A data set has the five-number summary minimum 12, Q1 = 30, median 41, Q3 = 48, and maximum 80. Using the 1.5 × IQR rule, which statement is correct?

The maximum is an outlier; the minimum is not.

The minimum is an outlier, but the maximum is not.

Neither the minimum nor the maximum is an outlier.

Both the minimum and the maximum are outliers.

Across 30 school districts, the correlation between library books per student and average math score is r = 0.82. A student concludes, "If a district buys more library books, its math scores will go up." Which teacher response best helps the student critique this reasoning?

Note that the data are observational, so a lurking variable such as district funding could raise both books and scores.

Explain that r = 0.82 means that 82% of the variation in district math scores is caused by the number of library books.

Explain that the conclusion is wrong because only a correlation of exactly 1 can show causation.

Agree with the conclusion, because a correlation above 0.8 is strong enough by itself to establish cause and effect.

Score 40–49 50–59 60–69 70–79 80–89 90–99 Frequency 3 5 14 9 16 7 The table shows the scores of 54 students on a unit test. Which statement correctly identifies the class containing the median and the modal class?

The median is in 70–79 and the modal class is 80–89

The median is in 80–89 and the modal class is 80–89

The median is in 60–69 and the modal class is 80–89

The median is in 70–79 and the modal class is 70–79

A student conjectures that the product of two increasing functions is always increasing on their common domain. Which pair of functions is a counterexample on the stated domain?

f(x) = x and g(x) = x − 4 on all real numbers

f(x) = x + 1 and g(x) = x + 1 on [0, ∞)

f(x) = eˣ and g(x) = 2ˣ + 1 on all real numbers

f(x) = √x and g(x) = x³ on [0, ∞)

Praxis Mathematics (5165)

Free Praxis Mathematics 5165 Practice Questions

10 sample questions for the Praxis Mathematics (5165), with the correct answer and an explanation for each. No account needed. Try each one before you open the answer.

Sections covered on the exam: Number and Quantity and Algebra, Functions and Calculus, Geometry, Statistics and Probability, Teaching Mathematics.

Question 1 · Number and Quantity and Algebra
Which of the following is the quotient (7 + 4i)/(2 − 3i) written in the form a + bi?
- A.7/2 − (4/3)i
- B.2 + (29/13)i
- C.−2/5 − (29/5)i
- D.2/13 + (29/13)i
Show answer and explanation
Answer: D. 2/13 + (29/13)i
Multiply numerator and denominator by the conjugate 2 + 3i. The numerator is (7 + 4i)(2 + 3i) = 14 + 21i + 8i + 12i² = 2 + 29i, and the denominator is 2² + 3² = 13, so the quotient is 2/13 + (29/13)i. The answer 2 + (29/13)i comes from treating i² as +1, which makes the real part 26; −2/5 − (29/5)i comes from computing the denominator as 4 − 9 instead of 4 + 9.
Question 2 · Number and Quantity and Algebra
A chemist has 12 liters of a 15% acid solution. How many liters of a 40% acid solution must be added to obtain a mixture that is 25% acid?
- A.3 liters
- B.18 liters
- C.4.8 liters
- D.8 liters
Show answer and explanation
Answer: D. 8 liters
The acid amounts give 0.15(12) + 0.40x = 0.25(12 + x), so 1.8 + 0.40x = 3 + 0.25x, and 0.15x = 1.2 gives x = 8. Check: 1.8 + 3.2 = 5 liters of acid in 20 liters is 25%. The answer 3 liters comes from writing the right side as 0.25(12), forgetting that adding solution increases the total volume.
Question 3 · Functions and Calculus
Let f(x) = kx² − 1 for x ≤ 2 and f(x) = 3x + k for x > 2. For what value of k is f continuous for all real numbers?
- A.k = 7
- B.k = 7/3
- C.k = 5/3
- D.k = 7/5
Show answer and explanation
Answer: B. k = 7/3
Each piece is a polynomial, so the only concern is x = 2, where the left piece gives 4k − 1 and the right-side limit is 6 + k. Setting 4k − 1 = 6 + k gives 3k = 7, so k = 7/3. The answer k = 7 comes from evaluating kx² at x = 2 as 2k instead of 4k, and k = 7/5 comes from a sign error that writes the right side as 6 − k.
Question 4 · Statistics and Probability
A data set has the five-number summary minimum 12, Q1 = 30, median 41, Q3 = 48, and maximum 80. Using the 1.5 × IQR rule, which statement is correct?
- A.The minimum is an outlier, but the maximum is not.
- B.Neither the minimum nor the maximum is an outlier.
- C.The maximum is an outlier; the minimum is not.
- D.Both the minimum and the maximum are outliers.
Show answer and explanation
Answer: C. The maximum is an outlier; the minimum is not.
IQR = 48 − 30 = 18 and 1.5 × 18 = 27, so the fences are 30 − 27 = 3 and 48 + 27 = 75. The maximum, 80, is above 75, so it is an outlier, while the minimum, 12, is above 3, so it is not. Using fences of Q1 − IQR = 12 and Q3 + IQR = 66 (forgetting the factor 1.5) wrongly suggests the low end is borderline.
Question 5 · Teaching Mathematics
Across 30 school districts, the correlation between library books per student and average math score is r = 0.82. A student concludes, "If a district buys more library books, its math scores will go up." Which teacher response best helps the student critique this reasoning?
- A.Note that the data are observational, so a lurking variable such as district funding could raise both books and scores.
- B.Explain that r = 0.82 means that 82% of the variation in district math scores is caused by the number of library books.
- C.Explain that the conclusion is wrong because only a correlation of exactly 1 can show causation.
- D.Agree with the conclusion, because a correlation above 0.8 is strong enough by itself to establish cause and effect.
Show answer and explanation
Answer: A. Note that the data are observational, so a lurking variable such as district funding could raise both books and scores.
No variable was assigned by the researchers, so a strong association cannot rule out confounding variables like funding or family income. Here r² = 0.82² ≈ 0.67, meaning about 67% of the variation in scores is accounted for by the linear relationship with books, which is still not a causal statement. Interpreting r as a causal percentage confuses r with r² and association with causation, and even r = 1 would not prove causation.
Question 6 · Number and Quantity and Algebra
The equation z³ = 8i has three complex solutions. Which of the following is one of them?
- A.1 + √3 i
- B.2i
- C.−√3 + i
- D.√3 − i
Show answer and explanation
Answer: C. −√3 + i
Write 8i = 8(cos 90° + i sin 90°). The cube roots have modulus ∛8 = 2 and arguments (90° + 360°k)/3 = 30°, 150°, 270°, giving √3 + i, −√3 + i, and −2i. The value 2i is tempting, but (2i)³ = 8i³ = −8i, and 1 + √3 i has argument 60°, so its cube has argument 180° and equals −8.
Question 7 · Number and Quantity and Algebra
Month Change in price
March increase of 20%
April decrease of 25%
May increase of 10%
An item's price changed each month as shown. What is the overall percent change from the start of March to the end of May?
- A.An increase of 1%
- B.A decrease of 1%
- C.No change
- D.An increase of 5%
Show answer and explanation
Answer: B. A decrease of 1%
Successive changes multiply: 1.20 × 0.75 × 1.10 = 0.99, so the final price is 99% of the original, a 1% decrease. Adding the percents gives 20 − 25 + 10 = 5%, which ignores that each change applies to a different base price.
Question 8 · Functions and Calculus
x 1 2 3 4 5
f(x) 3 −1 2 2 −4
A function f is continuous on [1, 5] and has the values shown. Which of the following statements must be true? I. f has at least three zeros in (1, 5). II. f has exactly three zeros in (1, 5). III. f(c) = 1 for some c in (2, 3).
- A.I, II, and III
- B.I only
- C.I and III only
- D.II and III only
Show answer and explanation
Answer: C. I and III only
The sign changes on [1, 2], [2, 3], and [4, 5], so by the Intermediate Value Theorem there is a zero in each, and I is true. Nothing rules out extra zeros, for example between 3 and 4, so II is not guaranteed. Since f(2) = −1 < 1 < 2 = f(3), the IVT also guarantees III.
Question 9 · Statistics and Probability
Score 40–49 50–59 60–69 70–79 80–89 90–99
Frequency 3 5 14 9 16 7
The table shows the scores of 54 students on a unit test. Which statement correctly identifies the class containing the median and the modal class?
- A.The median is in 70–79 and the modal class is 80–89
- B.The median is in 80–89 and the modal class is 80–89
- C.The median is in 60–69 and the modal class is 80–89
- D.The median is in 70–79 and the modal class is 70–79
Show answer and explanation
Answer: A. The median is in 70–79 and the modal class is 80–89
With 54 scores, the median is the average of the 27th and 28th values. Cumulative frequencies are 3, 8, 22, 31, …, so both the 27th and 28th scores fall in 70–79. The modal class is the one with the greatest frequency, 80–89 with 16 students; assuming the median sits in the tallest class is the mistake behind placing it in 80–89.
Question 10 · Teaching Mathematics
A student conjectures that the product of two increasing functions is always increasing on their common domain. Which pair of functions is a counterexample on the stated domain?
- A.f(x) = x + 1 and g(x) = x + 1 on [0, ∞)
- B.f(x) = eˣ and g(x) = 2ˣ + 1 on all real numbers
- C.f(x) = x and g(x) = x − 4 on all real numbers
- D.f(x) = √x and g(x) = x³ on [0, ∞)
Show answer and explanation
Answer: C. f(x) = x and g(x) = x − 4 on all real numbers
Both x and x − 4 are increasing, but their product x² − 4x has derivative 2x − 4, which is negative for x < 2, so the product decreases on (−∞, 2). The product eˣ(2ˣ + 1) is increasing because both factors are positive and increasing, and on [0, ∞) the products (x + 1)² and x^(7/2) are increasing because both factors are nonnegative there. The conjecture does hold when both functions are positive, which is why the counterexample must use negative values.

Month	Change in price
March	increase of 20%
April	decrease of 25%
May	increase of 10%

x	1	2	3	4	5
f(x)	3	−1	2	2	−4

Score	40–49	50–59	60–69	70–79	80–89	90–99
Frequency	3	5	14	9	16	7

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