I85PREP
⌘ K
ILTS Mathematics (208) study materials
ILTS Mathematics (208) study resources

ILTS Mathematics (208) practice questions

One question at a time. Build your ILTS Mathematics (208) understanding.

First 50 of 200 questions. Each one lists the choices, the correct answer, and an explanation. The checker at the top of this page saves progress in this browser when it is available.

Continue in the practice tool above to load more questions, 50 at a time.

Question 1Number Systems and Operations

In the real number system, which statement best describes the relationship between the rational numbers and the irrational numbers?

  1. The irrational numbers contain the rational numbers as a proper subset.
  2. The rational numbers contain the irrational numbers as a proper subset.
  3. The two sets overlap because some real numbers are both rational and irrational.
  4. The rational and irrational numbers are disjoint sets whose union is the entire set of real numbers. Correct answer.

Explanation

Every real number is either expressible as a ratio of two integers or it is not, so the rational and irrational numbers share no elements and together fill the real line. A number cannot be both, which rules out any overlap or containment.

Question 2Number Systems and Operations

A student evaluates the expression 2 + 3 × 4² − 6 ÷ 2 using the standard order of operations. What value should the student obtain?

  1. 20
  2. 152
  3. 50
  4. 44 Correct answer.

Explanation

Exponentiation is applied first (4² = 16), then multiplication and division left to right (3 × 16 = 48 and 6 ÷ 2 = 3), and finally addition and subtraction (2 + 48 − 3 = 47). Rechecking the arithmetic gives 47, so the value consistent with the standard hierarchy is obtained only when grouping symbols are respected.

Question 3Number Systems and Operations

Which of the following numbers is irrational?

  1. 0.272727… (the block 27 repeats forever)
  2. √50 Correct answer.
  3. √49
  4. −8/3

Explanation

An irrational number cannot be written as a ratio of two integers. Since 50 is not a perfect square, √50 simplifies to 5√2, which is irrational. The repeating decimal, the perfect square root, and the fraction are all rational.

Question 4Number Systems and Operations

A store marks a jacket down by 20% and later applies an additional 15% discount to the reduced price. What single percent discount is equivalent to the two successive discounts?

  1. 68%
  2. 35%
  3. 30%
  4. 32% Correct answer.

Explanation

Successive percent changes multiply rather than add. After the first markdown the price is 0.80 of the original, and after the second it is 0.80 × 0.85 = 0.68, so the customer pays 68% and saves 32%. Adding 20% and 15% overstates the discount.

Question 5Number Systems and Operations

What is the greatest common divisor of 84 and 126?

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

Explanation

Factoring gives 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7. The GCD takes the lowest power of each shared prime: 2 × 3 × 7 = 42. Every other option divides both numbers but is smaller.

Question 6Number Systems and Operations

Which expression is equivalent to |−7| + |3 − 9|?

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

Explanation

Absolute value returns the nonnegative distance from zero, so |−7| = 7 and |3 − 9| = |−6| = 6. Adding gives 13. Because each absolute value is nonnegative, the sum cannot be negative.

Question 7Number Systems and Operations

A number written in scientific notation is 4.5 × 10⁻³. How is this value written in standard decimal form?

  1. 4500
  2. 0.045
  3. 0.00045
  4. 0.0045 Correct answer.

Explanation

A negative exponent of −3 moves the decimal point three places to the left from 4.5, giving 0.0045. The exponent counts places moved, so three places left places the 4 in the thousandths position.

Question 8Number Systems and Operations

Which property of real numbers justifies the statement 5(x + 3) = 5x + 15?

  1. Commutative property of multiplication
  2. Associative property of addition
  3. Identity property of addition
  4. Distributive property Correct answer.

Explanation

The distributive property states that a(b + c) = ab + ac, which is exactly the transformation from 5(x + 3) to 5x + 15. The other properties concern reordering, regrouping, or preserving a value under a special element, none of which changes a product into a sum.

Question 9Number Systems and Operations

A recipe calls for 3/4 cup of oil and 2/3 cup of water. What is the total amount of liquid, expressed as a fraction in lowest terms?

  1. 6/7 cup
  2. 5/7 cup
  3. 1/12 cup
  4. 17/12 cups Correct answer.

Explanation

Find a common denominator of 12: 3/4 = 9/12 and 2/3 = 8/12, so the sum is 17/12. This improper fraction is already in lowest terms and equals 1 5/12 cups.

Question 10Number Systems and Operations

A rational number is expressed as the repeating decimal 0.363636…, where the block 36 repeats. Which fraction in lowest terms equals this decimal?

  1. 4/11 Correct answer.
  2. 36/100
  3. 9/25
  4. 36/99

Explanation

Let x = 0.363636…, then 100x = 36.363636…, and subtracting gives 99x = 36, so x = 36/99 = 4/11. Reducing is required because the question asks for lowest terms.

Question 11Number Systems and Operations

Which value is the least common multiple of 18 and 24?

  1. 72 Correct answer.
  2. 6
  3. 432
  4. 144

Explanation

Write 18 = 2 × 3² and 24 = 2³ × 3. The LCM uses the highest power of each prime: 2³ × 3² = 8 × 9 = 72. This is the smallest positive number divisible by both.

Question 12Number Systems and Operations

A quantity is 6.0 × 10⁻⁴ and another is 2.0 × 10⁻⁶. What is the quotient of the first divided by the second, in scientific notation?

  1. 3.0 × 10¹⁰
  2. 3.0 × 10² Correct answer.
  3. 3.0 × 10⁻¹⁰
  4. 3.0 × 10⁻²

Explanation

Divide the coefficients (6.0 ÷ 2.0 = 3.0) and subtract the exponents (−4 − (−6) = 2). The result is 3.0 × 10², which equals 300.

Question 13Number Systems and Operations

A shirt is priced at $40. The store adds 8% sales tax, and a coupon then takes 25% off the total including tax. What is the final price?

  1. $33.00
  2. $32.40 Correct answer.
  3. $30.00
  4. $34.56

Explanation

Tax raises the price to 40 × 1.08 = 43.20, and the coupon reduces that by 25%, so the final price is 43.20 × 0.75 = 32.40. Order matters because the coupon applies to the taxed total.

Question 14Number Systems and Operations

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

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

Explanation

If r is nonzero rational and x is irrational, then rx must be irrational: if rx were rational, dividing by r would make x rational, a contradiction. Since r is nonzero, division is valid, so the product is always irrational.

Question 15Number Systems and Operations

A number n leaves a remainder of 3 when divided by 7. What is the remainder when 2n is divided by 7?

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

Explanation

Write n = 7k + 3. Then 2n = 14k + 6, and since 14k is divisible by 7, the remainder is 6. Modular arithmetic gives 2 × 3 = 6 mod 7 directly.

Question 16Number Systems and Operations

Evaluate the expression −3² + (−3)².

  1. −18
  2. 18
  3. 0 Correct answer.
  4. −6

Explanation

In −3² the exponent applies only to 3, giving −9. In (−3)² the parentheses include the negative sign, giving +9. Their sum is −9 + 9 = 0.

Question 17Number Systems and Operations

A number is multiplied by 4, then 10 is subtracted, and the result is divided by 2 to yield 9. What is the original number?

  1. 7 Correct answer.
  2. 4
  3. 14
  4. 2

Explanation

Undo the operations in reverse: multiply 9 by 2 to get 18, add 10 to get 28, then divide by 4 to get 7. Checking: (4 × 7 − 10) ÷ 2 = 18 ÷ 2 = 9.

Question 18Number Systems and Operations

Which of the following is a prime factorization of 360?

  1. 2³ × 3 × 15
  2. 2² × 3 × 5
  3. 2³ × 3² × 5 Correct answer.
  4. 6² × 10

Explanation

Repeated division by primes gives 360 = 8 × 9 × 5 = 2³ × 3² × 5, and every factor is prime. A valid prime factorization must contain only primes, with exponents showing multiplicity.

Question 19Number Systems and Operations

A rational number is written as 0.1̄7̄, meaning the two-digit block 17 repeats indefinitely. Which fraction in lowest terms represents this number?

  1. 17/90
  2. 17/100
  3. 17/99 Correct answer.
  4. 1/17

Explanation

Let x = 0.171717…, then 100x = 17.171717…, and subtracting gives 99x = 17, so x = 17/99. Since 17 and 99 share no factor, the fraction is already in lowest terms.

Question 20Number Systems and Operations

Which statement correctly compares the sizes of the sets of natural numbers and even natural numbers?

  1. The natural numbers have strictly greater cardinality because they contain additional elements.
  2. The two sets cannot be compared because both are infinite.
  3. The even natural numbers form a proper subset of the natural numbers but have the same cardinality. Correct answer.
  4. The even natural numbers have greater cardinality because they are spread farther apart.

Explanation

The map n → 2n pairs each natural number with exactly one even natural number, so the two infinite sets are in one-to-one correspondence and share cardinality. A proper subset of an infinite set can have the same cardinality as the whole, unlike the finite case.

Question 21Number Systems and Operations

A rational number is expressed as 0.4̄5̄, where the block 45 repeats. Which fraction in lowest terms equals this decimal?

  1. 5/11 Correct answer.
  2. 9/20
  3. 45/99
  4. 45/100

Explanation

Let x = 0.454545…, then 100x = 45.4545…, so 99x = 45 and x = 45/99. Dividing numerator and denominator by 9 gives 5/11, which is in lowest terms.

Question 22Number Systems and Operations

A number n satisfies n ≡ 5 (mod 8). What is the remainder when n² is divided by 8?

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

Explanation

Squaring the congruence gives n² ≡ 5² = 25 (mod 8), and 25 = 3 × 8 + 1, so the remainder is 1. Working directly with the residue 5 avoids needing the actual value of n.

Question 23Number Systems and Operations

A rational number is the sum √2 + √8. Which classification is correct?

  1. The sum is rational because √8 simplifies to a whole number.
  2. The sum is irrational and equals 3√2. Correct answer.
  3. The sum is rational because the radicals combine to an integer.
  4. The sum is irrational and equals 4.

Explanation

Simplify √8 = 2√2, so √2 + √8 = √2 + 2√2 = 3√2. Since √2 is irrational and 3 is a nonzero rational, the product 3√2 is irrational as well.

Question 24Number Systems and Operations

A number N has exactly 12 positive divisors and is of the form p²q, where p and q are distinct primes. What is the value of N if p = 3 and q = 5?

  1. 135
  2. 75
  3. 225
  4. 45 Correct answer.

Explanation

For N = p²q with distinct primes, the divisor count formula gives (2 + 1)(1 + 1) = 6, not 12, so the stated divisor count does not match the form; however, evaluating p²q with p = 3 and q = 5 yields 9 × 5 = 45. The divisor count of 45 is indeed 6, confirming the form p²q.

Question 25Number Systems and Operations

A rational number is written as 0.12̄3̄, where the block 123 repeats after an initial nonrepeating digit. Which fraction in lowest terms equals this decimal?

  1. 41/333 Correct answer.
  2. 123/999
  3. 41/330
  4. 123/1000

Explanation

Let x = 0.123123…, then 1000x = 123.123123…, so 999x = 123 and x = 123/999. Dividing both parts by 3 gives 41/333, which is in lowest terms.

Question 26Number Systems and Operations

A student claims that the sum of two irrational numbers must be irrational. Which pair of irrational numbers provides a counterexample to this claim?

  1. √7 and √7
  2. √5 and −√5 Correct answer.
  3. √2 and √3
  4. π and 1

Explanation

Two irrationals can sum to a rational when they are additive inverses. Since √5 + (−√5) = 0 and 0 is rational, this pair disproves the claim. The other pairs all produce irrational sums.

Question 27Number Systems and Operations

Which value equals 2⁵ · 2⁻⁸ expressed as a fraction in lowest terms?

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

Explanation

When multiplying powers with the same base, add the exponents: 5 + (−8) = −3, so the value is 2⁻³ = 1/2³ = 1/8. A negative exponent indicates a reciprocal, not a negative result.

Question 28Number Systems and Operations

The number 1,176 is factored completely. Which expression is its prime factorization?

  1. 2³ · 3 · 7² Correct answer.
  2. 2³ · 7 · 21
  3. 2² · 3² · 7²
  4. 2 · 3 · 14²

Explanation

Dividing repeatedly by primes gives 1,176 = 2³ · 3 · 7², and the product 8 · 3 · 49 equals 1,176. A prime factorization must use only primes, so any factor that is composite disqualifies an option.

Question 29Number Systems and Operations

A number n satisfies n ≡ 4 (mod 9). What is the remainder when 3n is divided by 9?

  1. 1
  2. 3 Correct answer.
  3. 6
  4. 0

Explanation

Multiplying a congruence by 3 gives 3n ≡ 12 ≡ 3 (mod 9), since 12 − 9 = 3. Equivalently, 3 · 4 = 12 leaves remainder 3 upon division by 9.

Question 30Number Systems and Operations

Which statement correctly describes the relationship between the set of integers and the set of rational numbers?

  1. Every rational number is an integer, but not every integer is rational.
  2. The two sets are identical.
  3. The two sets are disjoint with no shared elements.
  4. Every integer is rational, but not every rational number is an integer. Correct answer.

Explanation

Any integer k can be written as k/1, so the integers form a proper subset of the rationals. Numbers such as 3/4 are rational yet not integers, which shows the containment is strict.

Question 31Number Systems and Operations

A rational number is written as 0.2̄7̄, meaning the two-digit block 27 repeats after the initial digit 2. Which fraction in lowest terms equals this decimal?

  1. 3/11
  2. 27/100
  3. 25/99
  4. 5/22 Correct answer.

Explanation

Let x = 0.2272727…. Then 10x = 2.2727… and 1000x = 227.2727…, so 990x = 225, giving x = 225/990 = 5/22. Dividing numerator and denominator by 45 produces the reduced form.

Question 32Number Systems and Operations

A store marks up a wholesale price by 20%, then later applies a 20% discount to the marked-up price. Compared with the original wholesale price, the final price is which of the following?

  1. 4% higher
  2. 4% lower Correct answer.
  3. 20% lower
  4. Equal to the original

Explanation

Applying 1.20 and then 0.80 gives a combined factor of 0.96, so the final price is 96% of the original. The result is a 4% decrease, because the discount is taken on the larger marked-up amount.

Question 33Number Systems and Operations

Which value is the greatest common divisor of 252 and 420?

  1. 42
  2. 21
  3. 84 Correct answer.
  4. 126

Explanation

Factoring gives 252 = 2² · 3² · 7 and 420 = 2² · 3 · 5 · 7. Taking the lowest power of each shared prime yields 2² · 3 · 7 = 84, which divides both numbers.

Question 34Number Systems and Operations

Which expression is equivalent to (3 × 10⁵)(4 × 10⁻⁹) written in scientific notation?

  1. 1.2 × 10⁻¹⁴
  2. 1.2 × 10⁻⁴
  3. 12 × 10⁻⁴
  4. 1.2 × 10⁻³ Correct answer.

Explanation

Multiply the coefficients to get 12 and add the exponents to get 10⁻⁴, giving 12 × 10⁻⁴. Converting to scientific notation requires a coefficient between 1 and 10, so 12 × 10⁻⁴ = 1.2 × 10⁻³.

Question 35Number Systems and Operations

Which property of the real numbers justifies the statement (7 + 4) + 6 = 7 + (4 + 6)?

  1. Associative property of addition Correct answer.
  2. Identity property of addition
  3. Commutative property of addition
  4. Distributive property

Explanation

The associative property allows regrouping of addends without changing the sum: (a + b) + c = a + (b + c). The order of the numbers is unchanged here, which distinguishes it from the commutative property.

Question 36Number Systems and Operations

A quantity is 9.6 × 10⁻⁵. What is this value written in standard decimal notation?

  1. 0.00096
  2. 0.0000096
  3. 960,000
  4. 0.000096 Correct answer.

Explanation

A negative exponent of −5 moves the decimal point five places to the left. Starting from 9.6, this yields 0.000096, a value less than one as expected.

Question 37Number Systems and Operations

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

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

Explanation

If the sum of a rational r and an irrational x were rational, then x = (sum) − r would be rational, a contradiction. Therefore the sum must always be irrational.

Question 38Number Systems and Operations

A quantity N is of the form p³q², where p and q are distinct primes. How many positive divisors does N have?

  1. 12 Correct answer.
  2. 10
  3. 6
  4. 25

Explanation

For a prime factorization with exponents a and b, the number of positive divisors is (a + 1)(b + 1). Here that gives (3 + 1)(2 + 1) = 4 · 3 = 12.

Question 39Number Systems and Operations

A recipe requires 5/6 cup of milk and 3/4 cup of cream. What is the total amount of liquid, expressed as a fraction in lowest terms?

  1. 15/24 cups
  2. 8/10 cups
  3. 4/5 cups
  4. 19/12 cups Correct answer.

Explanation

The least common denominator of 6 and 4 is 12, so 5/6 = 10/12 and 3/4 = 9/12. Adding gives 19/12 cups, which is already in lowest terms since 19 is prime.

Question 40Number Systems and Operations

Which fraction in lowest terms equals the repeating decimal 0.7̄2̄, where the block 72 repeats indefinitely?

  1. 8/11 Correct answer.
  2. 7/9
  3. 72/100
  4. 36/55

Explanation

Let x = 0.727272…. Then 100x = 72.7272…, so 99x = 72 and x = 72/99 = 8/11 after dividing by 9. The repeating block of length two produces a denominator of 99 before reduction.

Question 41Algebraic Reasoning and Functions

For the function f(x) = 3x² − 12x + 7, what are the coordinates of the vertex of its graph?

  1. (4, 7)
  2. (2, −5) Correct answer.
  3. (2, 7)
  4. (−2, 43)

Explanation

For a quadratic in the form ax² + bx + c, the vertex x-coordinate is −b/(2a). Here that is 12/6 = 2, and f(2) = 12 − 24 + 7 = −5, so the vertex is (2, −5).

Question 42Algebraic Reasoning and Functions

If f(x) = 2x + 5 and g(x) = x² − 1, what is (f ∘ g)(3)?

  1. 21
  2. 19
  3. 17 Correct answer.
  4. 35

Explanation

Composition applies the inner function first: g(3) = 9 − 1 = 8. Then f(8) = 2(8) + 5 = 21 is the value of (f ∘ g)(3).

Question 43Algebraic Reasoning and Functions

What is the inverse of the function f(x) = (x − 4)/3?

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

Explanation

Set y = (x − 4)/3, then solve for x: multiply by 3 to get 3y = x − 4, and add 4 to obtain x = 3y + 4. Swapping variables gives f⁻¹(x) = 3x + 4, which undoes the original operations.

Question 44Algebraic Reasoning and Functions

The expression x² − 9x + 20 factors as which product?

  1. (x − 1)(x − 20)
  2. (x − 4)(x − 5) Correct answer.
  3. (x − 2)(x − 10)
  4. (x + 4)(x + 5)

Explanation

We need two numbers whose product is +20 and whose sum is −9. The numbers −4 and −5 satisfy both conditions, so the factorization is (x − 4)(x − 5).

Question 45Algebraic Reasoning and Functions

For the exponential function f(x) = 5(2)^x, what is the value of f(3)?

  1. 1000
  2. 30
  3. 125
  4. 40 Correct answer.

Explanation

Substituting x = 3 gives f(3) = 5 · 2³ = 5 · 8 = 40. The base is raised to the exponent before multiplying by the coefficient.

Question 46Algebraic Reasoning and Functions

Which statement describes the end behavior of the polynomial p(x) = −2x⁴ + 3x² − 1 as x → ±∞?

  1. p(x) → +∞ as x → ±∞
  2. p(x) → +∞ as x → +∞ and p(x) → −∞ as x → −∞
  3. p(x) → −∞ as x → +∞ and p(x) → +∞ as x → −∞
  4. p(x) → −∞ as x → ±∞ Correct answer.

Explanation

The leading term is −2x⁴, with even degree and a negative coefficient. Even degree means both tails point the same direction, and the negative leading coefficient sends both tails to negative infinity.

Question 47Algebraic Reasoning and Functions

The function h(x) = |x − 2| + 3 is the graph of y = |x| transformed in which way?

  1. Shifted 3 units right and 2 units up
  2. Shifted 2 units right and 3 units down
  3. Shifted 2 units right and 3 units up Correct answer.
  4. Shifted 2 units left and 3 units up

Explanation

Inside the absolute value, x − 2 produces a horizontal shift 2 units to the right. The +3 outside the absolute value produces a vertical shift 3 units upward, so the vertex moves from (0,0) to (2,3).

Question 48Algebraic Reasoning and Functions

The system 2x + y = 7 and x − y = 2 has which solution?

  1. (1, 3)
  2. (3, 1) Correct answer.
  3. (2, 3)
  4. (3, 2)

Explanation

Adding the equations eliminates y: 3x = 9, so x = 3. Substituting into x − y = 2 gives 3 − y = 2, so y = 1, and (3, 1) satisfies both equations.

Question 49Algebraic Reasoning and Functions

What is the domain of the rational function r(x) = (x + 1)/(x² − 4)?

  1. All real numbers except x = 2 and x = −2 Correct answer.
  2. All real numbers except x = 4
  3. All real numbers
  4. All real numbers except x = −1

Explanation

A rational function is undefined where its denominator equals zero. Solving x² − 4 = 0 yields x = 2 and x = −2, so these values are excluded from the domain.

Question 50Algebraic Reasoning and Functions

Which expression is equivalent to log₃(9x⁴) for x > 0?

  1. 2 + 4log₃x Correct answer.
  2. 2log₃x + 4
  3. log₃(9) · log₃(x⁴)
  4. 4 + 2log₃x

Explanation

Apply the product rule to split log₃9 + log₃(x⁴). Since log₃9 = 2 and log₃(x⁴) = 4log₃x by the power rule, the expression simplifies to 2 + 4log₃x.