Number Types & Properties

2. Which of the following inequalities orders the numbers 0.2, 0.03, and from least to greatest?

2015 June(73C) Math Question 2

2. The graph below shows the number of students who were present on Thursday from each of the 5 groups in Ms. Meagan's class. What is the probability that a student selected at random from the class on Thursday is in Group 4 ?

2017 December(A10) Math Question 2

2. What is the sum of the complex numbers 3 – 4i and 5 + 3i?

2018 April(A09) Math Question 2

3. Which of the following inequalities orders the numbers 0.1, 0.04, and 1/6 from least to greatest?

2014 December(72G) Math Question 3

3. Which of the following integers is NOT a factor of 132?

2013 June(71C) Math Question 3

Which of the following matrices is equal to 5[1 2; 3 4]?

2024 June(G20) Math Question 3

4. Which of the following numbers has the greatest value?

2016 April(73E) Math Question 4

2011 April(67F) Math Question 5

6. The square root of a certain number is approximately 9.2371. The certain number is between what 2 integers?

2015 June(73C) Math Question 6

For the complex variable z, one of the following is a solution of z² = −9. Which one?

2025 October(J08) Math Question 10

Matrices A and B are given below. A = [−2, 6; 3, −8], B = [8, 5; 1, −4]. Which of the following matrices is A − B?

2024 September(H11) Math Question 10

2020 July(C01) Math Question 12

Which of the following matrices is equal to 3[−2, 1; 0, −5]?

2024 December(H31) Math Question 13

14. Which of the following integers is closest to

2017 June(74C) Math Question 14

15. What is the smallest integer greater than V85 ?

2018 June(A11) Math Question 15

2010 April(68G) Math Question 15

Which of the following numbers is equal to (10i + 2)/(2i)? (Note: i² = −1)

2025 October(J08) Math Question 15

Which of the following is a possible sum of the complex numbers √(−9) and √(−16)?

2023 September(G01) Math Question 15

16. What is the largest possible product for 2 even integers whose sum is 18 ?

Adding, subtracting, multiplying and dividing positive and negative numbers

2020 December(D03) Math Question 16

The number a is positive and even. The number b is negative and odd. The number a − b is:

2024 June(G20) Math Question 16

2014 April(71H) Math Question 17

Last semester, each of the 4 students listed below took 6 classes worth 1 credit each. In the 1st matrix, the row corresponding to each student gives the counts of the letter grades earned by that student. The second matrix gives the point value of each letter grade. The grade point average of each student is the total number of grade points earned by that student divided by that student's total number of class credits. To the nearest 0.01, what is Ann's grade point average?

2024 April(Z13) Math Question 17

18. For which of the following conditions will the sum of integers in and n always be an odd integer?

2017 April(74F) Math Question 18

2011 April(67F) Math Question 19

What is the least positive number that has a remainder of 3 when divided by 5 and a remainder of 4 when divided by 6?

2024 April(Z13) Math Question 20

Let y be an irrational number. Which of the following statements about the value of 15y must be true?

2024 September(H11) Math Question 20

Let y be an irrational number. Which of the following statements about the value of 5y is true?

2024 December(H31) Math Question 20

20. One of the following values for a makes the expression (2a+5)/(a2+1) undefined. Which one?

Complex Number

Undefined fraction

2021 April(D05) Math Question 20

21. What is the least positive number that has a remainder of 3 when divided by 5 and a remainder of 7 when divided by 9 ?

Remainders

2021 April(D05) Math Question 21

Let m and n be positive numbers such that m < n. Which of the following expressions has the greatest value?

2023 April(F11) Math Question 22

Let z = 4 + 5i and w = 3 + 7i, where i is the imaginary unit. What is the value of 2z + w?

2024 June(G20) Math Question 22

2010 December(69A) Math Question 23

Which of the following numbers is irrational?

35 February(Z14) Math Question 24

Each student in a particular classroom was given a string that was 72 inches long. Each student cut his or her string into pieces of equal length. Which of the following CANNOT be the length, in inches, of any student's pieces?

2024 June(G20) Math Question 25

Which of the following is the complex conjugate of 2 + 7i?

2024 April(Z13) Math Question 27

28. Which of the following operations will produce the largest result when substituted for the blank in the

2016 December(74H) Math Question 28

2009 December(68A) Math Question 28

What is the least positive number that has a remainder of 4 when divided by 6 and a remainder of 5 when divided by 7?

2024 June(G20) Math Question 28

2018 December(B05) Math Question 29

2012 December(71A) Math Question 30

Given that n is a rational number, which of the following statements about the number (−4√14 − n)/2 must be true?

2025 October(J08) Math Question 30

31. What is the 322nd digit after the decimal point in the repeating decimal 0.1357 ?

Repeating decimal

2021 June(D06) Math Question 31

Which of the following lists gives the numbers below arranged in order from least to greatest? 0.6, 0.08, 5/8

2024 April(Z13) Math Question 31

The number line shown below is marked in equal intervals. Two fractions are indicated on the number line. One of the following fractions corresponds to the point marked P. Which one?

2024 June(G20) Math Question 31

2019 April(Z15) Math Question 32

2012 June(70C) Math Question 32

2012 April(70G) Math Question 32

32. What is the complex conjugate of the number below?

Division of imaginary numbers (the use of conjugates)

2021 April(D05) Math Question 32

33. What is the least positive number that has a remainder of 3 when divided by 6 and a remainder of 6 when divided by 9 ?

2020 December(D03) Math Question 33

Given matrices A and B such that A = [−7 2 4; −1 0 −3] and B − A = [6 7 4; 1 −1 −4], what is matrix B?

35 February(Z14) Math Question 33

Given i² = −1, −2i(3 + 4i) − 3(2 + 5i) = ?

2024 September(H11) Math Question 34

35. For which of the equations below is its solution an integer?

2015 April(73G) Math Question 35

36. Which of the following arranges the numbers f, 1.8,
1.08, and 1.08 into ascending order? (Note: The
overbar notation shows that the digits under the bar
will repeat. For example, 1.73 == 1.737373 ... )

2019 June(B02) Math Question 36

2019 April(B04) Math Question 36

36. Among the following rational numbers, which has the greatest value?

2013 June(71C) Math Question 36

36. Given that i is the imaginary unit, which of the following complex numbers is equal to (7 + 6i)2 ?

2020 December(D03) Math Question 36

Given i² = −1, which of the following expressions is equivalent to (9i − 2)(3 − 5i)?

2024 December(H31) Math Question 36

38. Which of the following expressions, when evaluated, equals an irrational number?

2017 April(74F) Math Question 38

2010 December(69A) Math Question 40

2009 April(66F) Math Question 40

Given the relatively prime positive integers a, b, c, and d such that a > b > c > d, which of the following fractions has the greatest value?

2024 December(H31) Math Question 40

2011 June(69F) Math Question 41

2009 April(66F) Math Question 41

You want to list all the positive two-digit numbers for which the units digit is larger than the tens digit and for which the sum of the digits is 12. How many two-digit numbers should be on your list?

2023 September(G01) Math Question 41

What are all the values of z that make the equation below true? z² + z + 3 = 0

2024 December(H31) Math Question 42

43. What is the smallest positive integer having exactly
5 different positive integer divisors?

2019 June(B02) Math Question 43

2012 December(71A) Math Question 43

43. For i = square root of -1, (1 + 2i)^2 = ?

2016 April(73E) Math Question 43

2021 June(D06) Math Question 44

2010 December(69A) Math Question 45

What are the solutions of (x + 1)² + 1 = 0?

2024 September(H11) Math Question 45

2011 April(67F) Math Question 46

2020 July(C01) Math Question 46

2021 June(D06) Math Question 46

46. Let A, B, and C represent the digits in the hundreds, tens, and ones places, respectively, of a certain 3-digit whole number. Let D, E, and F represent the digits in the hundreds, tens, and ones places, respectively, of a different 3-digit whole number. The positive difference between the 2  numbers is greater than 100. Which of the following inequalities must be true?

Absolute value

2021 December(E23) Math Question 46

2009 December(68A) Math Question 47

For √(−1) = i, and for certain real numbers a and b, the equation (10 − i) + (8 + 6i) = a + bi is true. What is the value of ab?

2024 December(H31) Math Question 47

2013 December(71E) Math Question 48

2009 April(66F) Math Question 48

Matrix A has dimensions 3 × 2, and matrix B has dimensions 2 × 3. One of the following matrices is the matrix product AB. Which one?

2023 September(G01) Math Question 48

49. If x < y and y < 4, then what is the greatest possible integer value of x + y ?

2018 June(A11) Math Question 49

Let A be the greatest whole number that is less than √420. Let B be the least whole number that is greater than √56. What is A − B?

2024 June(G20) Math Question 50

2013 April(71G) Math Question 51

2012 June(70C) Math Question 51

35 February(Z14) Math Question 51

The product of 2 real numbers is a nonzero rational number. Which of the following statements CANNOT be true?

2024 June(G20) Math Question 51

52. Let x be a negative odd integer. The expression xy3 is a positive even integer whenever y is any member of which of the following sets?

2014 December(72G) Math Question 52

52. What is the 275th digit after the decimal point in the repeating decimal 0.6295 _____ ?

2021 April(Z04) Math Question 52

53. An integer is abundant if its positive integer factors, excluding the integer itself, have a sum that is greater than the integer. How many of the integers 6, 8, 10, and 12 are abundant?

2018 June(A11) Math Question 53

53. The product of 2 numbers is 25. If 1 of the numbers is the complex number 4 + 3i, what is the other number?

2014 April(71H) Math Question 53

53. Given that b is rational and i = √••−1 , the product of the expression (3 + bi) and which of the following expressions must be a rational number?

2021 December(E23) Math Question 53

2020 June(C02) Math Question 53

If −2 ≤ x ≤ 4 and −1 ≤ y ≤ 5, what is the maximum value of |x − y|?

2023 September(G01) Math Question 53

2013 April(71G) Math Question 54

2020 June(C02) Math Question 54

What is the maximum possible value of x/y² for x in the set {1, 1/2, 0} and y in the set {1/8, 1/4, 1/2}?

35 February(Z14) Math Question 54

2021 April(Z04) Math Question 55

Which of the following numbers is a solution to the equation 3x² + x + 5 = 0?

2023 April(F11) Math Question 55

2020 June(C02) Math Question 56

57. What is the distance, in coordinate units, between 2 + 6i and —4 + 3i in the complex plane?

Number Types & PropertiesThere are 108 ACT math questions under Number Types & Properties

Sub-topics

2015 June(73C) Math Question 2

2017 December(A10) Math Question 2