OpenLLM-France/wikimedia huggingface dataset:
# A-level Physics (Advancing Physics)/Stress, Strain & the Young Modulus/Worked Solutions

1. 10N of force are exerted on a wire with cross-sectional area 0.5mm$^2$. How much stress is being exerted on the wire?
0.5mm$^2$ = 0.5 x (10$^{−3}$)$^2$m$^2$ = 0.5 x 10$^{−6}$m$^2$
$\sigma ={\frac {10N}{0.5\times 10^{-6}m^{2}}}=20\ 000\ 000{\mbox{ Pa}}=20{\mbox{ MPa}}$
2. Another wire has a tensile strength of 70MPa, and breaks under 100N of force. What is the cross-sectional area of the wire just before breaking?
$70\times 10^{6}={\frac {100}{A}}$
$A={\frac {100}{70\times 10^{6}}}\approx 1.43\times 10^{-6}{\mbox{ m}}^{2}=1.43{\mbox{ }}\mu {\mbox{m}}^{2}$
3. What is the strain on a Twix bar (original length 10 cm) if it is now 12 cm long?
$E={\frac {12-10}{10}}={\frac {2}{10}}=0.2$
4. What is this strain, expressed as a percentage?
0.2 x 100 = 20%
5. 50N are applied to a wire with a radius of 1mm. The wire was 0.7m long, but is now 0.75m long. What is the Young's Modulus for the material the wire is made of?
$Y={\frac {({\frac {50}{\pi \times (1\times 10^{-}3)^{2}}})}{({\frac {0.75-0.7}{0.7}})}}\approx {\frac {16000000}{0.0714}}\approx 224000000{\mbox{ Pa}}=224{\mbox{ MPa}}$
6. Glass, a brittle material, fractures at a strain of 0.004 and a stress of 240 MPa. Sketch the stress-strain graph for glass.
7. (Extra nasty question which you won't ever get in an exam) What is the toughness of glass?
Toughness equals the area under the above graph. Since glass is brittle, we can assume that the gradient of the graph is constant, and, since the graph passes through the origin, it is a triangle. So:
$A_{triangle}={\frac {{\mbox{base }}\times {\mbox{ height}}}{2}}={\mbox{ Toughness }}={\frac {{\mbox{strain }}\times {\mbox{ stress}}}{2}}={\frac {0.004\times 240\times 10^{6}}{2}}=480000{\mbox{ Jm}}^{-3}=480{\mbox{ kJm}}^{-3}$
# Probabilities

The GRE frequently tests probability, which is the odds of a particular condition occurring.

## Rules

The probability of an event happening MORE THAN ONCE is equal to:
    (odds the 1st time)(odds the second 2nd)(etc.)
The probability of an event happening AT LEAST ONCE over many tries is:
1 - (odds of failure the first time)(odds of failure the second time)(etc.)
Probability is the likelihood of an event occurring. It is generally calculated as the number of desired outcomes desired by the total number of outcomes. Probability is sometimes expressed as a fraction, or as a decimal. When expressed as a decimal, a probability of “1” equals 100% chance, while a probability of .5 equals a 1 in 2 chance.
For example, when rolling a 6-sided die with faces numbered 1 to 6, the probability of rolling a 6 is 1 in 6, or 1.6666 repeating. (Assuming that the die is not imbalanced).
For the odds of rolling two 6s in a row on a 6-sided die, the formula is (odds the first time)(odds the second time).
The odds of rolling a six at least once over 2 tries on a 6-sided die. The formula is 1 - (combined odds of failure).

## Practice

1. Brian is sending out job applications to law firms. His odds of being hired at Firm A are 1 in 3; his odds at Firm B are 1 in 5, and his odds at Firm C are 1 in 7. What are the odds that Brian is hired by at least one of the law firms?
2. Melanie is shooting baskets at the gym. She has a 1 in 3 chance of making a three-point shot. What are the odds of Melanie making 2 shots in a row?
3. Darren has a 90% chance of being late for class each day. What are the odds that he shows up on time three times in a row?

## Comments

The distinction between the odds of something happening "every time" versus "at least once" is sometimes known as "straight" versus "cumulative" probability.
Probability can be one of the most complex areas of mathematics; however, GRE probability is usually relatively simple. The questions the GRE asks with regard to probability can almost always be solved with the above equations.

## Answers to Practice Questions

1. 57 out of 105
Since this is the odds of something happening at least once over multiple tries, the formula is 1 - (odds of failure the first time)(odds of failure the second time)(etc.).
The odds of success for Brian's applications are ${\tfrac {1}{3}}$, ${\tfrac {1}{5}}$ and ${\tfrac {1}{7}}$. Thus, the odds of failure for Brian's applications are ${\tfrac {2}{3}}$, ${\tfrac {4}{5}}$, and ${\tfrac {6}{7}}$.
Brian's odds of being hired by at least one of the firms are thus (1 - odds of failure), which works out to (1 - ${\tfrac {2}{3}}$(${\tfrac {4}{5}}$)(${\tfrac {6}{7}}$) or ${\tfrac {57}{105}}$
2. 1 out of 9
Since this is the odds of something happening more than once in a row, the formula is (odds the first time)(odds the second time)(etc.). The odds are: ${\tfrac {1}{3}}$(${\tfrac {1}{3}}$)
= ${\tfrac {1}{9}}$
3. 1 in 1000
Since this is the odds of something happening more than once in a row, the formula is (odds the first time)(odds the second time)(etc.). The odds are: ${\tfrac {1}{10}}$(${\tfrac {1}{10}}$)(${\tfrac {1}{10}}$)
= ${\tfrac {1}{1000}}$
# A Guide to the GRE/Pythagorean Theorem
## The Pythagorean Theorem

In a right triangle with sides a, b and c, a2 + b2 = c2, c being the longest side.
In this triangle, a2 + b2 = c2 and thus:
a =
b =
c =
There are a limited number of “Pythagorean Triples” or groups of integers which fit the formula.
The GRE tends to have questions which work out to integers; thus, keep an eye out for the “3-4-5” triangle and the “5-12-13” triangle.
The 3-4-5 Triangle
The squares of 3 and 4 add up to the square of 5, thus, the “3-4-5” triangle is common on the GRE. It may also be in the form of a “6-8-10” triangle or a “9-12-15” triangle, or another triangle similarly increased.
The 5-12-13 Triangle
In a similar manner the 3-4-5 triangle, this triangle has a series of integers which comport with the Pythagorean Theorem. This triple may also be increased to 10-24-26 and so on.
There are two special right triangles, the proportions of which can be inferred by their angle measurements.
A 45º-45º-90º triangle has sides in the ratio of 1 - 1 -
Thus if the shorter sides are 10, the longer side is
A 30º-60º-90º triangle has sides in the ratio of 1 -- 2.
Thus if the shortest side has a length of 5, the two other sides have lengths ofand 10.
Indeed, the proportions of every right triangle can be determined by the measurement of one of its angles based on this theorem. Mathematicians and engineers have developed tables of the proportions for these angles and have designated functions for them, such as sin and cos. This field of mathematics is known as “trigonometry” and is not tested on the GRE.

### Practice

1.
What is the value of q?
2.
Solve for r.
3.
Determine the value of s.

### Answers to Practice Questions

1.
In a right triangle with sides a, b and c, a2 + b2 = c2, where c is the longest side. Thus, q is equal to or
2.
In a right triangle with two equal sides, the longer side equals the shorter side multiplied by Thus, the shorter side equals the longer side multiplied by This means r equals 8 times this proportion, which equals or
3. 24
In a right triangle with sides a, b and c, a2 + b2 = c2, where c is the longest side. Thus, s equals which equals or 24. This is an example of the “Pythagorean Triple” of 5-12-13, in this case multiplied by 2.
# A Guide to the GRE/Quadrilaterals
## Quadrilaterals

The angles of a quadrilateral add up to 360º. There are 6 key types:
| | Square | Rectangle |
| Four equal, 90º angles | ✓ | ✓ |
| Four equal sides | ✓ | ✗ |
| | Rhombus | Parallelogram |
| Opposing sides are parallel | ✓ | ✓ |
| Diagonal angles are equal | ✓ | ✓ |
| Adjacent angles add up to 180º | ✓ | ✓ |
| All four sides are equal | ✓ | ✗ |
Trapezoid
* One set of parallel sides
Kite
* Two sets of adjacent equal sides
# Rates & Distances

Distance, rate and time problems are commonly tested on the GRE. These problems are a common type of algebra problem encountered on the test, which test setting up equations and working with variables.

## Rule

Distance equals rate multiplied by time.
If there are different rates for different periods of time, distance equals each rate multiplied by each unit of time.
For example, if a ship is traveling at 20 nautical miles per hour for 1.1 hours, the ship will travel a total of 22 nautical miles.

## Practice

1. A train travels 360 miles over the course of 8 hours. How fast does the train travel?
2. A car travels at 45 miles per hour. How long will it take for the car to travel 225 miles?
3. If a truck travels 175 miles over the course of 3.5 hours, how fast is the truck traveling?

## Comments

A common problem on the GRE is the so-called "brain-lapse mistake", where the person knows how to solve the problem but merely "spaces" mentally and puts the wrong number in the wrong part of the equation.
Rate and time problems, along with problems involving ratios and percentages are a common source of brian-lapse mistakes and thus test-takers who struggle with this particular problem should make sure to double-check rate and time problems on their GRE.

## Answers to Practice Questions

1. 45 miles per hour
Distance equals rate multiplied by time, thus:
360 = r(8)
                Take our initial equation.
45 = r
                Divide both sides by 8.
2. 5 hours
Distance equals rate multiplied by time; thus
t(45) = 225
                Take the initial equation.
t = 5
                Divide both sides by 45.
3. 50 miles per hour
Distance equals rate multiplied by time, thus:
r(3.5) = 175
                Take the initial equation.
r = 50
                Divide both sides by 3.5.
# Reading Practice 3

The Cathedral of St. Vincent de Paul presents a noteworthy example of a mixture between Moorish and Middle- Eastern architectural styles as well as the 5 traditional European Gothic style - a medley which naturally resulted from the history of the structure. Built in Tunis, the capital of Tunisia, in 1897, the cathedral harkens back to an era which saw a great 10 influx of French and Italians into Tunis, as the city was at the time a French protectorate - thus, its construction drew upon the skills and preferences of the locals, both French and Italian as well as 15 Tunisian. The result is a unique construction, featuring the domes and arches typical of the Islamic mosque, and greeted with palm trees at its front entrance, but adorned with crosses, both at 20 the façade and at its steeple. Though the Roman Catholic population has sharply declined in Tunis since Tunisian independence, the Archdiocese of Tunis still maintains the building and uses it as 25 the seat of its presence in the city.
1. According to the passage, which one of the following is true about the Cathedral of St. Vincent de Paul?
(A) When it was built, Tunis was a protectorate of France. (B) It was the first Roman Catholic cathedral constructed in Tunis. (C) It has features which are also associated with Islamic mosques.
2. Which of the following can be inferred about Tunis, based on the passage?
(A) Its Roman Catholic population has decreased since the Cathedral of St. Vincent de Paul was built. (B) It is the capital of Tunisia. (C) It has many instances of Gothic and other European architectural styles which harken back to its days as a French protectorate.

## Answers to Practice Questions

1. (A), (C)
Choice (A) can be inferred from lines 10-11, which explain that Tunis was a French protectorate at the time. Answer (C) can be similarly inferred from lines 15-18, which describe the features of the cathedral.
Answer (B), however, cannot be inferred. While the passage states that there was a great influx of Roman Catholics to Tunis, it does not state that the cathedral was the first in the region.
2. (A), (B)
Choice (A) can be inferred from lines 20-23 and their discussion of Tunisian independence. Lines 7-8 establish choice (B) - that Tunis was the capital of Tunisia. The third choice, however, cannot be inferred. The passage doesn't establish that there are many instances of Gothic and other European architectural styles. Only one is discussed - the cathedral in question.
# A Guide to the GRE/Radicals
## Radicals

Radicals can be multiplied and divided, and perfect squares can be extracted.
A radical is a number which equals another number if multiplied by itself a given number of times. The most common type of radical is the square root. Square roots can be multiplied and divided, as indicated above, but they cannot be added or subtracted.does not necessarily equal , but does equal
It is generally frowned upon placing a radical in the denominator of a fraction. To remove a radical from the denominator, multiply both the numerator and the denominator by the radical. For example,

### Practice

In these problems, solve for x.
1.
2.
3.

### Answers to Practice Questions

1.
Take the initial equation.
Multiply both sides by
Divide both sides by 3.
2. y = 3
Take the initial equation.
Square both sides.
Work out the parentheses.
Subtract 5 from both sides.
3.
Take the initial equation.
Multiply both sides by
Divide both sides by 7.
# A Guide to the GRE/Reading Practice 2
## Reading Practice 2

Black Elk Speaks chronicles author John Neihardt's conversations with a shaman of the Oglala Sioux tribe in the early 20th century, and is most well known for its 5 subject's recollection of the Battle of Little Bighorn, or as the author calls it, “The Rubbing Out of Long Hair” in reference to the Sioux name for General Custer. At age 13, Black Elk had participated both in 10 this battle as well as the subsequent altercation at Wounded Knee. However, a focus on the book for these recounts would be misplaced, not only because they are told some forty years after the events occurred, 15 but also because the majority of the work is dedicated not to the Sioux Wars of the late 1800s, but to the spiritual and cultural traditions that Neihardt sought to capture, including the rituals in which the 20 author of the book participated firsthand. 1. The author of the passage holds which of the following views about the book Black Elk Speaks?
(A) It contains a reliable contemporary account of the incidents at Little Bighorn and Wounded Knee. (B) It is the most comprehensive chronicle of Sioux spiritual traditions of any work to date. (C) References to the book for its chronicles of the Sioux Wars miss a greater point of the book. (D) It is also known by the title “The Rubbing Out of Long Hair.” (E) Its author did not have any experience with Sioux culture.
2. Based on the passage, Neihardt's purpose in writing Black Elk Speaks was to do which of the following?
(A) interview a survivor of Little Bighorn and Wounded Knee (B) contrast the views of the Oglala Sioux to that of other Indian tribes (C) describe changes in Sioux life in the early 20th century from their traditional way of living (D) document aspects of traditional Sioux culture (E) capture the entire history of the Sioux Wars of the late 1800s

## Answers to Practice Questions

1. (C)
This can be inferred from the phrase in lines 11-13, which state that a focus on the accounts of the Sioux conflicts is misplaced.
Choice (A) is incorrect because the author never attests to the reliability of the accounts of Little Bighorn in the book; indeed, the author suggests they may be somewhat compromised given the great stretch of time. Answer (B) is incorrect because the passage makes no claim that the book is the most reliable account. Choice (D) is wrong because “The Rubbing Out of Long Hair” refers to Little Bighorn itself, not the book. (E) is directly contradicted by the passage - the author of Black Elk Speaks interviewed a Sioux shaman.
2. (D)
This comes from lines 17-20. Neihardt wasn't predominantly concerned with interviewing survivors of the Sioux Wars as stated in answers (A) and (E) or contrasting the Sioux with others as in answer (B) - indeed, no other Indian tribes are mentioned. Likewise, nothing explicitly mentions the changes discussed in answer (C). Neihardt's purpose was to document Sioux culture.
# Ratios

A ratio express the proportion of two commodities in relation to each other (as opposed to in relation to the whole, as a fraction or percentage does).

## Rules

Add both numbers in a ratio to get the total amount with respect to each.
A ratio expresses quantity in relation to the other. This is not to be mixed up with parts of the whole. If the ratio of Democrats to Republicans at a conference is 6 to 4, then Republicans make up 40% of the conference (4 divided by 6 + 4). If guitar players make up 70% of the rock band seminar and drummers make up 30%, then the ratio of guitarists to drummers is 7 to 3.

## Practice

1. Of the 50 lawyers at the meeting, 22 are bankruptcy lawyers; the remainder are divorce lawyers. What is the ratio of bankruptcy lawyers to divorce lawyers?
2. The ratio of stocks to bonds in the portfolio is 4 to 9. If there are 169 items in the portfolio, how many of them are bonds?
3. The ratio of ratio questions to non-ratio questions on the math quiz is 4 to 5. If there are a total of 36 questions, how many are ratio questions?

## Comments

As alluded, the most common mistake area in ratios is forgetting that the ratio is an expression of one quantity in relation to each other and not the total amount. Whenever presented with a ratio, a test taker must add the two numbers to get the whole amount.

## Answers to Practice Questions

1. 11 to 14
The ratio is 22 to (number of divorce lawyers) = 22 to (50 - 22) = 22 to 28 = 11 to 14
2. 117
The ratio of stocks to bonds is 4 to 9. Thus, the fraction of the portfolio which is bonds is =
Always remember to add both numbers in a ratio to get the total amount.
multiplied by 169 is 117.
3. 16
The ratio of ratio questions to non-ratio questions is 4 to 5. Thus, the fraction of the quiz which is math questions is = multiplied by 36 is 16.
# Cell Charts
## Description

Cell charts indicate corresponding data of particular categories. To obtain totals or combinations, add the appropriate cells.
Often the cells will not give the total amount and the test taker will be asked to determine this. Pay attention to what data the cell is giving with regard to what data category.

## Practice
### Car Washes and Laundromats in Five Cities

| | Population | Car Washes | Laundromats |
| Arlington | 365,000 | 50 | 41 |
| Dallas | 1,198,000 | 84 | 90 |
| Fort Worth | 741,000 | 35 | 44 |
| Irving | 216,000 | 29 | 38 |
| Plano | 260,000 | 40 | 41 |
1. Which cities have the greatest and least number of car washes per capita?
2. Dallas makes up what percentage of the car washes and laundromats amongst the five cities?

## Answers to Practice Questions

1. Per capita is calculated by the number of items divided by the number of people. It can thus be calculated and compared by dividing columns 3 and 4 by column 2 (or a reduced version of it). Plano by far has the highest number of car washes per person, while Dallas has the lowest. Irving has the highest number of laundromats per capita, while Dallas again has the lowest.
2. Dallas has 84 of the 238 car washes, or about 35%, and it has 90 of the 254 car washes, again, about 35%.
# Circles
## Rules

The area of a circle is π multiplied by the radius squared. The circumference of a circle is π multiplied by the diameter (twice the radius).
A circle is a set of points of equal distance from a given point. The number π has been developed by mathematicians to express the ratio of the circumference of a circle to its diameter. π ≈ 3.14.
An arc within a circle is equal to the fraction of the circumference that the arc's angle is a fraction of 360º, or the entire circle.
In the circle to the left, arc AB has a length of one fourth of the circumference of the circle

## Practice

1. A circle has an area of 1. What is its radius?
2.
Square GHIJ is inscribed within the circle. If the square has an area of 64, what is the area of the circle?
3.
D is the center of the circle, and points E and F are both on the circle. If EF and DF are both equal to 5, what is the length of arc EF?

## Answers to Practice Questions

1.
The area of a circle is multiplied by the radius squared.
= 1 Take the initial equation.
= Divide both sides by
= = = Take the square root of both sides.
2.
If the area of the square is 64, then the sides of the square are 8. If the sides of the square equal 8, the diagonal - and thus the diameter of the circle equals
3.
The triangle is an equilateral triangle, meaning its angles are 60º. This means the arc has a length of one sixth of the circumference of the circle, or which works out to
# A Guide to the GRE/Content Purposes

On the GRE, some questions will ask “in which sentence does the author ...” and state a vague summary of the function of the sentence.
Practice identifying the roles of particular sentences.
Folklore in ancient civilizations often served to explain facts of the world as well as merely to entertain. The creation of these stories has persisted into modern
5 times - Americans once told stories of a great lumberjack, Paul Bunyan, who felled the trees of the great plains, and created the “Badlands” of South Dakota by tapping his axe during a nap. In Origins of the
10 Cyclops Myth, Matthew Williams argues that the ancient Greeks in fact did believe in or at least prepend the existence of one- eyed giants - cyclopses - in the Mediterranean. According to Williams,
15 Greeks likely encountered elephant skulls and skeletons in their voyages, and the idea of a single-eyed creature came from the presence of the larger aperture in the face of the skull.

## Questions

* In which sentence does the author given an example of modern folklore? What is this example?
* In which sentence does the author present a reason why the Greeks may have believed in the actual existence of the cyclops? What is this reason?

## Answers to Practice Questions

* This example occurs in the second sentence. The example is Paul Bunyan, who the author discusses to give an instance of contemporary folklore as well as its use to explain something - the great plains and the badlands.
* This occurs in the last sentence. After the author articulates a theory found in what is apparently a scholarly work, the author explains that Greeks believed that there were one-eyed giants because they found elephant skulls on their voyages which had large single holes (“apertures”) in them. They believed, according to this theory, that these were the skulls of one-eyed giants.
# Coordinates & Graphing
## Rules

In a standard coordinate plane, points are given locations of (x, y), where “x” is the horizontal coordinate and y is the vertical coordinate.
The x coordinate is also called the “abscissa”, while the y coordinate is called the “ordinate.”
The slope of a line graphed on the coordinate plane is equal to rise divided by run. If a line is expressed by an equation, the slope of the line is the number in front of the x when the equation is in the form y = ax + b.
For example, if the equation for a line is y = 2x + 1, the slope is 2 and the y intercept is 1. If the line is in the form y - +2x = 1, rearrange it so that it is in the above format - y = mx + b.
To determine when a line crosses the x or y axis, set x or y equal to zero.
For example, a line with the equation y = 6x - 5 crosses the y axis at (0, -5).

## Practice

1. The equation for a line is 2y - 4x - 5 = 0. What is the slope of the line?
2. What is the slope of a line which passes through points (3, -2) and (6,3)?

## Comments

What is the equation for the line above?

## Answers to Practice Questions

1. 2
When a line is in the form y = mx + b, m is the slope of the line and b is the y intercept. If a line is not in this format, it should be converted into this format.
2y - 4x - 5 = 0 Take the initial equation.
2y - 4x = 5 Add 5 to both sides.
2y = 4x + 5 Add 4x to both sides.
y = 2x + Divide both sides by 2. The slope of the line is 2.
2.
The slope of this line is rise over run, or This equals or
3. y = 3x + 1
When a line is in the form y = mx + b, m is the slope of the line and b is the y intercept.
The line crosses the y axis at (0,1) and thus its intercept is 1. It passes through the point (1,4) after passing through (0,1) meaning it has a rise of 3 for every unit of run. Thus, the slope of the line is 3 and its equation is y = 3x + 1.