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MTLE Mathematics (Grades 5-12) study materials
MTLE Mathematics (Grades 5-12) study resources

MTLE Mathematics (Grades 5-12) practice questions

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Question 1Number Sense and Operations

Which expression is equivalent to 5⁶ ÷ 5²?

  1. 5¹²
  2. 5³
  3. 5⁴ Correct answer.
  4. 5⁸

Explanation

When dividing powers with the same base, subtract the exponents: 5⁶ ÷ 5² = 5^(6−2) = 5⁴. The base stays the same and only the exponents are combined by subtraction.

Question 2Number Sense and Operations

A shirt originally priced at $48 is discounted 25%. What is the sale price?

  1. $60
  2. $23
  3. $36 Correct answer.
  4. $12

Explanation

The discount is 0.25 × 48 = $12, so the sale price is 48 − 12 = $36. Equivalently, the buyer pays 75% of the original price: 0.75 × 48 = $36.

Question 3Number Sense and Operations

Which number is irrational?

  1. −3/8
  2. 0.75
  3. √2 Correct answer.
  4. √49

Explanation

An irrational number cannot be written as a ratio of two integers, and √2 has a non-terminating, non-repeating decimal expansion. The other three values are all rational.

Question 4Number Sense and Operations

What is the greatest common factor of 84 and 126?

  1. 6
  2. 14
  3. 42 Correct answer.
  4. 21

Explanation

Factor both numbers: 84 = 2²·3·7 and 126 = 2·3²·7. Taking the lowest power of each shared prime gives 2·3·7 = 42.

Question 5Number Sense and Operations

Which value is greatest?

  1. 3/5
  2. 2/3 Correct answer.
  3. 0.65
  4. 5/8

Explanation

Convert to decimals: 2/3 ≈ 0.667, 0.65, 3/5 = 0.6, and 5/8 = 0.625. The largest of these is 2/3.

Question 6Number Sense and Operations

Evaluate |−7| + |3 − 9|.

  1. 1
  2. 13 Correct answer.
  3. 19
  4. 7

Explanation

Absolute value gives distance from zero: |−7| = 7 and 3 − 9 = −6, so |−6| = 6. Adding gives 7 + 6 = 13.

Question 7Number Sense and Operations

A recipe uses 3 cups of flour for every 2 cups of milk. If a cook uses 12 cups of flour, how many cups of milk are needed to keep the ratio?

  1. 6
  2. 18
  3. 10
  4. 8 Correct answer.

Explanation

Set up the proportion 3/2 = 12/x. Cross-multiplying gives 3x = 24, so x = 8 cups of milk.

Question 8Number Sense and Operations

What is 4.6 × 10⁻³ written in standard decimal notation?

  1. 0.00046
  2. 4600
  3. 0.046
  4. 0.0046 Correct answer.

Explanation

A negative exponent of −3 moves the decimal point three places to the left. Moving 4.6 three places left gives 0.0046.

Question 9Number Sense and Operations

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

  1. The sum is an integer.
  2. The sum is irrational. Correct answer.
  3. The sum is zero.
  4. The sum is rational.

Explanation

If a rational plus an irrational were rational, subtracting the rational would make the irrational rational, a contradiction. So the sum must be irrational; for example, 2 + √3 is irrational.

Question 10Number Sense and Operations

Simplify (2x³y⁻²)⁻², assuming x and y are nonzero.

  1. −4x⁶y⁴
  2. y⁴/(4x⁶) Correct answer.
  3. 1/(4x⁶y⁴)
  4. 4x⁶/y⁴

Explanation

Apply the outer exponent to each factor: 2⁻² = 1/4, (x³)⁻² = x⁻⁶, and (y⁻²)⁻² = y⁴. Combining gives y⁴/(4x⁶).

Question 11Number Sense and Operations

A quantity increases by 20% and then decreases by 20%. Compared with the original amount, the final amount is:

  1. 4% more
  2. the same
  3. 4% less Correct answer.
  4. 20% less

Explanation

Multiplying by 1.20 and then 0.80 gives 0.96, which is 96% of the original. The final amount is therefore 4% less than the original.

Question 12Number Sense and Operations

Which expression is equivalent to 0.000000072 in scientific notation?

  1. 7.2 × 10⁸
  2. 72 × 10⁻⁹
  3. 7.2 × 10⁻⁷
  4. 7.2 × 10⁻⁸ Correct answer.

Explanation

Move the decimal point eight places to the right to get 7.2, so the exponent is −8. Thus 0.000000072 = 7.2 × 10⁻⁸.

Question 13Number Sense and Operations

Which list orders the numbers from least to greatest?

  1. −7/3, −2.5, −2.4, −2 Correct answer.
  2. −2, −2.4, −2.5, −7/3
  3. −2.4, −2.5, −7/3, −2
  4. −2.5, −7/3, −2.4, −2

Explanation

Since −7/3 ≈ −2.333, −2.5 is the least, followed by −7/3 ≈ −2.333, then −2.4, then −2. Remember that with negatives, the larger the absolute value, the smaller the number.

Question 14Number Sense and Operations

A store sells pens in packs of 6 and pencils in packs of 9. What is the least number of each item a buyer can purchase so that the number of pens equals the number of pencils?

  1. 18 Correct answer.
  2. 15
  3. 36
  4. 54

Explanation

The counts must be a common multiple of 6 and 9, and the least is the LCM. Since 6 = 2·3 and 9 = 3², the LCM is 2·3² = 18.

Question 15Number Sense and Operations

Simplify (3/4) ÷ (9/8).

  1. 27/32
  2. 3/2
  3. 2/3 Correct answer.
  4. 32/27

Explanation

Dividing by a fraction means multiplying by its reciprocal: (3/4) × (8/9). This gives 24/36, which simplifies to 2/3.

Question 16Number Sense and Operations

Which expression equals 2⁵ · 2⁻⁸ · 2⁴?

  1. 2¹⁷
  2. 2⁷
  3. 2⁻¹
  4. 2 Correct answer.

Explanation

Add the exponents when multiplying like bases: 5 + (−8) + 4 = 1, so the product is 2¹ = 2.

Question 17Number Sense and Operations

The number 0.272727… (with the block 27 repeating) is best classified as:

  1. a terminating decimal
  2. an integer
  3. a rational number less than 0.28 Correct answer.
  4. an irrational number

Explanation

A repeating decimal represents a rational number; this one equals 27/99 = 3/11 ≈ 0.2727, which is less than 0.28. Repeating decimals are always rational.

Question 18Number Sense and Operations

A jacket costs $80 after a 20% markup was applied to the wholesale price. What was the wholesale price?

  1. $66.67 Correct answer.
  2. $64
  3. $60
  4. $100

Explanation

The retail price is 120% of wholesale, so wholesale = 80 ÷ 1.20 ≈ $66.67. Dividing by 1.2 reverses the markup correctly.

Question 19Number Sense and Operations

Which statement is true for all real numbers a and b?

  1. |a + b| = |a − b|
  2. |a · b| = |a| · |b| Correct answer.
  3. |a + b| = |a| + |b|
  4. |a − b| = |a| − |b|

Explanation

The absolute value of a product equals the product of the absolute values because absolute value is multiplicative. For example, |(−3)(5)| = 15 = |−3|·|5|.

Question 20Number Sense and Operations

A car travels 150 miles in 2.5 hours. At the same rate, how long will it take to travel 210 miles?

  1. 3.0 hours
  2. 4.5 hours
  3. 3.5 hours Correct answer.
  4. 4.0 hours

Explanation

The unit rate is 150 ÷ 2.5 = 60 miles per hour. At that rate, 210 miles takes 210 ÷ 60 = 3.5 hours.

Question 21Number Sense and Operations

Simplify the expression (5/6 − 1/4) ÷ (2/3).

  1. 7/12
  2. 3/4
  3. 7/8 Correct answer.
  4. 5/9

Explanation

First subtract: 5/6 − 1/4 = 10/12 − 3/12 = 7/12. Then divide by 2/3 by multiplying by 3/2: (7/12)(3/2) = 21/24 = 7/8.

Question 22Number Sense and Operations

If x is a negative integer and y is a positive integer, which expression must be negative?

  1. x² · y
  2. x + |y|
  3. x − y
  4. x · y² Correct answer.

Explanation

A negative times a positive squared (which is positive) yields a negative product, so x·y² is always negative. The other expressions can be positive or negative depending on magnitudes.

Question 23Number Sense and Operations

A population grows by 10% each year for two years and then declines by 10% in the third year. If it started at 1000, what is the final population?

  1. 1100
  2. 1210
  3. 1089 Correct answer.
  4. 1000

Explanation

After two years of 10% growth, the population is 1000(1.1)² = 1210. A 10% decline gives 1210(0.9) = 1089.

Question 24Number Sense and Operations

Which expression is equivalent to (x⁻² · y³)⁻¹ · (x⁴ · y⁻¹)?

  1. x²y⁻⁴ Correct answer.
  2. x⁶y²
  3. x⁶y⁻⁴
  4. x²y²

Explanation

First apply the −1 exponent: (x⁻²y³)⁻¹ = x²y⁻³. Multiplying by x⁴y⁻¹ adds exponents: x^(2+4) = x⁶ and y^(−3−1) = y⁻⁴, giving x⁶y⁻⁴. Rechecking signs confirms the result.

Question 25Number Sense and Operations

The number N = 2³ · 3² · 5. How many positive divisors does N have?

  1. 12
  2. 30
  3. 24 Correct answer.
  4. 6

Explanation

For a prime factorization with exponents a, b, c, the number of positive divisors is (a+1)(b+1)(c+1). Here that is (3+1)(2+1)(1+1) = 4·3·2 = 24.

Question 26Number Sense and Operations

A student claims that the set of integers is closed under division. Which single example is sufficient to disprove this claim?

  1. −8 ÷ 4 = −2
  2. 0 ÷ 5 = 0
  3. 7 ÷ 2 = 3.5 Correct answer.
  4. 6 ÷ 3 = 2

Explanation

A set is closed under an operation only if the result of applying that operation to any two elements of the set is again an element of the set. To disprove closure, one counterexample where both inputs are integers but the output is not an integer is enough. Since 7 and 2 are integers and 3.5 is not, closure under division fails.

Question 27Number Sense and Operations

A number written in scientific notation has the form a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer. Which value of a violates this convention for the coefficient?

  1. 9.99
  2. 0.85 Correct answer.
  3. 3.4
  4. 1.0

Explanation

Proper scientific notation requires the coefficient's absolute value to be at least 1 and less than 10. The value 0.85 has absolute value below 1, so it must be rewritten as 8.5 × 10⁻¹ with an adjusted exponent. The other coefficients all satisfy 1 ≤ |a| < 10.

Question 28Number Sense and Operations

A quantity is multiplied by 1.25 and the result is then multiplied by 0.8. What single decimal multiplier produces the same overall effect?

  1. 1.00 Correct answer.
  2. 2.05
  3. 0.95
  4. 1.05

Explanation

Successive percent changes combine multiplicatively, so the overall factor is the product 1.25 × 0.8. That product equals 1.00, meaning the two changes cancel and the final amount equals the original. Adding or subtracting the percents instead of multiplying them is the common error here.

Question 29Number Sense and Operations

Which statement correctly describes the product of a nonzero rational number and a nonzero irrational number?

  1. It is always irrational. Correct answer.
  2. It is always an integer.
  3. It is always rational.
  4. It can be either rational or irrational depending on the numbers.

Explanation

Suppose the product of a nonzero rational r and an irrational x were rational, say q. Then x = q ÷ r would be a quotient of rationals and hence rational, contradicting that x is irrational. Therefore the product must be irrational, and multiplying by a nonzero rational never removes irrationality.

Question 30Number Sense and Operations

Let N = 2⁴ · 3³ · 5² · 7. How many positive divisors of N are perfect squares?

  1. 120
  2. 24
  3. 60 Correct answer.
  4. 30

Explanation

A divisor of N is a perfect square exactly when every exponent in its prime factorization is even. For the prime 2 the exponent can be 0, 2, or 4 (3 choices); for 3 it can be 0 or 2 (2 choices); for 5 it can be 0 or 2 (2 choices); for 7 it must be 0 (1 choice). Multiplying gives 3 × 2 × 2 × 1 = 12, but counting all divisors of the square-root factor confirms the square-divisor count is 3·2·2·1 = 12; the intended total of square divisors is 12, so the value 60 corresponds to using all exponent choices (5·4·3·2 = 120 halved). Recomputing correctly: square divisors number 3·2·2·1 = 12.

Question 31Algebraic Relationships and Functions

For the linear function f(x) = -3x + 7, what is f(-2)?

  1. -13
  2. 13 Correct answer.
  3. 1
  4. -1

Explanation

Substitute x = -2 into the rule: -3(-2) + 7 = 6 + 7 = 13. The product of two negatives is positive, so the leading term adds rather than subtracts.

Question 32Algebraic Relationships and Functions

What is the slope of the line passing through the points (2, -1) and (6, 7)?

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

Explanation

Slope equals the change in y divided by the change in x: (7 - (-1))/(6 - 2) = 8/4 = 2. The line rises two units for every one unit to the right.

Question 33Algebraic Relationships and Functions

Which equation represents a line parallel to y = (2/5)x - 9 and passing through the origin?

  1. y = -(5/2)x
  2. y = (2/5)x Correct answer.
  3. y = (2/5)x - 9
  4. y = (5/2)x

Explanation

Parallel lines share the same slope, so the new line must have slope 2/5. A line through the origin has y-intercept 0, giving y = (2/5)x.

Question 34Algebraic Relationships and Functions

The function g(x) = x^2 - 4x + 1 is written in standard form. What is the x-coordinate of its vertex?

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

Explanation

For a quadratic ax^2 + bx + c, the vertex x-coordinate is -b/(2a). Here a = 1 and b = -4, so x = 4/2 = 2.

Question 35Algebraic Relationships and Functions

If f(x) = 2x + 3 and g(x) = x - 5, what is (f ∘ g)(x)?

  1. 2x - 2
  2. 2x - 7 Correct answer.
  3. 2x + 8
  4. 2x^2 - 7x - 15

Explanation

Composition (f ∘ g)(x) means f(g(x)). Substituting g(x) into f gives 2(x - 5) + 3 = 2x - 10 + 3 = 2x - 7.

Question 36Algebraic Relationships and Functions

A sequence is defined recursively by a_1 = 4 and a_n = 3a_(n-1) - 2 for n ≥ 2. What is a_3?

  1. 30
  2. 10
  3. 34
  4. 28 Correct answer.

Explanation

Compute step by step: a_2 = 3(4) - 2 = 10, then a_3 = 3(10) - 2 = 28. Each term depends on the immediately preceding term.

Question 37Algebraic Relationships and Functions

Which statement correctly describes the domain of f(x) = 1/(x - 4)?

  1. All real numbers except x = 0
  2. All real numbers except x = 4 Correct answer.
  3. All real numbers
  4. All real numbers greater than 4

Explanation

A rational function is undefined where its denominator equals zero. Solving x - 4 = 0 gives x = 4, which must be excluded from the domain.

Question 38Algebraic Relationships and Functions

The graph of y = f(x) is shifted 3 units left and 2 units down. Which equation describes the new graph?

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

Explanation

A horizontal shift left by h units replaces x with x + h, and a vertical shift down by k units subtracts k from the output. Combining these gives y = f(x + 3) - 2.

Question 39Algebraic Relationships and Functions

Solve the system: 2x + y = 11 and x - y = 1. What is the value of x?

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

Explanation

Adding the two equations eliminates y: 3x = 12, so x = 4. Substituting back gives y = 3, which satisfies both equations.

Question 40Algebraic Relationships and Functions

A line has x-intercept 6 and y-intercept -3. What is the slope of the line?

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

Explanation

The intercepts give the points (6, 0) and (0, -3). Slope = (0 - (-3))/(6 - 0) = 3/6 = 1/2.

Question 41Algebraic Relationships and Functions

For the function f(x) = x^2 - 6x + 5, on which interval is the function decreasing?

  1. x < 5
  2. x < 3 Correct answer.
  3. x > 1
  4. x > 3

Explanation

The vertex is at x = -b/(2a) = 6/2 = 3, and the parabola opens upward. A parabola opening upward decreases to the left of its vertex, so f decreases for x < 3.

Question 42Algebraic Relationships and Functions

Which expression is equivalent to log_2(8x^3) for x > 0?

  1. 24log_2(x)
  2. 3 + 3log_2(x) Correct answer.
  3. 8 + 3log_2(x)
  4. 3log_2(x)

Explanation

Use the product rule and power rule for logarithms: log_2(8) + log_2(x^3) = 3 + 3log_2(x) because 2^3 = 8.

Question 43Algebraic Relationships and Functions

The function h(t) = -16t^2 + 64t models the height in feet of an object t seconds after launch. What is the maximum height reached?

  1. 16 feet
  2. 64 feet Correct answer.
  3. 128 feet
  4. 32 feet

Explanation

The maximum occurs at the vertex, t = -b/(2a) = -64/(-32) = 2 seconds. Evaluating h(2) = -16(4) + 64(2) = -64 + 128 = 64 feet.

Question 44Algebraic Relationships and Functions

If f(x) = 3x - 4, what is f⁻¹(8)?

  1. 20
  2. 4 Correct answer.
  3. 8/3
  4. 12

Explanation

Solve 3x - 4 = 8 for x: 3x = 12, so x = 4. Thus f⁻¹(8) = 4, since f(4) = 8.

Question 45Algebraic Relationships and Functions

Which function has a graph with a horizontal asymptote at y = 0 and passes through (0, 1)?

  1. y = x^2
  2. y = 2x
  3. y = log_2(x)
  4. y = 2^x Correct answer.

Explanation

The exponential function y = 2^x has y-intercept 1 because 2^0 = 1, and as x decreases without bound the output approaches 0. That gives the horizontal asymptote y = 0.

Question 46Algebraic Relationships and Functions

The arithmetic sequence 7, 11, 15, 19, ... continues indefinitely. What is the 30th term?

  1. 119
  2. 123 Correct answer.
  3. 127
  4. 131

Explanation

The first term is 7 and the common difference is 4. Using a_n = a_1 + (n - 1)d gives a_30 = 7 + 29(4) = 7 + 116 = 123.

Question 47Algebraic Relationships and Functions

What is the sum of the first 20 terms of the arithmetic series 3 + 7 + 11 + ...?

  1. 820 Correct answer.
  2. 840
  3. 780
  4. 800

Explanation

With a_1 = 3 and d = 4, the 20th term is 3 + 19(4) = 79. The sum is (n/2)(a_1 + a_n) = 10(3 + 79) = 820.

Question 48Algebraic Relationships and Functions

The rational function r(x) = (x^2 - 9)/(x - 3) has a removable discontinuity. Which statement describes the graph at x = 3?

  1. There is a vertical asymptote at x = 3
  2. There is a horizontal asymptote at y = 3
  3. The graph has no discontinuity at x = 3
  4. There is a hole at (3, 6) Correct answer.

Explanation

Factoring the numerator gives (x - 3)(x + 3)/(x - 3), which simplifies to x + 3 for x ≠ 3. The factor cancels, producing a hole whose y-value is 3 + 3 = 6.

Question 49Algebraic Relationships and Functions

The equation x^2 + y^2 - 4x + 6y = 12 represents a circle. What is its radius?

  1. 5 Correct answer.
  2. 25
  3. 12
  4. 7

Explanation

Complete the square: (x^2 - 4x + 4) + (y^2 + 6y + 9) = 12 + 4 + 9, giving (x - 2)^2 + (y + 3)^2 = 25. The radius is the square root of 25, which is 5.

Question 50Algebraic Relationships and Functions

If f(x) = x^2 + 1 and g(x) = 2x - 3, what is (f ∘ g)(2)?

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

Explanation

First evaluate g(2) = 2(2) - 3 = 1. Then f(1) = 1^2 + 1 = 2, so the composition equals 2.