ACT Math Quiz: Real Numbers

What this quiz covers

This quiz focuses on Real Numbers, giving you a quick way to practice the rules, question types, and explanations that matter most for ACT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $-\dfrac{7}{3},\ -\sqrt{5},\ -2.2,\ -2$ (correct answer)
2. $-\sqrt{5},\ -\dfrac{7}{3},\ -2.2,\ -2$
3. $-\dfrac{7}{3},\ -2.2,\ -\sqrt{5},\ -2$
4. $-2,\ -2.2,\ -\sqrt{5},\ -\dfrac{7}{3}$

Explanation: To order negative numbers, convert each to a decimal and remember that the more negative a value is, the smaller it is. Here $-\dfrac{7}{3} \approx -2.333$, $-\sqrt{5} \approx -2.236$, $-2.2$ is already decimal, and $-2$ is the largest, so from least to greatest the order is $-\dfrac{7}{3},\ -\sqrt{5},\ -2.2,\ -2$. The ordering $-\sqrt{5},\ -\dfrac{7}{3},\ -2.2,\ -2$ swaps the first two by treating $-2.236$ as more negative than $-2.333$. The ordering $-\dfrac{7}{3},\ -2.2,\ -\sqrt{5},\ -2$ places $-2.2$ before $-\sqrt{5}$, missing that $-2.236$ lies further left on the number line than $-2.2$. The ordering $-2,\ -2.2,\ -\sqrt{5},\ -\dfrac{7}{3}$ runs from greatest to least, the reverse of what was asked. Convert every value to a decimal first and read the list off the number line from left to right, checking the direction the question requests.

Question 2

1. $-\frac{7}{3}$
2. $\frac{7}{3}$ (correct answer)
3. $\frac{3}{7}$
4. $7$

Explanation: The absolute value of a number is its distance from zero on the number line, always positive or zero. The absolute value of $-\frac{7}{3}$ is the distance from $-\frac{7}{3}$ to 0, which is $\frac{7}{3}$ units. Therefore, $|-\frac{7}{3}| = \frac{7}{3}$.

Question 3

1. $-0.5$
2. $0.1$ (correct answer)
3. $-0.1$
4. $0.05$

Explanation: We need to identify the greatest among -0.5, 0.1, -0.1, and 0.05. On the number line, positive numbers are greater than negative numbers, and among positive numbers, larger values are to the right. Comparing the positive values: 0.1 > 0.05, and both are greater than the negative values -0.5 and -0.1. Therefore, 0.1 is the greatest.

Question 4

1. $\sqrt{-1}$
2. $\ln(-1)$
3. $4^{0.5}$ (correct answer)
4. $\frac{1}{0}$

Explanation: We need to identify which expression represents a real number. √(-1) is undefined in real numbers (square root of negative), ln(-1) is undefined (logarithm of negative), 4^0.5 = √4 = 2 is a positive real number, and 1/0 is undefined (division by zero). Only 4^0.5 represents a real number.

Question 5

1. $3.14 < \dfrac{22}{7} < \pi$
2. $3.14 < \pi < \dfrac{22}{7}$ (correct answer)
3. $\pi < 3.14 < \dfrac{22}{7}$
4. $\dfrac{22}{7} < \pi < 3.14$

Explanation: This is an ordering of real numbers question testing number sense with irrational numbers. Choice B (3.14 < π < 22/7) is correct — converting to decimals: 3.14 = 3.1400..., π ≈ 3.14159..., 22/7 ≈ 3.14286. The correct order from least to greatest is: 3.14 < π < 22/7. Choice A (3.14 < 22/7 < π) places 3.14 correctly but swaps π and 22/7 — a very common error since 22/7 is often introduced as a shorthand for π, but it is actually slightly larger than π. Choice C (π < 3.14 < 22/7) incorrectly places π below 3.14, reversing their actual relationship — π ≈ 3.14159, which is greater than 3.14. Choice D (22/7 < π < 3.14) inverts the entire order, placing 22/7 as the smallest when it is actually the largest. Pro tip: Convert all three to decimals before comparing: 22 ÷ 7 ≈ 3.142857. This removes any ambiguity. Remember: 22/7 is a common approximation for π, but it overestimates π by about 0.001.

Question 6

1. $1.7$
2. $1.73$ (correct answer)
3. $1.8$
4. $1.83$

Explanation: To approximate $\sqrt{3}$, we find perfect squares near 3. Since $1^2 = 1$ and $2^2 = 4$, $\sqrt{3}$ is between 1 and 2. More precisely, $1.7^2 = 2.89$ and $1.8^2 = 3.24$, so $\sqrt{3}$ is between 1.7 and 1.8. Calculating: $1.73^2 = 2.9929 \approx 3$, so $\sqrt{3} \approx 1.73$.

Question 7

1. $3.14 < \dfrac{22}{7} < \pi$
2. $3.14 < \pi < \dfrac{22}{7}$ (correct answer)
3. $\pi < 3.14 < \dfrac{22}{7}$
4. $\dfrac{22}{7} < \pi < 3.14$

Explanation: This is an ordering of real numbers question testing number sense with irrational numbers. Choice B (3.14 < π < 22/7) is correct — converting to decimals: 3.14 = 3.1400..., π ≈ 3.14159..., 22/7 ≈ 3.14286. The correct order from least to greatest is: 3.14 < π < 22/7. Choice A (3.14 < 22/7 < π) places 3.14 correctly but swaps π and 22/7 — a very common error since 22/7 is often introduced as a shorthand for π, but it is actually slightly larger than π. Choice C (π < 3.14 < 22/7) incorrectly places π below 3.14, reversing their actual relationship — π ≈ 3.14159, which is greater than 3.14. Choice D (22/7 < π < 3.14) inverts the entire order, placing 22/7 as the smallest when it is actually the largest. Pro tip: Convert all three to decimals before comparing: 22 ÷ 7 ≈ 3.142857. This removes any ambiguity. Remember: 22/7 is a common approximation for π, but it overestimates π by about 0.001.

Question 8

1. $0$
2. $-0.6$
3. $-0.4$
4. $-0.8$ (correct answer)

Explanation: We need to identify the smallest among -0.8, -0.6, -0.4, and 0. On the number line, these values are arranged as: -0.8 < -0.6 < -0.4 < 0. Among negative numbers, the one with greater absolute value is smaller, and all negative numbers are smaller than zero. Therefore, -0.8 is the smallest.

Question 9

1. $-0.25$
2. $0.25$
3. $-0.75$
4. $0.5$ (correct answer)

Explanation: We need to identify the greatest among -0.25, 0.25, -0.75, and 0.5. On the number line, positive numbers are greater than negative numbers, and among positive numbers, larger values are greater. Comparing: 0.5 > 0.25 > -0.25 > -0.75. Therefore, 0.5 is the greatest.

Question 10

1. 3
2. 36 (correct answer)
3. 72
4. 108

Explanation: The correct answer is B (36). The LCM of 9 and 12 is found by listing multiples: multiples of 12 are 12, 24, 36... and 36 ÷ 9 = 4, so 36 is divisible by both. Alternatively: LCM = (9 × 12) ÷ GCF(9,12) = 108 ÷ 3 = 36. A (3) is the GCF of 9 and 12, not the LCM — a classic confusion between the two concepts. C (72) doubles the correct answer — possibly from computing 36 × 2 or using the wrong formula. D (108) is the product of 9 × 12, forgetting to divide by the GCF. Remember: LCM × GCF = product of the two numbers.

Question 11

1. 7
2. 8
3. 9 (correct answer)
4. 10

Explanation: This is a number line counting question testing careful decimal approximation and boundary exclusion. Choice C (9) is correct — convert the bounds: −17/3 ≈ −5.667 and 15/4 = 3.75. The integers strictly between these values are: −5, −4, −3, −2, −1, 0, 1, 2, 3. Count: 9 integers. Note that −6 < −5.667 so −6 is NOT between the bounds; and 4 > 3.75 so 4 is also NOT included. Choice A (7) undercounts, perhaps starting from −4 (misidentifying −17/3 as approximately −4.something). Choice B (8) miscounts by one, possibly including or excluding a boundary value incorrectly. Choice D (10) overcounts, perhaps starting from −6 (rounding −5.667 down to −6 and including it as "between"). Pro tip: Always convert fractions to decimals first. −17/3 = −5.666..., which means −6 is NOT between (it's outside the lower bound) and −5 IS between. The word "between" means strictly greater than and strictly less than — do not include values equal to the bounds (and these bounds aren't integers anyway).

Question 12

1. $-8$
2. $8$ (correct answer)
3. $-16$
4. $0$

Explanation: We need to find the absolute value of -8. The absolute value represents the distance from zero on the number line, which is always positive. The distance from -8 to 0 is 8 units. Therefore, |-8| = 8. Choice A incorrectly gave -8, not understanding that absolute value removes the negative sign.

Question 13

1. 2.2
2. 2.24 (correct answer)
3. 2.5
4. 2.7

Explanation: We need to approximate $\sqrt{5}$. Since $2^2 = 4$ and $3^2 = 9$, $\sqrt{5}$ lies between 2 and 3. More precisely, since $2.2^2 = 4.84$ and $2.3^2 = 5.29$, $\sqrt{5}$ is approximately 2.24. Using a calculator confirms $\sqrt{5} \approx 2.236$. Therefore, 2.24 is the best approximation. Choice A gave 2.2, which is slightly too small since $2.2^2 = 4.84 < 5$.

Question 14

1. $-3, -\frac{3}{4}, \frac{-1}{2}, 0.75$ (correct answer)
2. $\frac{-1}{2}, -3, -\frac{3}{4}, 0.75$
3. $-3, \frac{-1}{2}, -\frac{3}{4}, 0.75$
4. $-3, -\frac{3}{4}, 0.75, \frac{-1}{2}$

Explanation: We need to order the numbers -3, -1/2, 0.75, and -3/4 from least to greatest. Converting to decimal form: -3 = -3.0, -1/2 = -0.5, 0.75 = 0.75, -3/4 = -0.75. On the number line from left to right: -3 < -0.75 < -0.5 < 0.75. Therefore, the order from least to greatest is -3, -3/4, -1/2, 0.75. Choice B incorrectly placed -1/2 first, not recognizing that -3 is the smallest negative number.

Question 15

1. $4$
2. $-4$ (correct answer)
3. $\frac{3}{4}$
4. $2.5$

Explanation: We need to identify which number is smallest among 4, -4, 3/4, and 2.5. Converting to decimal form: 4 = 4.0, -4 = -4.0, 3/4 = 0.75, 2.5 = 2.5. On the number line from left to right: -4 < 0.75 < 2.5 < 4. The leftmost position represents the smallest value. Therefore, -4 is the smallest number. Choice A incorrectly selected 4, confusing greatest with smallest.

Question 16

1. $\sqrt{5}$
2. $0.333\ldots$ (correct answer)
3. $\pi$
4. $e$

Explanation: A rational number can be expressed as a fraction p/q where p and q are integers and q ≠ 0. √5 is irrational, 0.333... = 1/3 is rational (repeating decimal), π is irrational, and e is irrational. The decimal 0.333... represents the fraction 1/3, making it rational.

Question 17

1. $\sqrt{10}$
2. $0.2$ (correct answer)
3. $\pi$
4. $\ln(5)$

Explanation: A rational number can be expressed as a fraction p/q where p and q are integers and q ≠ 0. √10 is irrational, 0.2 = 2/10 = 1/5 is rational (terminating decimal), π is irrational, and ln(5) is irrational. The decimal 0.2 terminates, making it rational.

Question 18

1. $-0.04$
2. $-\dfrac{1}{20}$
3. $0$ (correct answer)
4. $-\sqrt{0.0016}$

Explanation: To find the greatest number, we convert all to decimal form for comparison. Converting: -0.04 = -0.04, -1/20 = -0.05, 0 = 0, and -√0.0016 = -0.04. On the number line: -0.05 < -0.04 = -0.04 < 0, so 0 is the greatest value. Choice A incorrectly suggests a negative number could be greater than zero, but all negative numbers are less than zero.

Question 19

1. $-\frac{3}{5}, \frac{1}{4}, 0.8, -0.9$
2. $0.8, -\frac{3}{5}, \frac{1}{4}, -0.9$
3. $-0.9, -\frac{3}{5}, \frac{1}{4}, 0.8$
4. $0.8, \frac{1}{4}, -\frac{3}{5}, -0.9$ (correct answer)

Explanation: We need to order the numbers from greatest to least. Converting to decimal form: -3/5 = -0.6, 0.8 = 0.8, -0.9 = -0.9, 1/4 = 0.25. Ordering from greatest to least: 0.8 > 0.25 > -0.6 > -0.9. Therefore, the order is 0.8, 1/4, -3/5, -0.9. Choice B incorrectly placed -3/5 before 1/4, not recognizing that positive numbers are greater than negative numbers.

Question 20

1. $0, -\frac{1}{4}, 0.5, \frac{1}{3}$
2. $0.5, 0, \frac{1}{3}, -\frac{1}{4}$
3. $-\frac{1}{4}, \frac{1}{3}, 0, 0.5$
4. $-\frac{1}{4}, 0, \frac{1}{3}, 0.5$ (correct answer)

Explanation: We need to order -1/4, 0.5, 0, and 1/3 from least to greatest. Convert to decimals: -1/4 = -0.25, 0.5 = 0.5, 0 = 0, 1/3 ≈ 0.333. On the number line: -0.25 < 0 < 0.333 < 0.5. The correct order from least to greatest is -1/4, 0, 1/3, 0.5.