Which list orders the numbers −√2, −1.4111... (the digit 1 repeats), −99/70, and −1.41 from least to greatest?

−99/70, −√2, −1.4111..., −1.41

−99/70, −√2, −1.41, −1.4111...

−1.41, −1.4111..., −√2, −99/70

−√2, −99/70, −1.4111..., −1.41

Number of books read over the summer by 30 students Books Number of students 0 3 1 7 2 10 3 6 4 4 Which statement about the mean and median number of books read is correct?

The mean is about 2.03 and the median is 2.

The mean is about 12.2 and the median is 2.

The mean is about 2.03 and the median is 3.

The mean is 2.0 and the median is 10.

TExES Mathematics 4-8 (115)

Free TExES Math 4-8 (115) Practice Questions

10 sample questions for the TExES Mathematics 4-8 (115), 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 Concepts, Patterns and Algebra, Geometry and Measurement, Probability and Statistics, Mathematics.

Question 1 · Number Concepts
Which list orders the numbers −√2, −1.4111... (the digit 1 repeats), −99/70, and −1.41 from least to greatest?
- A.−99/70, −√2, −1.41, −1.4111...
- B.−1.41, −1.4111..., −√2, −99/70
- C.−99/70, −√2, −1.4111..., −1.41
- D.−√2, −99/70, −1.4111..., −1.41
Show answer and explanation
Answer: C. −99/70, −√2, −1.4111..., −1.41
As decimals, 99/70 ≈ 1.414286, √2 ≈ 1.414214, 1.4111... ≈ 1.411111, and 1.41 = 1.410000. For negative numbers the one with the largest absolute value is least, so −99/70 < −√2 < −1.4111... < −1.41. Listing −√2 first comes from treating 99/70 as smaller than √2, and the reversed list orders the absolute values instead of the negatives.
Question 2 · Patterns and Algebra
Which of the following relations define y as a function of x? I. y^2 = x + 4 II. y = √(x + 4) III. {(−2, 5), (0, 5), (3, −1), (6, 5)}
- A.I, II, and III
- B.I and II only
- C.I and III only
- D.II and III only
Show answer and explanation
Answer: D. II and III only
In relation I, solving gives y = ±√(x + 4), so an input such as x = 0 has two outputs (2 and −2) and the relation is not a function. Relation II uses only the principal square root, so each allowed input has one output. In relation III no x-value repeats; repeated y-values are allowed, so it is a function.
Question 3 · Geometry and Measurement
Which of the following sets of side lengths form a right triangle? I. 9 cm, 40 cm, 41 cm II. 7 in, 24 in, 26 in III. 1.5 m, 2 m, 2.5 m
- A.I and II only
- B.I, II, and III
- C.II and III only
- D.I and III only
Show answer and explanation
Answer: D. I and III only
By the converse of the Pythagorean theorem, test whether the squares of the two shorter sides add to the square of the longest. For I, 81 + 1,600 = 1,681 = 41^2, and for III, 2.25 + 4 = 6.25 = 2.5^2, so both are right triangles. For II, 49 + 576 = 625, which is not 26^2 = 676 (the triple is 7-24-25), and since 625 < 676 that triangle is obtuse.
Question 4 · Probability and Statistics
A quality-control inspector weighs a random sample of 5 bags of trail mix and records how many grams over the labeled weight each bag is: 2, 4, 4, 5, 10. What is the sample standard deviation of these values?
- A.2.0
- B.About 2.68
- C.3.0
- D.9.0
Show answer and explanation
Answer: C. 3.0
The mean is 25 ÷ 5 = 5, the squared deviations are 9, 1, 1, 0 and 25 with sum 36, and the sample variance is 36 ÷ (5 − 1) = 9, so s = √9 = 3. Dividing by 5 gives the population value of about 2.68, 9 is the variance rather than the standard deviation, and 2.0 is the mean absolute deviation.
Question 5 · Mathematics
Annual salary: $46,800, paid semimonthly (24 equal paychecks per year)
Federal income tax withheld per paycheck: $168.00
A Texas office employee's pay information is shown. What is the teacher's net pay for one paycheck?
- A.$1,535.32
- B.$1,563.60
- C.$1,632.82
- D.$1,404.30
Show answer and explanation
Answer: A. $1,535.32
Gross pay per paycheck is 46,800 ÷ 24 = $1,950.00. Deductions are Social Security 0.062 × 1,950 = $120.90, Medicare 0.0145 × 1,950 = $28.28, retirement 0.05 × 1,950 = $97.50 and federal tax $168.00, totaling $414.68, and Texas has no state income tax. Net pay = 1,950.00 − 414.68 = $1,535.32. Dividing the salary by 26 (biweekly) instead of 24 gives $1,404.30, and leaving out Medicare gives $1,563.60.
Question 6 · Number Concepts
Which of the following numbers are irrational? I. √8/√2 II. 0.101001000100001..., where each block of zeros is one longer than the block before it III. 2 + √3
- A.I and II only
- B.I, II, and III
- C.II and III only
- D.I and III only
Show answer and explanation
Answer: C. II and III only
Statement I simplifies to √(8/2) = √4 = 2, which is rational. The decimal in II never terminates and never repeats a fixed block, so it is irrational even though it follows a pattern. In III, a rational number plus an irrational number is irrational. Any choice that includes I overlooks that a quotient of two irrationals can be rational.
Question 7 · Patterns and Algebra
Let f(x) = (x − 1)^3 + 2. Which of the following statements are true? I. f^(−1)(x) = ∛(x − 2) + 1 II. The graph of f^(−1) is the reflection of the graph of f across the line y = x. III. f^(−1)(10) = 3
- A.I, II, and III
- B.II and III only
- C.I and III only
- D.I and II only
Show answer and explanation
Answer: A. I, II, and III
Swapping variables in y = (x − 1)^3 + 2 and solving gives y = ∛(x − 2) + 1, so statement I is true. Every inverse function's graph is the reflection of the original across y = x, so statement II is true. Statement III is true because f^(−1)(10) = ∛8 + 1 = 3; check: f(3) = 8 + 2 = 10.
Question 8 · Geometry and Measurement
From the top of a cliff 50 meters above sea level, an observer sees two boats due east of the cliff, in a straight line with its base. The angle of depression to the nearer boat is 32 degrees and the angle of depression to the farther boat is 20 degrees.
To the nearest meter, how far apart are the two boats?
- A.217 m
- B.57 m
- C.13 m
- D.235 m
Show answer and explanation
Answer: B. 57 m
Each angle of depression equals the angle of elevation from that boat, so each horizontal distance is 50 ÷ tan of the angle. The farther boat is 50 ÷ tan 20° ≈ 137.37 m from the base and the nearer boat is 50 ÷ tan 32° ≈ 80.02 m away, so the boats are about 57 m apart. The value 13 m comes from multiplying by the tangents instead of dividing, and 217 m adds the two distances.
Question 9 · Probability and Statistics
Number of books read over the summer by 30 students
Books Number of students
0 3
1 7
2 10
3 6
4 4
Which statement about the mean and median number of books read is correct?
- A.The mean is about 12.2 and the median is 2.
- B.The mean is about 2.03 and the median is 3.
- C.The mean is 2.0 and the median is 10.
- D.The mean is about 2.03 and the median is 2.
Show answer and explanation
Answer: D. The mean is about 2.03 and the median is 2.
The total number of books is 0 × 3 + 1 × 7 + 2 × 10 + 3 × 6 + 4 × 4 = 61, so the mean is 61 ÷ 30 ≈ 2.03. The median is the mean of the 15th and 16th values; the cumulative counts are 3, 10, 20, so both are 2. Dividing 61 by the 5 categories gives 12.2, and 10 is a frequency, not a data value.
Question 10 · Mathematics
A pair of shoes is marked $64.00. The store takes 30% off the marked price, and then a coupon takes an additional 10% off the sale price. Sales tax of 8.25% is applied to the final price. What is the total cost, to the nearest cent?
- A.$43.65
- B.$45.60
- C.$41.57
- D.$40.32
Show answer and explanation
Answer: A. $43.65
The discounts are applied in turn: 64.00 × 0.70 = 44.80, then 44.80 × 0.90 = 40.32. Tax is about 3.33, so the total is 40.32 × 1.0825 = $43.65 to the nearest cent. Combining the discounts as a single 40% gives $41.57, and $40.32 forgets the sales tax.

Books	Number of students
0	3
1	7
2	10
3	6
4	4

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