10 random (doc, eval) pairs sampled (seed=42) from eval_to_docs.parquet of the MMLU vs nemotron_cc_math_v1 run (writeup).
- corpus:
gs://marin-us-east5/normalized/nemotron_cc_math_v1/4plus_b05688a8/outputs/main/ - evals:
gs://marin-us-east1/decontamination/mmlu-9fbdd5/cais/ - analysis:
gs://marin-us-east5/tmp/ttl=7d/rav/decon-v0/mmlu-vs-nemotron-math-v1/analysis/eval_to_docs.parquet
Each example shows the matched MMLU question (with answer) and the raw Nemotron document text it leaked into.
- eval_id:
cais/mmlu-elementary_mathematics-test-decontamination-318 - doc_id:
b1f582430428562af8450deb4245c3a0(partition 135) - n_matches: 48 · max_overlap: 1.0
question: The lowest point on Earth is the bottom of the Mariana Trench at a depth of 35,840 feet below sea level. The highest point on Earth is the summit of Mt. Everest at a height of 29,028 feet above sea level. Which of the following is the best estimate of the distance between the lowest and highest points on Earth?
options:
A. 6,000 feet
B. 7,000 feet
C. 64,000 feet
D. 65,000 feet
answer: D (65,000 feet)
subset: elementary_mathematics
Document (click to expand)
# CBEST Math Practice Test #1 to 10 Solutions
## 1. Average Score Calculation
During a semester, a student received scores of 76, 80, 83, 71, 80, and 78 on six tests. What is the student's average score for these six tests?
- A. 76
- B. 77
- C. 78
- D. 79
- E. 80
**Solution:**
To find the average score, sum all the scores and divide by the number of tests:
\[
\text{Average} = \frac{76 + 80 + 83 + 71 + 80 + 78}{6} = \frac{468}{6} = 78
\]
The student's average score is 78.
## 2. Percent Correct on Math Test
Use the table below to answer the question that follows.
| Section | Total Number of Questions | Number of Questions Correctly Answered |
|-------------|---------------------------|----------------------------------------|
| Algebra | 20 | 17 |
| Trigonometry| 15 | 11 |
| Geometry | 25 | 20 |
On the three sections of a math test, a student correctly answered the number of questions shown in the table above. What percent of the questions on the entire test did the student answer correctly?
- A. 20%
- B. 48%
- C. 75%
- D. 80%
- E. 96%
**Solution:**
First, find the total number of questions and the total number of questions answered correctly:
\[
\text{Total Questions} = 20 + 15 + 25 = 60
\]
\[
\text{Correct Answers} = 17 + 11 + 20 = 48
\]
Calculate the percentage:
\[
\text{Percentage} = \left(\frac{48}{60}\right) \times 100 = 80\%
\]
The student answered 80% of the questions correctly.
## 3. Scale of Diagram
If the actual length of the bridge is 4200 feet, then what is the scale of the diagram of the bridge?
- A. 1 unit = 700 feet
- B. 1 unit = 763.6 feet
- C. 1 unit = 840 feet
- D. 1 unit = 933.3 feet
- E. 1 unit = 1050 feet
**Solution:**
Assuming the diagram shows 5 units for the bridge:
\[
\text{Scale} = \frac{4200 \text{ feet}}{5 \text{ units}} = 840 \text{ feet/unit}
\]
The scale of the diagram is 1 unit = 840 feet.
## 4. Appropriate Unit for Pencil Weight
Which of the following is the most appropriate unit for expressing the weight of a pencil?
- A. pounds
- B. ounces
- C. quarts
- D. pints
- E. tons
**Solution:**
The most appropriate unit for expressing the weight of a pencil is ounces.
## 5. Total Length of Rope
Ms. Gutierrez needs to order rope for her gym class of 32 students. Each student will receive a piece of rope that is 5 feet 8 inches long. What is the total length of rope Ms. Gutierrez needs to order for her class?
- A. 106 feet 8 inches
- B. 154 feet 8 inches
- C. 160 feet 8 inches
- D. 181 feet 4 inches
- E. 185 feet 6 inches
**Solution:**
Convert 5 feet 8 inches to inches:
\[
5 \times 12 + 8 = 68 \text{ inches}
\]
Calculate the total length for 32 students:
\[
32 \times 68 = 2176 \text{ inches}
\]
Convert back to feet and inches:
\[
2176 \div 12 = 181 \text{ feet } 4 \text{ inches}
\]
The total length of rope needed is 181 feet 4 inches.
## 6. Length of Shoreline
What is the total length of Clear Lake's shoreline?
- A. 22 miles
- B. 44 miles
- C. 48 miles
- D. 56 miles
- E. 84 miles
**Solution:**
Assuming the diagram shows a loop with segments adding up to 44 miles, the total length of Clear Lake's shoreline is 44 miles.
## 7. Perimeter of Glass Tabletop
A glass tabletop is supported by a rectangular pedestal. If the tabletop is 8 inches wider than the pedestal on each side, what is the perimeter of the glass tabletop?
- A. 92 inches
- B. 116 inches
- C. 176 inches
- D. 184 inches
- E. 232 inches
**Solution:**
Assuming the pedestal is 48 inches by 36 inches, the tabletop dimensions are:
\[
48 + 2 \times 8 = 64 \text{ inches (length)}
\]
\[
36 + 2 \times 8 = 52 \text{ inches (width)}
\]
Calculate the perimeter:
\[
2 \times (64 + 52) = 232 \text{ inches}
\]
The perimeter of the glass tabletop is 232 inches.
## 8. Monthly Cat Food Usage
Rob uses 1 box of cat food every 5 days. Approximately how many boxes of cat food does he use per month?
- A. 2 boxes
- B. 4 boxes
- C. 5 boxes
- D. 6 boxes
- E. 7 boxes
**Solution:**
Assuming a month has 30 days:
\[
\frac{30}{5} = 6 \text{ boxes}
\]
Rob uses approximately 6 boxes of cat food per month.
## 9. Developing Film
Tara can develop 2 rolls of film in about 18 minutes. At this rate, how long will it take her to develop 8 rolls of film?
- A. 42 minutes
- B. 1 hour 12 minutes
- C. 1 hour 20 minutes
- D. 1 hour 44 minutes
- E. 2 hours 24 minutes
**Solution:**
Calculate the time for 1 roll:
\[
\frac{18}{2} = 9 \text{ minutes per roll}
\]
Calculate the time for 8 rolls:
\[
8 \times 9 = 72 \text{ minutes} = 1 \text{ hour } 12 \text{ minutes}
\]
It will take Tara 1 hour 12 minutes to develop 8 rolls of film.
## 10. Probability of Drawing Marbles
Liliana has a bag of marbles. The bag contains 18 black, 15 red, 11 blue, and 8 white marbles. Liliana randomly takes a red marble from the bag and leaves the marble on a table. What is the probability that she will next take a red or a white marble from the bag?
- A. \(\frac{112}{2601}\)
- B. \(\frac{2}{17}\)
- C. \(\frac{11}{26}\)
- D. \(\frac{22}{51}\)
- E. \(\frac{23}{52}\)
**Solution:**
After removing one red marble, the bag contains:
- 17 red marbles
- 8 white marbles
- Total: 18 + 14 + 11 + 8 = 51 marbles
Calculate the probability:
\[
\frac{17 + 8}{51} = \frac{25}{51}
\]
The probability is \(\frac{25}{51}\), which simplifies to \(\frac{25}{51}\).
## 11. Seniors Attending Graduate School
At a college, approximately 2 out of 5 seniors go on to attend graduate school. If there are 750 seniors at the college, how many would be expected to attend graduate school?
- A. 75 seniors
- B. 107 seniors
- C. 150 seniors
- D. 214 seniors
- E. 300 seniors
**Solution:**
Calculate the expected number:
\[
\frac{2}{5} \times 750 = 300
\]
300 seniors are expected to attend graduate school.
## 12. Difference in Library Books
The Mills Library has 1,007,199 books. The Springvale Library has 907,082 books. Which of the following is the best estimate of how many more books the Mills Library has than the Springvale Library?
- A. 100,000 books
- B. 80,000 books
- C. 10,000 books
- D. 8,000 books
- E. 1,000 books
**Solution:**
Calculate the difference:
\[
1,007,199 - 907,082 = 100,117
\]
The best estimate is 100,000 books.
## 13. Estimate Product
Which of the following is the best estimate for \(4,286 \times 390\)?
- A. 12,000,000
- B. 1,600,000
- C. 1,200,000
- D. 16,000
- E. 12,000
**Solution:**
Estimate the product:
\[
4,286 \approx 4,000 \quad \text{and} \quad 390 \approx 400
\]
\[
4,000 \times 400 = 1,600,000
\]
The best estimate is 1,600,000.
## 14. Distance Between Extremes
The lowest point on Earth is the bottom of the Mariana Trench at a depth of 35,840 feet below sea level. The highest point on Earth is the summit of Mt. Everest at a height of 29,028 feet above sea level. Which of the following is the best estimate of the distance between the lowest and highest points on Earth?
- A. 6,000 feet
- B. 7,000 feet
- C. 64,000 feet
- D. 65,000 feet
- E. 66,000 feet
**Solution:**
Calculate the total distance:
\[
35,840 + 29,028 = 64,868 \text{ feet}
\]
The best estimate is 65,000 feet.
## 15. Test Score Interpretation
Kim's test scores indicate that:
- A. he scored as well as or better than 72 of the test takers.
- B. 28% of the test takers scored better than he did.
- C. he would perform adequately in a twelfth grade science class.
- D. he scored as well as or better than 88% of the test takers.
- E. he answered 72% of the questions correctly.
**Solution:**
Kim's percentile score is 88, meaning he scored as well as or better than 88% of the test takers.
## 16. Simplifying Expression
The expression \(-105 + (-14) + 34\) simplifies to which of the following?
- A. –57
- B. –75
- C. –85
- D. 143
- E. 153
**Solution:**
Simplify the expression:
\[
-105 - 14 + 34 = -119 + 34 = -85
\]
The expression simplifies to –85.
## 17. Temperature Difference
As part of a unit on weather, students recorded the outdoor temperature at 8:30 A.M. for five mornings. What was the difference between the week's highest and lowest morning temperatures?
| Day | Temperature (°C) |
|-----------|------------------|
| Monday | 7 |
| Tuesday | 12 |
| Wednesday | 4 |
| Thursday | –13 |
| Friday | –15 |
- A. –3°C
- B. –1°C
- C. 8°C
- D. 22°C
**Solution:**
Find the highest and lowest temperatures:
- Highest: 12°C
- Lowest: –15°C
Calculate the difference:
\[
12 - (-15) = 12 + 15 = 27°C
\]
The difference is 27°C, but the closest option is 22°C.
- eval_id:
cais/mmlu-high_school_physics-test-decontamination-132 - doc_id:
8ff4b76169a616eede86cb463697db65(partition 47) - n_matches: 16 · max_overlap: 1.0
question: What happens to the force of gravitational attraction between two small objects if the mass of each object is doubled and the distance between their centers is doubled?
options:
A. It is doubled.
B. It is quadrupled.
C. It is halved.
D. It remains the same.
answer: D (It remains the same.)
subset: high_school_physics
Document (click to expand)
# AP Physics 1 Practice Test 1
## Test Information
- **Question count**: 10 questions
- **Time**: 18 minutes
### Questions
1. **Gravitational Force**
What happens to the force of gravitational attraction between two small objects if the mass of each object is doubled and the distance between their centers is doubled?
- \( \text{A. It is doubled.} \)
- \( \text{B. It is quadrupled.} \)
- \( \text{C. It is halved.} \)
- \( \text{D. It remains the same.} \)
**Explanation**: The gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:
\[
F = G \frac{m_1 m_2}{r^2}
\]
If each mass is doubled, the force becomes \( G \frac{(2m_1)(2m_2)}{r^2} = 4G \frac{m_1 m_2}{r^2} \). If the distance is also doubled, it becomes:
\[
F = G \frac{4m_1 m_2}{(2r)^2} = G \frac{4m_1 m_2}{4r^2} = G \frac{m_1 m_2}{r^2}
\]
Thus, the force remains the same.
2. **Inclined Plane (Frictionless)**
If the surface of the incline is frictionless, how long will the block take to reach the bottom if it was released from rest at the top?
- \( \text{A. 0.5 s} \)
- \( \text{B. 1.0 s} \)
- \( \text{C. 1.4 s} \)
- \( \text{D. 2.0 s} \)
3. **Inclined Plane (Frictionless)**
If the surface of the incline is frictionless, with what speed will the block reach the bottom if it was released from rest at the top?
- \( \text{A. 8 m/s} \)
- \( \text{B. 10 m/s} \)
- \( \text{C. 14 m/s} \)
- \( \text{D. 18 m/s} \)
4. **Inclined Plane (With Friction)**
If the coefficient of friction between the block and the incline is 0.4, how much work is done by the normal force on the block as it slides down the full length of the incline?
- \( \text{A. 0 J} \)
- \( \text{B. 2.0 J} \)
- \( \text{C. 4.0 J} \)
- \( \text{D. 4.9 J} \)
5. **Orbiting Satellites**
Two satellites are in circular orbit around the Earth. The distance from Satellite 1 to Earth’s center is \( r_1 \), and the distance from Satellite 2 to the Earth’s center is \( r_2 \). What is the speed of Satellite 2?
6. **Wave Analysis**
The graph above shows a wave. What quantity of the wave does the indicated segment represent?
- \( \text{A. Frequency} \)
- \( \text{B. Wavelength} \)
- \( \text{C. Period} \)
- \( \text{D. Cannot be determined} \)
7. **Sound Wave Transition**
Sound waves travel at 350 m/s through warm air and at 3,500 m/s through brass. What happens to the wavelength of a 700 Hz acoustic wave as it enters brass from warm air?
- \( \text{A. It decreases by a factor of 20.} \)
- \( \text{B. It decreases by a factor of 10.} \)
- \( \text{C. It increases by a factor of 10.} \)
- \( \text{D. The wavelength remains unchanged.} \)
**Explanation**: The wavelength \( \lambda \) is given by \( \lambda = v/f \). In air, \( \lambda_{\text{air}} = 350/700 = 0.5 \) m, and in brass, \( \lambda_{\text{brass}} = 3500/700 = 5 \) m. Thus, the wavelength increases by a factor of 10.
8. **Pulley System**
In the figure above, the coefficient of sliding friction between the small block and the tabletop is 0.2. If the pulley is frictionless and massless, what will be the acceleration of the blocks once they are released from rest?
- \( \text{A. } 0.5g \)
- \( \text{B. } 0.6g \)
- \( \text{C. } 0.7g \)
- \( \text{D. } 0.8g \)
9. **Newton's Third Law on Ice**
Two people, one of mass 100 kg and the other of mass 50 kg, stand facing each other on an ice-covered (essentially frictionless) pond. If the heavier person pushes on the lighter one with a force \( F \), then
- \( \text{A. the force felt by the heavier person is } (-1/2)F \)
- \( \text{B. the force felt by the person is } -2F \)
- \( \text{C. the magnitude of the acceleration of the lighter person will be half of the magnitude of the acceleration of the heavier person} \)
- \( \text{D. the magnitude of the acceleration of the lighter person will be twice the magnitude of the acceleration of the heavier person} \)
**Explanation**: By Newton's Third Law, the force on both is equal and opposite. The acceleration \( a \) is given by \( F = ma \). Therefore, for the lighter person, \( a_1 = F/50 \), and for the heavier person, \( a_2 = F/100 \). Thus, \( a_1 = 2a_2 \).
10. **Electrostatics**
The figure above shows two positively charged particles. The \( +Q \) charge is fixed in position, and the \( +Q \) charge is brought close to \( +Q \) and released from rest. Which of the following graphs best depicts the acceleration of the \( +Q \) charge as a function of its distance \( R \) from \( +Q \)?
- Graph options not provided in text.
This content is derived from AP Physics 1 practice materials, focusing on physics concepts and problem-solving.
- eval_id:
cais/mmlu-college_mathematics-test-decontamination-63 - doc_id:
5aded0ea598e35885934f112cb0ce4f1(partition 54) - n_matches: 19 · max_overlap: 0.8260869565217391
question: In a game two players take turns tossing a fair coin; the winner is the first one to toss a head. The probability that the player who makes the first toss wins the game is
options:
A. 1/4
B. 1/3
C. 1/2
D. 2/3
answer: D (2/3)
subset: college_mathematics
Document (click to expand)
# A Problem in Probability
**Date: Jun 6, 2004**
### User: lhuyvn
Hi members,
I have traveled this forum sometimes, but this is my first question. I hope to get your help so that I can prepare better for my GRE Math test.
Following is my question:
In a game, two players take turns tossing a fair coin; the winner is the first one to toss a head. The probability that the player who makes the first toss wins the game is:
- A) \( \frac{1}{4} \)
- B) \( \frac{1}{3} \)
- C) \( \frac{1}{2} \)
- D) \( \frac{2}{3} \)
- E) \( \frac{3}{4} \)
Thanks in advance,
LuuTruongHuy
---
### User: maverick280857
Here's my solution:
Suppose \( A \) starts first. Then the different possibilities are tabulated thus:
- \( AH \) (A gets a head, game stops)
- \( AT, BT, AH \) (A gets tails, B gets tails, A gets heads, game stops)
- \( AT, BT, AT, BT, AH \) (A gets tails, B gets tails, A gets tails, B gets tails, A gets heads, game stops)
- and so on....
So the probability is given by the sum:
\[ \frac{1}{2} + \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} + \ldots \]
The \( k \)th term is
\[ \left(\frac{1}{2}\right)^p \]
where \( p = 2k+1 \) for \( k = 0, 1, 2, \ldots \) (note that there are \( 2k+1 \) continued products in the \( k \)th term).
The game goes on as long as \( A \) and \( B \) get tails and stops as soon as \( A \) gets a head, since \( A \) was the one who started the game first.
This is an infinite sum, the value of which is given by
\[ \text{SUM} = \frac{\frac{1}{2}}{1 - \frac{1}{4}} = \frac{2}{3} \]
I think \( \frac{2}{3} \) should be the answer, but I could be wrong (as usual) ;-)
Someone please correct me if I'm wrong. If any part of the solution is wrong or not clear, please let me know. (I have assumed that you are familiar with addition and multiplication in probability and also with geometric progressions, especially containing an infinite number of terms—the kinds that appear in such problems.)
Cheers,
Vivek
---
### User: uart
The answer is \( \frac{2}{3} \).
The first player has a probability of \( \frac{1}{2} \) that both he will take the first toss and that he will win on that toss. The second player only has a probability of \( \frac{1}{4} \) that he will both take his first toss and win on that toss. The first player then has a probability of \( \frac{1}{8} \) that he will both require his second toss and win on that toss. Continuing on like this, the first player has a probability of \( \frac{1}{2} + \frac{1}{8} + \frac{1}{32} + \ldots \) and the second player has a probability of \( \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \ldots \) of winning.
---
### User: Hurkyl
For those who like clever answers, you can skip the infinite series. 😊
Suppose the first player's first flip is a tails. Now, if you look at how the game proceeds, it is identical to the original game, except the first and second players are reversed.
So if \( p \) is the probability that the first player in the game wins, then once the first player flips a tails, the second player has a probability \( p \) of winning. (and probability 0 of winning otherwise)
Since there's a \( \frac{1}{2} \) chance the first player will flip tails, the second player has a probability \( \frac{p}{2} \) of winning, and the first player a probability \( p \).
Thus, \( p = \frac{2}{3} \).
---
### User: lhuyvn
Thank you all for the very nice answers!
---
- eval_id:
cais/mmlu-college_mathematics-test-decontamination-78 - doc_id:
2fa7078033e91569d14c42b349e6a7cb(partition 89) - n_matches: 13 · max_overlap: 0.5909090909090909
question: A total of x feet of fencing is to form three sides of a level rectangular yard. What is the maximum possible area of the yard, in terms of x ?
options:
A. x^2/9
B. x^2/8
C. x^2/4
D. x^2
answer: B (x^2/8)
subset: college_mathematics
Document (click to expand)
# Perimeter vs. Area
**Question:**
A total of \( x \) feet of fencing is to form three sides of a level rectangular yard. What is the maximum possible area of the yard, in terms of \( x \)?
**Answer:**
The problem can be approached by setting up the equations for the perimeter and area. Let \( L \) be the length and \( W \) be the width of the yard. Since only three sides are fenced, the equation for the fencing is:
\[ x = 2W + L \]
The area \( A \) of the rectangle is given by:
\[ A = L \times W \]
Substituting \( L = x - 2W \) from the fencing equation into the area equation gives:
\[ A = (x - 2W) \times W = xW - 2W^2 \]
To find the maximum area, take the derivative of \( A \) with respect to \( W \) and set it to zero:
\[ \frac{d}{dW}(xW - 2W^2) = x - 4W = 0 \]
Solving for \( W \), we get:
\[ W = \frac{x}{4} \]
Substituting back to find \( L \):
\[ L = x - 2\left(\frac{x}{4}\right) = \frac{x}{2} \]
Thus, the maximum area is:
\[ A = \left(\frac{x}{2}\right)\left(\frac{x}{4}\right) = \frac{x^2}{8} \]
**Discussion:**
- **moo5003** (Aug 18, 2008): Initially guessed \( \frac{x^2}{9} \) assuming a square, but realized that maximizing area involves setting \( W = \frac{x}{4} \) and \( L = \frac{x}{2} \), resulting in \( \frac{x^2}{8} \).
- **symbolipoint** (Aug 18, 2008): Confirmed the approach, noting the quadratic form with a negative coefficient, indicating a maximum point.
- **dynamicsolo** (Aug 18, 2008): Agreed with the solution and noted that the maximal area for fencing problems is not always a square unless specific symmetry conditions are met.
- **kimhs** (Jan 24, 2011): Provided an alternative method using line intercepts, arriving at the same maximum area of \( \frac{x^2}{8} \).
This problem is relevant for understanding optimization in calculus, particularly in contexts like the Math GRE.
- eval_id:
cais/mmlu-elementary_mathematics-test-decontamination-145 - doc_id:
442da4063a33a85a68289c0379fce29d(partition 205) - n_matches: 20 · max_overlap: 1.0
question: A shape is mad eof 12 right triangles of equal size. Each right triangle has a base of 4 cm and a height of 5 cm. What is the total area, in square centimeters, of the shape?
options:
A. 10
B. 60
C. 120
D. 240
answer: C (120)
subset: elementary_mathematics
Document (click to expand)
# Caddell Prep: Educational Content
## 2018 Grade 6 Math Regents
### Problem 1
An equation is shown below:
\[ 12 - 9 + c = 12 \]
What value of \( c \) makes the equation true?
- A. 0
- B. 3
- C. 9
- D. 12
### Problem 2
Kate has a coin collection. She keeps 7 of the coins in a box, which is only 5% of her entire collection. What is the total number of coins in Kate’s coin collection?
- A. 12
- B. 14
- C. 120
- D. 140
### Problem 3
What is the greatest common factor of 36 and 90?
- A. 6
- B. 18
- C. 36
- D. 180
### Problem 4
The relationship between Robert’s age, \( r \), and Julia’s age, \( j \), can be represented by the equation:
\[ r = j + 3 \]
Which table of values represents the relationship between Robert’s age and Julia’s age?
- A.
- B.
- C.
- D.
### Problem 14
All the students in the sixth grade either purchased their lunch or brought their lunch from home on Monday.
- 24% of the students purchased their lunch.
- 190% students brought their lunch from home.
How many students are in the sixth grade?
- A. 76
- B. 166
- C. 214
- D. 250
### Problem 16
Joe walks on a treadmill at a constant rate. The equation below describes the relationship between \( t \), the time he walks in hours, and \( d \), the distance he walks in miles:
\[ d = 4t \]
Which graph represents the relationship between the amount of time Joe walks and the distance he walks?
- A.
- B.
- C.
- D.
### Problem 19
There are 230 calories in 4 ounces of a type of ice cream. How many calories are in 6 ounces of that ice cream?
- A. 232
- B. 345
- C. 345
- D. 460
### Problem 22
A shape is made of 12 right triangles of equal size. Each right triangle has a base of 4 cm and a height of 5 cm. What is the total area, in square centimeters, of the shape?
- A. 10
- B. 60
- C. 120
- D. 240
### Problem 25
Pat bounces a basketball 25 times in 30 seconds. At that rate, approximately how many times will Pat bounce the ball in 150 seconds?
- A. 120
- B. 125
- C. 144
- D. 145
### Problem 26
Which expression is equivalent to \( 5(4x + 3) - 2x \)?
- A. \( 18x + 15 \)
- B. \( 18x + 3 \)
- C. \( 7x + 8 \)
- D. \( 2x + 8 \)
### Problem 27
Mark graphed points on the coordinate plane to represent the locations of his school and a bank. Mark wants to add the location of the library on the coordinate plane. The distance from the library to the school is the same as the distance from the bank to the school. Which ordered pair could be the coordinates of the library?
- A. (2, 4)
- B. (2, 8)
- C. (4, 4)
- D. (6, 8)
### Problem 28
A student draws the net below to show the dimensions of a container that is shaped like a right rectangular prism. What is the surface area, in square inches, of the container?
- A. 19
- B. 30
- C. 38
- D. 62
### Problem 29
Which two expressions are equivalent?
- A. \( x + x + x \) and \( x^3 \)
- B. \( 14x + 10 - 2x \) and \( 16x + 10 \)
- C. \( 12x + 16x \) and \( 4(3x + 4x) \)
- D. \( 12x^2 + 5x + 10 \) and \( 17x^2 + 10 \)
### Problem 30
A machine fills boxes at a constant rate. At the end of 35 minutes, it has filled 5 boxes. Which table represents the relationship between the number of minutes the machine fills boxes and the number of boxes it has filled?
- A.
- B.
- C.
- D.
### Problem 31
Which expression represents the perimeter of the figure below?
- A. \( 5x + 2y \)
- B. \( x + y + z \)
- C. \( 5x + 2y + z \)
- D. \( (5 + 2 + 1)(x + y + z) \)
### Problem 32
The elevations, in feet, of three cities are marked on the number line. The point 0 on the number line represents sea level. Which statement must be true?
- A. City P and City Q are above sea level.
- B. City Q and City R are below sea level.
- C. City P is above sea level and City Q is below sea level.
- D. City P is above sea level and City R is below sea level.
### Problem 33
A basketball player attempts 15 baskets in a game. He makes 9 of the attempted baskets. Which ratio describes the number of baskets the player made to the number of baskets the player attempted?
- A. \( \dfrac{3}{5} \)
- B. \( \dfrac{5}{3} \)
- C. \( \dfrac{2}{5} \)
- D. \( \dfrac{5}{2} \)
### Problem 34
Which number line shows a graph of the inequality \( x < -25 \)?
- A.
- B.
- C.
- D.
### Problem 35
The coordinates of the points below represent the vertices of a rectangle:
- \( P: (2, 2) \)
- \( Q: (6, 2) \)
- \( R: (6, 5) \)
- \( S: (2, 5) \)
What is the perimeter, in units, of rectangle PQRS?
- A. 8
- B. 12
- C. 14
- D. 16
### Problem 36
Carol has \( 1\dfrac{5}{8} \) cups of yogurt to make smoothies. Each smoothie uses \( \dfrac{1}{3} \) cup of yogurt. What is the maximum number of smoothies that Carol can make with the yogurt?
- A. 1
- B. 4
- C. 5
- D. 7
### Problem 37
Which expression is equivalent to \( 60 - 3y - 9 \)?
- A. \( 3(17 - y) \)
- B. \( 3(20 - y) - 3 \)
- C. \( 17(3 - y) \)
- D. \( 20(3 - 3y) - 9 \)
### Problem 38
A grocery store sells a bag of 5 lemons for \$2.00. What is the unit cost of each lemon in the bag?
- A. \$2.50
- B. \$0.60
- C. \$0.40
- D. \$0.10
- eval_id:
cais/mmlu-high_school_statistics-test-decontamination-179 - doc_id:
df575796823c7cf95f25c3503ca5f1e9(partition 49) - n_matches: 14 · max_overlap: 0.96875
question: The probability is 0.2 that a value selected at random from a normal distribution with mean 600 and standard deviation 15 will be above what number?
options:
A. 0.84
B. 603.8
C. 612.6
D. 587.4
answer: C (612.6)
subset: high_school_statistics
Document (click to expand)
# AP Statistics Practice Test 9
## Test Information
- **Questions:** 10
- **Time:** 23 minutes
### Questions
1. Eighteen trials of a binomial random variable \( X \) are conducted. If the probability of success for any one trial is 0.4, write the mathematical expression you would need to evaluate to find \( P(X = 7) \). Do not evaluate.
- **Expression:**
\[
P(X = 7) = \binom{18}{7} (0.4)^7 (0.6)^{11}
\]
2. Two variables, \( x \) and \( y \), seem to be exponentially related. The natural logarithm of each \( y \) value is taken and the least-squares regression line of \( \ln(y) \) on \( x \) is determined to be \( \ln(y) = 3.2 + 0.42x \). What is the predicted value of \( y \) when \( x = 7 \)?
- **Predicted Value:**
\[
\ln(y) = 3.2 + 0.42 \times 7 = 6.14
\]
\[
y = e^{6.14} \approx 464.05
\]
3. You need to construct a 94% confidence interval for a population proportion. What is the upper critical value of \( z \) to be used in constructing this interval?
- **Upper Critical Value:**
\( z = 1.88 \)
4. Which of the following best describes the shape of the histogram at the left?
- **Answer:** This requires visual inspection of the histogram. Options typically include:
- Approximately normal
- Skewed left
- Skewed right
- Approximately normal with an outlier
- Symmetric
5. The probability is 0.2 that a value selected at random from a normal distribution with mean 600 and standard deviation 15 will be above what number?
- **Calculation:**
Using the z-score formula:
\[
z = \frac{X - \mu}{\sigma}
\]
For \( P(Z > z) = 0.2 \), \( z \approx 0.84 \).
\[
0.84 = \frac{X - 600}{15}
\]
\[
X = 0.84 \times 15 + 600 = 612.6
\]
6. Which of the following are examples of continuous data?
- **Examples:**
I. The speed your car goes
III. The average temperature in San Francisco
IV. The wingspan of a bird
- **Answer:** I, III, and IV only
7. Use the following computer output for a least-squares regression for Questions below.
- **Equation of the Least-Squares Regression Line:**
\[
\hat{y} = 22.94 + 0.5466x
\]
8. Use the following computer output for a least-squares regression for Questions below.
- **P-value for the t-test of the hypothesis \( H_0: \beta = 0 \) versus \( H_A: \beta \neq 0 \):**
\[
0.01 < P < 0.05
\]
9. "A hypothesis test yields a P-value of 0.20." Which of the following best describes what is meant by this statement?
- **Explanation:**
The probability of getting a finding at least as extreme as that obtained by chance alone if the null hypothesis is true is 0.20.
10. A random sample of 25 men and a separate random sample of 25 women are selected to answer questions about attitudes toward abortion. The answers were categorized as "pro-life" or "pro-choice." Which of the following is the proper null hypothesis for this situation?
- **Null Hypothesis:**
The variables "gender" and "attitude toward abortion" are independent.
- eval_id:
cais/mmlu-elementary_mathematics-test-decontamination-130 - doc_id:
b67199fa24d987a7d801e8acb769fdbb(partition 5) - n_matches: 7 · max_overlap: 1.0
question: The distance between Miriam’s house and Debbie’s house is 444.44 meters. Which statement about the values of the digits in the distance, in meters, between their houses is true?
options:
A. The value of the 4 in the tenths place is 1/10 the value of the 4 in the tens place.
B. The value of the 4 in the hundredths place is 1/10 the value of the 4 in the ones place.
C. The value of the 4 in the hundreds place is 10 times greater than the value of the 4 in the ones place.
D. The value of the 4 in the tenths place is 10 times greater than the value of the 4 in the hundredths place.
answer: D (The value of the 4 in the tenths place is 10 times greater than the value of the 4 in the hundredths place.)
subset: elementary_mathematics
Document (click to expand)
# Fifth Grade (Grade 5) Place Value Questions
## Write the Number in Standard Form
Write the number in standard form.
- $7 + 0.2 + 0.05 + 0.009$
1. 7.259
2. 72.59
3. 725.9
4. 7,259
## Rounding Decimals
A gas station sold 300.5849 gallons of gas in a day. How many gallons of gas did the gas station sell, rounded to the nearest hundredth?
- A. 300
- B. 300.58
- C. 300.585
- D. 300.59
## Identify Place Values
In the number 8,592,436, which digit is in the thousands place?
- 1. 2
- 2. 5
- 3. 7
- 4. 8
## Value of a Digit
What is the value of the 2 in the following number? 529,307,604,000
- 1. Hundred billion
- 2. Ten billion
- 3. Ten million
- 4. One million
## Rounding Numbers
Round the following number to the nearest 100.
- 126
1. 100
2. 150
3. 200
## Writing Numbers in Standard Form
Write five million, two hundred seventy-five thousand, three hundred twelve in standard form.
- 5,275,312
## Understanding Place Values
The distance between Mary's house and Diana's house is 555.55 meters. Which statement about the values of the digits in the distance, in meters, between their houses is true?
- 1. The value of the 5 in the tenths place is 10 times greater than the value of the 5 in the hundredths place.
- 2. The value of the 5 in the tenths place is 1/10 the value of the 5 in the tens place.
- 3. The value of the 5 in the hundredths place is 1/10 the value of the 5 in the ones place.
- 4. The value of the 5 in the hundreds place is 10 times greater than the value of the 5 in the ones place.
## Rearranging Digits
Rearrange digits below to make the largest number possible.
- 129378
1. 987,321
2. 123,789
3. 879,123
4. 978,123
## Writing Numbers in Expanded Form
Write the number 678,425 in expanded form.
- 1. 600,000 + 70,000 + 8,000 + 400 + 20 + 5
- 2. 60,000 + 70,000 + 8,000 + 400 + 20 + 5
- 3. 600,000 + 70,000 + 800 + 400 + 20 + 5
- 4. 600,000 + 70,000 + 8,000 + 400 + 5
## Rounding to the Nearest Hundred
Round 5,634 to the nearest hundred.
- 1. 5,700
- 2. 5,630
- 3. 5,600
- 4. 5,640
## Rounding to the Nearest Thousand
Round 25,329 to the nearest thousand.
- 1. 25,000
- 2. 26,000
- 3. 25,329
- 4. 25,300
## Identifying Place Values
Which digit in the numeral 563,789 is in the ten thousands place?
- 1. 5
- 2. 6
- 3. 3
- 4. 9
## Value of a Digit
What is the value of 6 in 6,209?
- 1. 6
- 2. 600
- 3. 6,000
- 4. 600,000
## Identifying Place Values
Identify which number is in the tens place in the number 127,985,630.
- 1. 1
- 2. 3
- 3. 8
- 4. 5
## Understanding Decimal Operations
2 hundreds 7 hundredths - 5 thousandths = 200.065
## Writing Numbers in Standard Form
What is this number in standard form?
- 600,000 + 40,000 + 2,000 + 40 + 7
1. 642,407
2. 6,424,700
3. 642,047
4. 64,247
## Writing Numbers in Word Form
A grain of sand is measured. It had a diameter of 0.049 millimeter. What is 0.049 written in word form?
- 1. forty-nine
- 2. forty-nine tenths
- 3. forty-nine hundredths
- 4. forty-nine thousandths
## Understanding Decimal Operations
9 ones 3 thousandths - 2 one 5 hundredths = 6.953
## Identifying Place Values
Identify which number is in the hundred millions place in the number 127,985,630.
- 1. 1
- 2. 3
- 3. 8
- 4. 5
## Rounding Decimals
A gas station sold 658.5849 gallons of gas in a day. How many gallons of gas did the gas station sell, rounded to the nearest tenth?
- 1. 660
- 2. 658.6
- 3. 700
- 4. 658.58
- eval_id:
cais/mmlu-college_mathematics-test-decontamination-78 - doc_id:
64ec55a1036b55511959b6ac35496c8c(partition 76) - n_matches: 13 · max_overlap: 0.7222222222222222
question: A total of x feet of fencing is to form three sides of a level rectangular yard. What is the maximum possible area of the yard, in terms of x ?
options:
A. x^2/9
B. x^2/8
C. x^2/4
D. x^2
answer: B (x^2/8)
subset: college_mathematics
Document (click to expand)
# Perimeter vs. Area
## Question:
A total of \(x\) feet of fencing is to form three sides of a level rectangular yard. What is the maximum possible area of the yard, in terms of \(x\)?
### Problem Explanation:
A student considers the maximum area of a yard using only three sides of fencing. The student proposes that if the yard is approximately square, the maximum area could be \(\frac{x^2}{9}\). They explore that extending one side may yield a larger area since only one side would need fencing.
### Approach:
1. Let \(A\) be the area, \(x\) the total fencing, \(L\) the length, and \(W\) the width.
2. The area is given by \(A = L \times W\).
3. The fencing constraint is \(x = 2W + L\).
4. Rearranging gives \(L = \frac{x}{2} - 2W\).
5. Substituting into the area formula: \(A = W\left(\frac{x}{2} - 2W\right)\).
6. Simplifying: \(A = \frac{xW}{2} - 2W^2\).
7. To maximize, take the derivative with respect to \(W\) and set to zero: \(\frac{dA}{dW} = \frac{x}{2} - 4W = 0\).
8. Solving gives \(W = \frac{x}{4}\).
9. Substitute back to find \(L = \frac{x}{2}\).
10. The maximum area is \(A = \left(\frac{x}{2}\right)\left(\frac{x}{4}\right) = \frac{x^2}{8}\).
### Discussion:
- **moo5003**: This method maximizes the area at \(W = \frac{x}{4}\) and \(L = \frac{x}{2}\), resulting in an area of \(\frac{x^2}{8}\).
- **symbolipoint**: Confirms the approach, noting the quadratic nature and the negative coefficient leading to a maximum.
- **dynamicsolo**: Agrees and highlights that a square solution occurs only with full perimeter fencing or symmetric partitions.
- **kimhs**: Provides an alternative method using line intercepts, confirming the same maximum area of \(\frac{x^2}{8}\).
### Conclusion:
For a rectangular yard fenced on three sides with total fencing \(x\), the maximum area is \(\frac{x^2}{8}\), with dimensions \(L = \frac{x}{2}\) and \(W = \frac{x}{4}\).
- eval_id:
cais/mmlu-elementary_mathematics-test-decontamination-78 - doc_id:
93b6b4fcfc07f11f1c8946a6da881ca9(partition 179) - n_matches: 1 · max_overlap: 0.8235294117647058
question: Write the ratio of 2 cups to 3 qt as a fraction in simplest form.
options:
A. 3 over 2
B. 2 over 3
C. 2 over 12
D. 1 over 6
answer: D (1 over 6)
subset: elementary_mathematics
Document (click to expand)
## Ratio Test: Math Quiz!
**14 Questions | By NCrabtree | Last updated: Jan 27, 2021 | Total Attempts: 282**
### Questions and Answers
1. **Matt sold 35 tickets to the school play and Renee sold 45 tickets. What is the ratio of the number of tickets Matt sold to the number of tickets Renee sold?**
- A. \( \frac{9}{7} \)
- B. \( \frac{7}{16} \)
- C. \( \frac{7}{9} \)
- D. \( \frac{16}{7} \)
2. **Which ratios are equal to 4:32?**
- A. 2 : 96; 2 : 16
- B. 12 : 96; 2 : 16
- C. 12 : 96; 12 : 16
- D. 2 : 96; 12 : 16
3. **The American flag is usually made with its width and length in the ratio of 10 to 19. Which dimensions are in the correct ratio for the flag?**
- A. 30 ft by 53 ft
- B. 27 ft by 53 ft
- C. 27 in. by 57 in.
- D. 30 in. by 57 in.
4. **Find a ratio equivalent to \( \frac{9}{21} \).**
- A. \( \frac{9}{21} \)
- B. \( \frac{24}{9} \)
- C. \( \frac{15}{40} \)
- D. \( \frac{15}{45} \)
5. **Write the ratio of 96 runners to 216 swimmers in simplest form.**
- A. \( \frac{96}{216} \)
- B. \( \frac{16}{36} \)
- C. \( \frac{4}{9} \)
- D. \( \frac{9}{4} \)
6. **Write the ratio of 2 cups to 3 qt as a fraction in simplest form.**
- A. \( \frac{3}{2} \)
- B. \( \frac{2}{3} \)
- C. \( \frac{2}{12} \)
- D. \( \frac{1}{6} \)
7. **Find a ratio equivalent to 3:25.**
- A. 1:9
- B. 9:225
- C. 9:75
- D. 75:9
8. **Which fraction shows the ratio of 3 dogs to 5 dogs?**
- A. \( \frac{6}{10} \)
- B. \( \frac{9}{10} \)
- C. 27:40
- D. 12:15
9. **Find a ratio equivalent to \( \frac{27}{81} \).**
- A. 3:10
- B. 9:9
- C. 1:3
- D. 5:9
10. **Maria tossed a coin 20 times and got 12 heads. What is the first step to find the ratio of the number of tails to the total number of tosses?**
- A. Divide 12 by 20
- B. Subtract 12 from 20
- C. Multiply 12 by 20
- D. Add 12 to 20
11. **12/24 equals 50/100.**
- A. True
- B. False
12. **1/3 equals 2/9.**
- A. True
- B. False
13. **2/3 equals 24/36.**
- A. True
- B. False
14. **16 to 3 equals 27 to 5.**
- A. True
- B. False
### Related Topics
- Proportion
- Fraction
- Decimal
### More Ratio Quizzes
- Math Quiz: Ratio And Proportion Practice Paper Questions!
- Ratios And Unit Rates
- Ratios, Proportions & Rate Vocabulary Quiz
- eval_id:
cais/mmlu-high_school_physics-test-decontamination-81 - doc_id:
c5e60cc3a8a23a256c893439f711fc45(partition 196) - n_matches: 36 · max_overlap: 0.7659574468085106
question: A cannon is mounted on a truck that moves forward at a speed of 5 m/s. The operator wants to launch a ball from a cannon so the ball goes as far as possible before hitting the level surface. The muzzle velocity of the cannon is 50 m/s. At what angle from the horizontal should the operator point the cannon?
options:
A. 5°
B. 41°
C. 45°
D. 49°
answer: D (49°)
subset: high_school_physics
Document (click to expand)
### Idealist Group Title
A cannon is mounted on a truck that moves forward at a speed of 5 m/s. The operator wants to launch a ball from a cannon so the ball goes as far as possible before hitting the level surface. The muzzle velocity of the cannon is 50 m/s. What angle from the horizontal should the operator point the cannon?
**Comments and Responses:**
1. **Mashy Group Title:**
You've already chosen the best response.
*Medals: 0*
You must know what is angle to get the maximum range for a two-dimensional projectile motion.
(one year ago)
2. **Idealist Group Title:**
You've already chosen the best response.
*Medals: 0*
I don't get it. Can you show me the work?
(one year ago)
3. **Mashy Group Title:**
You've already chosen the best response.
*Medals: 0*
First, you tell me what is the general expression for RANGE?
(one year ago)
4. **Idealist Group Title:**
You've already chosen the best response.
*Medals: 0*
All I know is that the cannon isn't stationary.
(one year ago)
5. **Mashy Group Title:**
You've already chosen the best response.
*Medals: 0*
You need to know a little bit about 2D projectile motion!
(one year ago)
---
**Discussion on the Problem:**
The problem can be approached by considering the relative motion of the cannonball. The initial velocity of the cannonball is a combination of the muzzle velocity and the velocity of the truck. The maximum range in projectile motion is achieved when the launch angle is 45 degrees relative to the horizontal, assuming no air resistance and level ground. However, because the cannon is on a moving truck, the optimal angle might differ slightly.
**Mathematical Analysis:**
The initial velocity of the cannonball relative to the ground is given by:
\[
\vec{v}_0 = \vec{v}_{\text{truck}} + \vec{v}_{\text{muzzle}}
\]
Where:
- \(\vec{v}_{\text{truck}} = 5 \, \text{m/s}\) (horizontal)
- \(\vec{v}_{\text{muzzle}} = 50 \, \text{m/s}\) at angle \(\theta\)
The horizontal and vertical components of the muzzle velocity are:
\[
v_{\text{muzzle},x} = 50 \cos \theta
\]
\[
v_{\text{muzzle},y} = 50 \sin \theta
\]
Thus, the total velocity components are:
\[
v_{x} = 5 + 50 \cos \theta
\]
\[
v_{y} = 50 \sin \theta
\]
The range \(R\) is given by:
\[
R = \frac{v_x v_y}{g}
\]
Substituting the expressions for \(v_x\) and \(v_y\):
\[
R = \frac{(5 + 50 \cos \theta)(50 \sin \theta)}{g}
\]
To maximize \(R\), take the derivative with respect to \(\theta\) and set it to zero. Solving this will give the optimal angle \(\theta\).