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ACT Math Help: Real Numbers

Review real example questions for Real Numbers in ACT Math.

Question 1 / 10

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Order the following from least to greatest: $-2$ , $-\dfrac{7}{3}$ , $-\sqrt{5}$ , $-2.2$ .

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Question 1

Order the following from least to greatest: $-2$ , $-\dfrac{7}{3}$ , $-\sqrt{5}$ , $-2.2$ .

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

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

What is the absolute value of $-\frac{7}{3}$ ?

$-\frac{7}{3}$
$\frac{7}{3}$ (correct answer)
$\frac{3}{7}$
$7$

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

Which number is greatest? $-0.5$ , $0.1$ , $-0.1$ , $0.05$

$-0.5$
$0.1$ (correct answer)
$-0.1$
$0.05$

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

Which expression represents a real number? $\sqrt{-1}$ , $\ln(-1)$ , $4^{0.5}$ , $\frac{1}{0}$

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

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

Which of the following lists the numbers $3.14$ , $\pi$ , and $\frac{22}{7}$ in order from least to greatest? (Note: $\pi \approx 3.14159...$ )

$3.14 < \dfrac{22}{7} < \pi$
$3.14 < \pi < \dfrac{22}{7}$ (correct answer)
$\pi < 3.14 < \dfrac{22}{7}$
$\dfrac{22}{7} < \pi < 3.14$

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

What is the approximate value of $\sqrt{3}$ ?

$1.7$
$1.73$ (correct answer)
$1.8$
$1.83$

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

Which of the following lists the numbers $3.14$ , $\pi$ , and $\frac{22}{7}$ in order from least to greatest? (Note: $\pi \approx 3.14159...$ )

$3.14 < \dfrac{22}{7} < \pi$
$3.14 < \pi < \dfrac{22}{7}$ (correct answer)
$\pi < 3.14 < \dfrac{22}{7}$
$\dfrac{22}{7} < \pi < 3.14$

Question 8

Which number is smallest? $-0.8$ , $-0.6$ , $-0.4$ , $0$

$0$
$-0.6$
$-0.4$
$-0.8$ (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

Which number is greatest? $-0.25$ , $0.25$ , $-0.75$ , $0.5$

$-0.25$
$0.25$
$-0.75$
$0.5$ (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

What is the least common multiple (LCM) of 9 and 12?

3
36 (correct answer)
72
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.