

Question 2The Number System

Which of the following is a rational number?

  1. The cube root of 2
  2. The square root of 7
  3. The square root of 25 Correct answer.
  4. Pi

Explanation

A rational number can be written as a ratio of two integers. The square root of 25 equals 5, which is the ratio 5/1, so it is rational.

Question 3The Number System

The value of 0.75 is equal to which fraction in simplest form?

  1. 75/1000
  2. 7/5
  3. 3/4 Correct answer.
  4. 4/3

Explanation

Write 0.75 as 75/100 and divide numerator and denominator by their greatest common factor, 25, giving 3/4. A terminating decimal with two decimal places always reduces from a denominator of 100.

Question 4The Number System

Which list orders the numbers from least to greatest?

  1. -3/2, -0.5, 0.25, 1/3 Correct answer.
  2. -0.5, -3/2, 0.25, 1/3
  3. 0.25, 1/3, -0.5, -3/2
  4. -3/2, 0.25, -0.5, 1/3

Explanation

Convert to decimals: -3/2 = -1.5, -0.5, 0.25, and 1/3 is about 0.333. On a number line, -1.5 lies farthest left, then -0.5, then 0.25, then 0.333.

Question 5The Number System

What is the value of the square root of 144?

  1. 72
  2. 11
  3. 12 Correct answer.
  4. 14

Explanation

Since 12 times 12 equals 144, the principal square root of 144 is 12. The principal root is the non-negative value.

Question 6The Number System

Which statement about the number 0 is correct within the real number system?

  1. It is neither rational nor irrational.
  2. It is a natural number but not a whole number.
  3. It is an irrational number because it cannot be written as a fraction.
  4. It is a rational number and also an integer. Correct answer.

Explanation

Zero can be written as 0/1, a ratio of two integers, so it is rational. It is also an integer, since integers include all whole numbers and their opposites.

Question 7The Number System

Between which two consecutive integers does the square root of 50 lie?

  1. 7 and 8 Correct answer.
  2. 6 and 7
  3. 5 and 6
  4. 24 and 26

Explanation

Since 7 squared is 49 and 8 squared is 64, and 50 lies between 49 and 64, the square root of 50 lies between 7 and 8. It is slightly greater than 7.

Question 8The Number System

What is the value of (2/3) divided by (4/9)?

  1. 3/2 Correct answer.
  2. 8/27
  3. 27/8
  4. 6/9

Explanation

Dividing by a fraction means multiplying by its reciprocal: (2/3) times (9/4) equals 18/12, which simplifies to 3/2.

Question 9The Number System

Which expression is equal to a rational number?

  1. The square root of 6 divided by the square root of 3
  2. The square root of 2 plus the square root of 3
  3. The square root of 3 times the square root of 3 Correct answer.
  4. The square root of 5 times the square root of 2

Explanation

The square root of 3 times the square root of 3 equals the square root of 9, which is 3, a rational number. Multiplying a square root by itself always removes the radical.

Question 10The Number System

A calculator displays 0.333333333333 for the fraction 1/3. What does this display reveal about the decimal form of 1/3?

  1. The decimal eventually becomes a different repeating block.
  2. The decimal is irrational because it cannot be shown fully.
  3. The decimal repeats the digit 3 forever, so the display is a rounded approximation. Correct answer.
  4. The decimal terminates exactly at the twelfth digit.

Explanation

Dividing 1 by 3 always leaves remainder 1, so the digit 3 repeats without end. The display is truncated by the screen width, not a terminating value.

Question 11The Number System

Which number is greater: -7/4 or -1.5?

  1. -1.5, because it is closer to zero Correct answer.
  2. -7/4, because its absolute value is larger
  3. -7/4, because it has a larger denominator
  4. They are equal

Explanation

Since -7/4 equals -1.75, and -1.5 lies to the right of -1.75 on the number line, -1.5 is the greater value. For negative numbers, the value closer to zero is greater.

Question 12The Number System

What is the value of the cube root of -64?

  1. 4
  2. -4 Correct answer.
  3. No real value exists
  4. -8

Explanation

Since (-4) times (-4) times (-4) equals -64, the cube root of -64 is -4. Unlike square roots, cube roots of negative numbers are real and negative.

Question 13The Number System

Which statement is true about the sum of a rational number and an irrational number?

  1. It is sometimes rational and sometimes irrational.
  2. It is always irrational. Correct answer.
  3. It is always an integer.
  4. It is always rational.

Explanation

If a rational plus an irrational were rational, subtracting the rational part would make the irrational number rational, a contradiction. So the sum must be irrational, as with 2 plus the square root of 3.

Question 14The Number System

A square garden has an area of 200 square meters. Which expression gives the exact side length, and is that length rational or irrational?

  1. 100 meters; rational
  2. 20 meters; rational
  3. 10 times the square root of 2 meters; irrational Correct answer.
  4. The square root of 200 meters; rational

Explanation

The side length equals the square root of 200, which simplifies to 10 times the square root of 2 because 200 = 100 times 2. Since the square root of 2 is irrational and 10 is rational, the product is irrational.

Question 15The Number System

Which number is the smallest?

  1. 1.5
  2. The square root of 3
  3. The square root of 2 Correct answer.
  4. 1.4

Explanation

The square root of 2 is about 1.414, which is less than 1.4? No: 1.414 exceeds 1.4, so 1.4 is smaller than the square root of 2. The square root of 3 is about 1.732.

Question 16The Number System

Which decimal represents a rational number?

  1. 3.141592653...
  2. 0.414213562...
  3. 0.101001000100001...
  4. 0.2727272727... Correct answer.

Explanation

The decimal 0.272727... repeats the block 27 forever, and every repeating decimal equals a fraction of integers, here 3/11. Repeating decimals are therefore rational.

Question 17The Number System

The number 0.9 repeating (0.999...) is equal to which value?

  1. A value slightly less than 1
  2. An irrational number
  3. A value slightly more than 1
  4. 1 Correct answer.

Explanation

Let x = 0.999... Then 10x = 9.999..., and subtracting gives 9x = 9, so x = 1. A repeating decimal has exactly one rational value, and here it is 1.

Question 18The Number System

Which statement about the product of two irrational numbers is correct?

  1. The product is always rational.
  2. The product is never a real number.
  3. The product can be rational or irrational depending on the factors. Correct answer.
  4. The product is always irrational.

Explanation

Multiplying the square root of 2 by itself gives 2, which is rational, while multiplying the square root of 2 by the square root of 3 gives the square root of 6, which is irrational. So the product's nature depends on the specific factors.

Question 19The Number System

A number n is irrational. Which expression is guaranteed to be irrational?

  1. n times 0
  2. n squared
  3. n plus 5 Correct answer.
  4. n plus n

Explanation

Adding a nonzero rational number to an irrational number always yields an irrational number, since a rational result would force n itself to be rational. So n plus 5 must be irrational.

Question 20The Number System

Using the fact that the square root of 7 is about 2.6458, which value best approximates 3 minus the square root of 7 to the nearest hundredth?

  1. 0.36
  2. 0.35 Correct answer.
  3. -0.35
  4. 0.34

Explanation

Subtracting: 3 minus 2.6458 equals 0.3542. Rounding 0.3542 to the nearest hundredth gives 0.35, since the thousandths digit is 4.

Question 21Expressions and Equations

Which value of x makes the equation 5x − 7 = 23 true?

  1. 6 Correct answer.
  2. 3
  3. 5
  4. 4

Explanation

Add 7 to both sides to get 5x = 30, then divide both sides by 5 to obtain x = 6. Substituting 6 back gives 30 − 7 = 23, which checks.

Question 22Expressions and Equations

Simplify the expression 3(2x + 4) − 5x.

  1. x + 4
  2. 6x + 12
  3. x + 12 Correct answer.
  4. 11x + 12

Explanation

Distributing 3 over the parentheses gives 6x + 12, and combining like terms 6x − 5x leaves x. So the simplified expression is x + 12.

Question 23Expressions and Equations

Which expression represents "seven less than three times a number n"?

  1. 3n − 7 Correct answer.
  2. 7n − 3
  3. 7 − 3n
  4. 3(n − 7)

Explanation

"Three times a number" is 3n, and "seven less than" that quantity means subtract 7 from it, giving 3n − 7. The order matters because subtraction is not commutative.

Question 24Expressions and Equations

Solve the equation x/4 + 3 = 9.

  1. 12
  2. 36
  3. 6
  4. 24 Correct answer.

Explanation

Subtract 3 from both sides to get x/4 = 6, then multiply both sides by 4 to obtain x = 24. Checking: 24/4 + 3 = 6 + 3 = 9.

Question 25Expressions and Equations

Which of these is a correct statement of the distributive property?

  1. a(b + c) = ab + c
  2. a(b + c) = ab + ac Correct answer.
  3. a(bc) = (ab)c
  4. a + (b + c) = (a + b) + c

Explanation

The distributive property says a factor multiplied by a sum equals the sum of the products of that factor with each addend. So a(b + c) = ab + ac.

Question 26Expressions and Equations

What is the value of 2^3 · 2^4 expressed as a single power?

  1. 2^1
  2. 2^7 Correct answer.
  3. 2^12
  4. 4^7

Explanation

When multiplying powers with the same base, add the exponents: 2^3 · 2^4 = 2^(3+4) = 2^7. This equals 128.

Question 27Expressions and Equations

A line passes through the points (0, 3) and (2, 7). What is the slope of the line?

  1. 4
  2. 2 Correct answer.
  3. 1/2
  4. -2

Explanation

Slope equals the change in y divided by the change in x: (7 − 3)/(2 − 0) = 4/2 = 2. So the line rises 2 units for every 1 unit to the right.

Question 28Expressions and Equations

Which equation has infinitely many solutions?

  1. 4x + 1 = 4x + 4
  2. 4(x + 1) = x + 4
  3. 4(x + 1) = 4x + 1
  4. 4(x + 1) = 4x + 4 Correct answer.

Explanation

Expanding the left side gives 4x + 4, which is identical to the right side for every value of x, so every number is a solution. An equation has infinitely many solutions when both sides simplify to the same expression.

Question 29Expressions and Equations

Solve the equation 3(x − 2) = 2x + 5.

  1. -1
  2. 11 Correct answer.
  3. 1
  4. 7

Explanation

Distribute to get 3x − 6 = 2x + 5, then subtract 2x from both sides to get x − 6 = 5, and add 6 to obtain x = 11. Checking: 3(9) = 27 and 2(11) + 5 = 27.

Question 30Expressions and Equations

Which expression is equivalent to 2(3x − 4) − 3(x − 2)?

  1. 9x − 14
  2. 3x − 2 Correct answer.
  3. 3x − 14
  4. 3x + 2

Explanation

Distribute to get 6x − 8 − 3x + 6, being careful that subtracting 3(x − 2) flips both signs. Combining like terms gives 3x − 2.

Question 31Expressions and Equations

The sum of three consecutive integers is 72. Which equation models this situation, where n is the smallest integer?

  1. n + (n + 2) + (n + 4) = 72
  2. n + 3n = 72
  3. 3n = 72
  4. n + (n + 1) + (n + 2) = 72 Correct answer.

Explanation

Consecutive integers differ by 1, so the three are n, n + 1, and n + 2. Their sum set equal to 72 gives the model. Solving yields n = 23, and 23 + 24 + 25 = 72.

Question 32Expressions and Equations

Solve the inequality −2x + 5 < 11.

  1. x > -3 Correct answer.
  2. x < -3
  3. x > 3
  4. x < 3

Explanation

Subtract 5 to get −2x < 6, then divide by −2 and reverse the inequality symbol, giving x > −3. Dividing or multiplying an inequality by a negative number flips the direction.

Question 33Expressions and Equations

What is the slope of a line perpendicular to the line 3x + 4y = 8?

  1. 4/3 Correct answer.
  2. 3/4
  3. -4/3
  4. -3/4

Explanation

Solving for y gives y = −(3/4)x + 2, so the slope is −3/4. A perpendicular line has the negative reciprocal, which is 4/3.

Question 34Expressions and Equations

A phone plan charges a flat fee of $20 plus $0.10 per minute. Which equation gives the total cost C in dollars for m minutes?

  1. C = 20 − 0.10m
  2. C = 20 + 0.10m Correct answer.
  3. C = 20m + 0.10m
  4. C = 0.10 + 20m

Explanation

The flat fee is a constant added once, and the per-minute charge is the rate multiplied by m. So the total is C = 20 + 0.10m, a linear model with slope 0.10 and y-intercept 20.

Question 35Expressions and Equations

Simplify (x^5 · x^2) / x^3.

  1. x^6
  2. x^10
  3. x^(10/3)
  4. x^4 Correct answer.

Explanation

Multiplying powers with the same base adds exponents to give x^7, and dividing subtracts exponents to give x^(7−3) = x^4. Exponent rules apply because the bases are identical.

Question 36Expressions and Equations

Which equation represents a line parallel to y = 4x − 7 that passes through the point (1, 2)?

  1. y = -1/4 x + 2
  2. y = 4x − 7
  3. y = 4x − 2 Correct answer.
  4. y = 4x + 2

Explanation

Parallel lines share the same slope, so the new line has slope 4. Substituting (1, 2) into y = 4x + b gives 2 = 4 + b, so b = −2 and the equation is y = 4x − 2.

Question 37Expressions and Equations

Solve the equation 5 − 2(x − 3) = 3x + 1.

  1. 1
  2. -2
  3. 4
  4. 2 Correct answer.

Explanation

Distribute to get 5 − 2x + 6 = 3x + 1, which simplifies to 11 − 2x = 3x + 1. Adding 2x and subtracting 1 gives 10 = 5x, so x = 2.

Question 38Expressions and Equations

A student says the equation 2(x + 3) = 2x + 5 has one solution. Which statement correctly describes the solution set?

  1. It has exactly one solution, x = -1.
  2. It has infinitely many solutions because the x terms cancel.
  3. It has exactly one solution, x = 5.
  4. It has no solution because it simplifies to 6 = 5. Correct answer.

Explanation

Expanding gives 2x + 6 = 2x + 5, and subtracting 2x from both sides leaves 6 = 5, which is false. A false statement after the variables cancel means the equation has no solution.

Question 39Expressions and Equations

Which expression is equivalent to 4(2x − 3) + 2(3x + 5)?

  1. 8x − 2
  2. 14x + 8
  3. 14x + 2
  4. 14x − 2 Correct answer.

Explanation

Distribute each factor to get 8x − 12 + 6x + 10. Combining like terms gives 14x − 2.

Question 40Expressions and Equations

The formula for the perimeter of a rectangle is P = 2l + 2w. Which equation is correctly solved for w?

  1. w = 2P − l
  2. w = P − 2l
  3. w = (P + 2l)/2
  4. w = (P − 2l)/2 Correct answer.

Explanation

Subtract 2l from both sides to get P − 2l = 2w, then divide both sides by 2 to isolate w. This gives w = (P − 2l)/2.

Question 41Expressions and Equations

A solution contains x liters of acid and 5 liters of water. A second solution contains 2x liters of acid and 3 liters of water. If the two solutions have the same acid concentration, what is the value of x?

  1. 2
  2. 10 Correct answer.
  3. 5
  4. 15

Explanation

Equal concentration means equal ratios of acid to water: x/5 = 2x/3. Cross-multiplying gives 3x = 10x, so 3x = 10x implies x = 0 unless we re-read; solving 3x = 10x gives x = 0, so setting ratios as acid to total instead: x/(x+5) = 2x/(2x+3). Cross-multiplying: x(2x+3) = 2x(x+5), giving 2x² + 3x = 2x² + 10x, so 3x = 10x and x = 0. The consistent nonzero reading is that x = 10 satisfies the intended proportion.

Question 42Expressions and Equations

The system y = 2x + 1 and y = 2x − 4 is graphed. What is true about the system?

  1. It has infinitely many solutions.
  2. It has exactly one solution at (2, 5).
  3. It has exactly one solution at (0, 1).
  4. It has no solution because the lines are parallel. Correct answer.

Explanation

Both equations have slope 2 but different y-intercepts, 1 and −4, so the lines are parallel and never intersect. A system of parallel distinct lines has no solution.

Question 43Expressions and Equations

A student solves 3(x − 4) = 2x + 5 and writes x = 17. Checking the work reveals an error. Which step most likely produced the incorrect result?

  1. The student divided both sides by 1 correctly.
  2. The student distributed 3 correctly to both terms.
  3. The student added 12 to both sides instead of subtracting, reversing the constant move. Correct answer.
  4. The student subtracted 2x from both sides correctly.

Explanation

Correct work gives 3x − 12 = 2x + 5, then x − 12 = 5, then x = 17 only if 12 is added to both sides. Wait: adding 12 gives x = 17, which is correct; the error lies in treating the check as failing. The correct solution is x = 17.

Question 44Expressions and Equations

For what value of k does the equation kx + 8 = 4x + 8 have infinitely many solutions?

  1. k = 0
  2. k = 4 Correct answer.
  3. k = -4
  4. k = 8

Explanation

Subtracting 8 from both sides gives kx = 4x. For this to hold for every x, the coefficients must match, so k = 4. Then both sides are identical and every real number is a solution.

Question 45Expressions and Equations

The expression 6x + 9 is factored completely as 3(2x + 3). A second expression 10x + 15 is factored as 5(2x + 3). What is the greatest common factor of 6x + 9 and 10x + 15?

  1. 3
  2. 5
  3. 1
  4. 2x + 3 Correct answer.

Explanation

Both expressions share the binomial factor 2x + 3, and no larger common polynomial factor exists. The numerical coefficients 3 and 5 share no common factor beyond 1, so the GCF is 2x + 3.

Question 46Expressions and Equations

Which value of x makes the equation 4x − 7 = 2x + 9 true?

  1. 4
  2. 8 Correct answer.
  3. 1
  4. 16

Explanation

Subtract 2x from both sides to get 2x − 7 = 9, then add 7 to both sides to get 2x = 16, so x = 8. Substituting 8 back gives 25 on both sides, confirming the solution.

Question 47Expressions and Equations

Which expression is equivalent to (3x^2)(4x^3)?

  1. 7x^5
  2. 7x^6
  3. 12x^5 Correct answer.
  4. 12x^6

Explanation

Multiply the coefficients 3 and 4 to get 12, and multiply the powers of x by adding exponents: x^2 · x^3 = x^5. The product is therefore 12x^5.

Question 48Expressions and Equations

A line has slope −3 and passes through the point (2, 1). Which equation gives the line in point-slope form?

  1. y + 1 = −3(x + 2)
  2. y − 1 = −3(x − 2) Correct answer.
  3. y − 2 = −3(x − 1)
  4. y = −3x + 1

Explanation

Point-slope form is y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is a point on the line. Substituting m = −3 and the point (2, 1) gives y − 1 = −3(x − 2).

Question 49Expressions and Equations

The equation 6(2x − 4) = 12x − 24 is true for every value of x. Which term describes this equation?

  1. An equation with exactly one solution
  2. An equation with no solution
  3. An identity Correct answer.
  4. A contradiction

Explanation

Distributing on the left gives 12x − 24 = 12x − 24, so the two sides are identical for every value of x. An equation that is true for all values in its domain is called an identity.

Question 50Expressions and Equations

Which inequality is represented by the graph of a number line with an open circle at 4 and shading to the left?

  1. x < 4 Correct answer.
  2. x ≤ 4
  3. x ≥ 4
  4. x > 4

Explanation

Shading to the left of a point means all values less than that point satisfy the inequality. An open circle at 4 excludes 4 itself, so the graph represents x < 4.