OpenLLM-France/wikimedia huggingface dataset:
# Prealgebra for Two-Year Colleges/Appendix (procedures)/Subtracting whole numbers with borrowing
## Addition

Addition operations are denoted by the + sign and the - (-)= +. The addition operator (plus sign) will take any two numbers, called addends, as operands to work on. The result is called the sum of the two numbers.
The operation of addition is commutative. This means that the addition of two numbers will give the same sum regardless of the order in which the numbers are added.
Example:
* $7+5=12$ and $5+7=12$
* $3+2+4=9$ and $4+3+2=9$

### Vertical Addition

To add vertically, stack the numbers on top.
Now add the numbers in the first or right column. If you get a number with tens in it, put what's in the tens place on the tens place in the problem. You should have:
Add the tens column now.
Repeat in a similar fashion for each additional column (hundreds, thousands, etc.)until you finish the problem.

### Exercises

In Exercises 1–25, find the sum.
* 1. $5+3$
* 2. $8+7$
* 3. $9+2$
* 4. $6+3$
* 5. $1+4$
* 6. $2+17$
* 7. $12+11$
* 8. $53+8$
* 9. $41+9$
* 10. $84+12$
* 11. $16+17$
* 12. $7,576+5,345$
* 13. $2,345+3,245$
* 14. $8,952+9,423$
* 15. $2,783+2,389$
* 16. $189,583+1,574,822$
* 17. $1.5+2.7$
* 18. $5.4+3.9$
* 19. $8.3+9.2$
* 20. $2.23+4.89$
* 21. $534.4+34.675$
* 22. $348.904+23,498.2$
* 23. $1.673+48,210.38$
* 24. $10.4823+94.29478$
* 25. $128.52+2,070.24$

### Answers

* 1. $8$
* 2. $15$
* 3. $11$
* 4. $9$
* 5. $5$
* 6. $19$
* 7. $23$
* 8. $61$
* 9. $50$
* 10. $96$
* 11. $33$
* 12. $12,921$
* 13. $5,590$
* 14. $18,375$
* 15. $5,172$
* 16. $1,764,405$
* 17. $4.2$
* 18. $9.3$
* 19. $17.5$
* 20. $7.12$
* 21. $569.075$
* 22. $23,847.104$
* 23. $48,212.053$
* 24. $104.77708$
* 25. $2,198.76$

## Subtraction

Subtraction, as you probably also have seen, uses the minus (-) sign. The generic subtraction operator will take any two numbers as operands. The result is called the difference of the two numbers.
Subtraction is not a commutative operation. Changing the order of the operands will likely give a different (not the same) result.
Example:
* $7-5=2$ and $5-7=-2$
* $3-2-4=-3$ and $4-3-2=-1$

### Exercises

In Exercises 26–50, find the difference.
* 26. $5-3$
* 27. $8-7$
* 28. $9-2$
* 29. $6-3$
* 30. $1-4$
* 31. $2-17$
* 32. $12-11$
* 33. $53-8$
* 34. $41-9$
* 35. $84-12$
* 36. $16-17$
* 37. $7,576-5,345$
* 38. $2,345-3,245$
* 39. $8,952-9,423$
* 40. $2,783-2,389$
* 41. $1,574,822-189,583$
* 42. $2.7-1.5$
* 43. $5.4-3.9$
* 44. $8.3-9.2$
* 45. $2.23-4.89$
* 46. $10.38-1.673$
* 47. $534.4-34.675$
* 48. $348.904-23,498.2$
* 49. $10.4823-94.29478$
* 50. $2,070.24-128.52$

### Answers

* 26. $2$
* 27. $1$
* 28. $7$
* 29. $3$
* 30. $-3$
* 31. $-15$
* 32. $1$
* 33. $45$
* 34. $32$
* 35. $72$
* 36. $-1$
* 37. $2,231$
* 38. $-900$
* 39. $-471$
* 40. $394$
* 41. $1,385,239$
* 42. $1.2$
* 43. $1.5$
* 44. $-0.9$
* 45. $-2.66$
* 46. $8.707$
* 47. $499.725$
* 48. $-23,149.296$
* 49. $-83.81248$
* 50. $1,941.72$
# Prealgebra for Two-Year Colleges/Workbook AIE/Length, area, and volume

Length, Weight, and Volume
Materials Needed: Textbook

## Why

Real-world problems almost always contain units. Some of the most common units in the U.S. are English units of length, weight, and volume.

## Learning Objectives

* Explain the difference between length, weight, and volume, and identify appropriate English units for each.
* Know the conversion factors for common English units of length, weight, and volume.
* Give examples of situations in which a given operation (+, –, , ) is appropriate.

## Warm-up

You should have the following facts memorized. Complete the table. Length : ___12__ in = 1 ft ___3__ ft = 1 yd ___5280__ ft = 1 mi Weight : ___16__ oz = 1 lb ___2000__ lbs = 1 ton Time: ___60__ sec = 1 min ___60__ min = 1 hr ___24__ hr = 1 day ___30__ dys = 1 mon ___12__ mon = 1 yr ___52__ wks = 1 yr ___365__ dys = 1 yr Volume/Capacity : ___8__ fl oz = 1 cup ___2__ cups = 1 pt ___2__ pt = 1 qt ___4__ qt = 1 gal

## Activity

1) Does the situation describe a length, weight, or volume? Write L, W, or V. a) Filling you car with gasV b) Going on a dietW c) Measuring cough syrupV d) Figuring postageW e) Driving to CincinnatiL f) Building a fenceL 2) Write an appropriate English unit for measuring each of the following: a) The distance from San Francisco to Washington, D.C. miles b) The weight of an aircraft carrier tons c) The size of a soda bottle quart d) The size of a sheet of paper inches e) The amount of flour in a loaf of bread cup f) The amount of Cheerios in a box oz 3) Because of construction CalTrans needs to install 100 yds of concrete barriers along the edge of Highway 4. Each barrier is 1 yd 2 ft long. How many barriers are needed? 100 yds = 300 ft; 1 yd 2 ft = 5 ft 100/5 = 20 barriers
4) Write a realistic problem situation for 2 tons 850 lbs – 1 ton 1430 lbs. Solve the problem you wrote. A fully-loaded pickup truck weighs 2 tons 850 lbs. If the emptied truck weighs 1 ton 1420 lbs (a fact I found in the owner’s manual), then how much does the cargo weigh? 2 tons 850 lbs – 1 ton 1430 lbs = 1420 lbs
5) Write a realistic problem situation for 1 cup 4 fl oz  13. Solve the problem you wrote. I have 13 friends coming over on a hot day. I want to make enough lemonade to be able to serve each of them 1 cup 4 fl oz. How much lemonade do I need to make? 1 cup 8 fl oz  13 = 13 cups 52 fl oz = 19 cups 4 fl oz
6) Write a realistic problem situation for 3 lbs 12 oz  12. Solve the problem you wrote. At Costco, a box of 12 cans of cat food weighs 3 lbs 12 oz. How much does each can weigh? 3 lbs 12 oz  12 = 5 oz
7) Write a realistic problem situation for 24 ft 9 in + 18 ft 6 in + 24 ft 9 in + 18 ft 6 in. Solve the problem you wrote. I have a garden that is 24 ft 9 in long and 18 ft 6 in wide. What length of fencing material should I buy if I want the fence to go all the way around the garden? 24 ft 9 in + 18 ft 6 in + 24 ft 9 in + 18 ft 6 in = 84 ft 30 in = 86 ft 6 in
# Prealgebra for Two-Year Colleges/Appendix (procedures)/Reducing fractions

Equivalent fractions represent the same number, although they are written differently. For example,
    ${\frac {2}{4}}$ and ${\frac {1}{2}}$
represent the same amount. The latter fraction has a smaller denominator, so we say the fraction has been reduced.
You can reduce a fraction by dividing the numerator and denominator by the same number. For example,
    ${\frac {75}{100}}={\frac {75{\color {Red}\div 5}}{100{\color {Red}\div 5}}}={\frac {15}{20}}$
and
    ${\frac {15}{20}}={\frac {15{\color {Red}\div 5}}{20{\color {Red}\div 5}}}={\frac {3}{4}}$.
When we ask you to simplify a fraction or reduce a fraction such as 75/100, we want you to find the equivalent fraction with the smallest possible denominator. In this case, the fully reduced form is 3/4. The only number that we could divided into both 3 and 4 would be 1, but that would not reduce the fraction any further.
    ${\frac {3}{4}}={\frac {3{\color {Red}\div 1}}{4{\color {Red}\div 1}}}={\frac {3}{4}}$.
You are also allowed to multiply the numerator and denominator by the same number. We usually call this finding equivalent fractions.
# Prealgebra for Two-Year Colleges/To the instructor/Socratic method

The suggested method of using this workbook is the Socratic method. In a typical class setting, the professor asks a question and calls on a student who may or may not have volunteered an answer. The professor either then continues to ask the student questions or moves on to another student.
The employment of the Socratic method has some uniform features but can also be heavily influenced by the temperament of the teacher. The method begins by calling on a student at random, and asking about how to begin to solve a problem. The first step is to ask the student to paraphrase the problem, in order to ensure that the student has read and has a basic understanding problem.
One hallmark of Socratic questioning is that typically there is more than one "correct" answer. The primary goal of the Socratic method is for students to understand their reasoning in solving a problem, as opposed to memorizing a procedure without understanding why it works.
# Prealgebra for Two-Year Colleges/Appendix (procedures)/Subtracting fractions

Find a common denominator and then subtract the numerators. Reduce the result if you can.
# Prealgebra for Two-Year Colleges/Appendix (procedures)/Rounding
## Rounding

To round a number, you need to look at the numbers right of the decimal. If the first number right of the decimal is bigger than or equal to 5 we add 1 to the number left of the decimal this is known as rounding up. If the first number right of the decimal place is less than 5 we simply rewrite the number again excluding all the numbers right of the decimal place; this is known as rounding down. Note that after the number is rounded up or down all of the numbers right of the decimal are discarded and only the numbers left of the decimal are written.

## Examples

1.49999999 rounds to 1
1.50000000 rounds to 2
1234.56 rounds to 1235
# Prealgebra for Two-Year Colleges/Appendix (procedures)/Reading Decimal Numbers

The word and is used only for the decimal point; And is also used to separate a whole number and a fraction. A comma is used at every third place, starting at the decimal point and moving left. Frequently a comma or space is used at every third place moving to the right of the decimal point.
Additionally, no comma is used before and. Also, all decimals end in ths except unitary decimals that end in th.
| Decimal numeral | Reading thereof |
| 2,697,787.84 | two million, six hundred ninety-seven thousand, seven hundred eighty-seven and eighty-four hundredths |
| 2,009 | two thousand, nine |
| 1,987 | one thousand, nine hundred eighty-seven |
| 0.684 | six hundred eighty-four thousandths |
| 17.04 | seventeen and four hundredths |
| 0.1 | one tenth |
| 4.3 | four and three tenths |
| 0.000,1 | one ten-thousandth |
| 5.000,008 | five and eight millionths |
| 0.000,73 | seventy-three hundred-thousandths |

## References and notes

* ↑ Business Mathematics, 10th Edition. Authors are Charles D. Miller, Stanley A. Salzman, and Gary Clendenen. Published in 2006 by Pearson Education, Inc. ISBN 0-321-27782-1
* ↑ Any thorough English arithmetic text discusses the reading of decimal numerals.
# Spanish/Tenses/Imperative
### The Imperative Mood

* Affirmative commands
* Negative commands
# Spanish/Tenses/Indicative
### The Indicative Mood

* Present
* Preterite
* Imperfect
* Future
* Conditional
* Present
* Preterite
* Imperfect
* Future
* Conditional
* Present
* Preterite
* Imperfect
* Future
* Conditional
* Present
* Preterite
* Imperfect
* Future
* Conditional
# Spanish/Tener ganas de ...

To feel like doing something
# Spanish/Tener que ...

Tener que in Spanish grammar is used to convey the idea of having to do something.
It is relatively simple to form and these are the following steps to take:
1) Take the appropriate form of tener:
To do this you need to decide which tense you are going to use: the present tense(for actions that one need to do now), the imperfect (for a past, continuous action), the perfect tense (for past, complete actions) and the future.
For learning purposes it is useful to look at the present tense:
| Person | yo | tú | él | nosotros | vosotros | ellos |
| Verb | tengo | tienes | tiene | tenemos | tenéis | tienen |
2)To all of these verbs you add que to get the desired effect.
3)Moreover, for other tenses, you simply use the appropriate form of tener and again add que.
# Spanish/Tenses/Imperative/Affirmative
## Regular Affirmative Spanish Commands

Spanish commands can vary based on if they are singular or plural and their formality.

## Tú Commands

An affirmative tú command is the same as the Present Third Person Singular (Él, Ella, Usted) conjugation in present tense.
For Example:
| Ar | Ar | ER | ER | Ir | Ir |
| Stem + a | Stem + a | Stem + e | Stem + e | Stem + e | Stem + e |
| Habla | Speak | Come | Eat | Escribe | Write |

## Usted Commands
### Ar

An Ar usted command is formed by dropping the "o" from the Present First Person Singular conjugation (the yo form) and then adding an "e". For Example:
| Hablar - To speak | Hablar - To speak | Comprar - To buy | Comprar - To buy |
| Yo Form | Command | Yo Form | Command |
| Hablo | Hable | Compro | Compre |

### Er and Ir

An Er/Ir usted command is formed by dropping the "o" from the Present First Person Singular conjugation (the yo form) and then adding an "a". For example:
| Beber - To drink | Beber - To drink | Vivir - To live | Vivir - To live |
| Yo Form | Command | Yo Form | Command |
| Bebo | Beba | Vivo | Viva |

## Ustedes Commands

An ustedes command is formed by adding an "n" to the end of an usted command.
For example:
| Hablar - To speak | Hablar - To speak | Compartir - To share | Compartir - To share |
| Ud. Command | Uds. Command | Ud. Command | Uds. Command |
| Hable | Hablen | Comparta | Compartan |
# Spanish/Tenses/Imperative/Negative

| | -ar | -ar | -er/-ir | -er/-ir |
| | conjugation | example | conjugation | example |
| | no + subjunctive | hablar | no + subjunctive | no comer |
| Tú | no + stem + es | no hables | no + stem + as | no comas |
| Él | no + stem + e | no hable | no + stem + a | no coma |
| Nosotros | no + stem + emos | no hablemos | no + stem + amos | no comamos |
| Vosotros | no + stem + éis | no habléis | no + stem + áis | no comáis |
| Ellos | no + stem + en | no hablen | no + stem + an | no coman |
Some things to note:
* Negative commands (ordering someone to not do something) are formed with a verb in its conjugated subjunctive form preceded by the word "no."
* There is no yo form in the imperative. One does not simply give commands to oneself.
# Spanish/Subjunctive

Verbs | Verb Tenses | Verbs List

## Subjuntivo

These four verb tenses refer to the speakers' feelings toward the action and are about things that should or could be. These tenses are unfamiliar to English speakers as they are no longer commonly used in English grammar. They are divided into present, past, and future just like the indicative tenses.
    Presente
* Presente de subjuntivo
    Pasado
* Imperfecto de subjuntivo
* Pluscuamperfecto de subjuntivo o, pasado perfecto subjuntivo
    Futuro
* Futuro subjuntivo (today, used only in legal documents)
# Spanish/Tenses/Gerund

| -AR | -AR | -AR | -ER / -IR | -ER / -IR | -ER / -IR |
| conjugation | example | translation | conjugation | example | translation |
| -AR | hablar | to speak | -ER / -IR | comer | to eat |
| stem+ando | hablando | speaking | stem+iendo | comiendo | eating |
Some things to note:
* Verbs that are -Er/-Ir double vowel verbs (caer, creer, leer, huir, etc.) will substitute a 'y' for the 'i' in the ending: cayendo, creyendo, leyendo, huyendo, etc. This does not include -guir verbs, as the 'u' exists only to elecit the hard 'g' sound and is not pronounced. This same change occurs in the third person of the preterit.
* With -Ir stem changing verbs (morir, dormir, sentir, vestir, pedir) the 'o' in the stem will change to a 'u' (muriendo, durmiendo), or the 'e' to an 'i' (sintiendo, vistiendo, pidiendo). This same change occurs in the third person of the preterit and the nosotros and vosotros of the present subjunctive.
* Ir has the irregular gerund, yendo, and poder the irregular gerund pudiendo.
* These are not nouns. Me gusta nadando is incorrect. While in English, we say "I like swimming," you cannot do so in Spanish. "Swimming" can be either a verb or noun in English, meaning "moving through water" (a verb) or the act of doing so (a noun). In Spanish, you say "me gusta nadar". "I like to swim."
* These forms are often used with the verb estar, for use in continuous tenses. When used as such, it is estar that is conjugated to agree with the subject. The gerund does not change. Ellas están hablando is correct, while ellas están hablandas is not.
* -ER and -IR verbs undergo the same change to make the gerund.
# Spanish/Tenses/Indicative/Perfect/Conditional

| | -AR / -ER / -IR | -AR / -ER / -IR | -AR / -ER / -IR |
| | conjugation | example | translation |
| | -AR / -ER / -IR | hablar | to speak |
| Yo | habría + past participle | habría hablado | I would have spoken |
| Tú | habrías + past participle | habrías hablado | You would have spoken |
| Él | habría + past participle | habría hablado | He would have spoken |
| Nosotros | habríamos + past participle | habríamos hablado | We would have spoken |
| Vosotros | habríais + past participle | habríais hablado | You all would have spoken |
| Ellos | habrían + past participle | habrían hablado | They would have spoken |
Some things to note:
* This tense (and all perfect tenses) uses a form of the irregular helping verb haber (meaning to have done). As this is the future perfect indicative, it takes the simple future indicative form of haber.
* The past participle of a regular verb is formed by adding -ado to the stem of an -AR verb, and -ido to the stem of an -ER or -IR verb.
* As the structure for this tense is a form of haber with the past participle, there are no irregular verbs in this tense. Any irregularity would come from a verb with an irregular past participle, but the haber + past participle structure still holds true.
* While the past participle is oftentimes the same word as the adjective form of a verb (las palabras habladas - the spoken words), the past participle itself is not an adjective, and therefore does not have to "agree" with the subject. Regardless of the subject's gender or number, the past participle remains the same. Ellas habrían hablado, NOT ellas habrían habladas.
* Be careful not to confuse the past participle with the gerund. habría hablado, not habría hablando.
# Puzzles/Easy Sequence 8/Solution

This series is represented by the months. If the month has 31 days, it gets a one. If not, it gets a zero.
| Month | January | February | March | April | May | June | July | August | September | October | November | December |
| Length | 31 | 28 or 29 | 31 | 30 | 31 | 30 | 31 | 31 | 30 | 31 | 30 | 31 |
| Sequence | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 |
Alternatively, it can be described as the remainder after dividing by 2 of the length of each month in a common year (as opposed to a leap year).
# Puzzles/Easy Sequence 10/Solution

If one indexes the sequence from t, then the function for each term is trivially
    $a_{n}=\left\{{\begin{matrix}1\ \mathrm {if} \ n>t\\0\ \mathrm {otherwise} \end{matrix}}\right.$
Because this is the end of the easy sequences, you get something a bit more tricky - another solution:
    $a_{n}=\left\{{\begin{matrix}1\ \mathrm {if} \ n<t\\1\ \mathrm {if} \ t<n<t+7\\1\ \mathrm {if} \ t+7<n\\0\ \mathrm {otherwise} \end{matrix}}\right.$
Part of what mathematicians do is look for patterns. But be warned, just because it *looks* like it's doing something doesn't mean it's going to keep doing it forever! Showing that something does actually follow a pattern is some of what mathematicians spend their time on. But don't worry about that now, go look at some more sequences!
A bit too trivial for my liking, what about
$a_{n}=(n+4)\ \mathbf {div} \ 6$
resulting in 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2...
What about any of these?
$a_{n}=(n+k)\ \mathbf {div} \ (k+2)$ with k >= 4
The sequence wasn´t supposed to necessarily change from 1 to 2 exactly after the ellipsis ("...").
another solution: $a_{n}=(k*f(n))\ \mathbf {div} \ f(n)$ with $k\in \ ]1,2[$
f(n) = f(n-1) ! for n > 1 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
# Puzzles/Easy Sequence 4/Solution

alternate subtracting and adding numbers starting with 1 and increasing by 1 each time.
3,2,4,1,5,0,6,-1,7,...
# Puzzles/Easy Sequence 6/Solution

They are powers of 3:
3 9 27 81 243 729 2,187 ...
# Puzzles/Easy Sequence 7/Solution

< Number sequences
They are the sum of two numbers before (also known as the Fibonacci-sequence):
2 3 5 8 13 21 34 55 89
That is, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13 and so on...
# Puzzles/Decision puzzles/Yet Another Weighing/Solution

Puzzles|Decision puzzles|Yet Another Weighing|Solution
One way to solve this would be to try to manually work out all the possible combinations and their sums. That would take a while.
Instead, notice that each item is half the weight of the previous item. Lets assume you have less items and work out the general rule for any number of items, where each is twice as much as the next item:
Without even listing them, we can see the pattern. The numbers 1, 3, 7, 15 follow the formula of 2$^n$ - 1, where n=1, 2, 3, 4 i.e. the number of items.
For 5 items, n=5, 2$^5$ - 1 = 31.
The pattern would continue for more items:
# Puzzles/Easy Sequence 5/Solution

< Number sequences
Problem 3 2 4 1 4 ?
Solution
Alternate subtracting and multiplying, starting with 1 and increasing by 1 each time.
3,2,4,1,4,-1,-6,...
# Puzzles/Easy Sequence 1/Solution

< Back to Problems
They are ascending powers of 2:
1 2 4 8 16 32 64 ...
Note : 2 raise power 0 (i.e 2^0) equals 1
# Puzzles/Easy Sequence 3/Solution

a). The ascending powers of 2 are added to the previous number.
We start with 1 and add 2^0 to get 2, then add 2^2 (4) to get 6, plus 2^3 (8) equals 14, + 2^4=30, + 2^5=62
1, 2, 4, 6, 14, 30, . . .
b). The sequence in question is 1, 2, 6, 14, ... (without the 4). We can describe this sequence as follows:
We obtain: 1, 2, 6, 14, 36, 94, ...
# Puzzles/Easy Sequence 2/Solution

Add 1 2 3 4... OR x(x-1)/2+1 :
1 2 4 7 11 16 22 29 ...
$a_{n}=a_{n-2}+a_{n-1}+1$:
1 2 4 7 12 20 33 ...
Add the three previous numbers:
1 2 4 7 13 24 44 ...
The numbers with an odd number of ones in their binary representation:
1 2 4 7 8 11 13 ... or 1 10 100 111 1000 1011 1101 ...
A formula for the $n$th number is: $a_{0}=1$, $a_{2n}=a_{n}+2n$, $a_{2n+1}=6n+3-a_{n}$.
# Startups in the Philippines/A Survival Guide for Startups in the Philippines
## Startup Stacks in the Philippines

Stacks commonly utilized by startups in the Philippines. Range from software development stacks; proprietary to open source; platforms; to office productivity suites of and enterprise services.

## Startup Fundraising in the Philippines

This section will cover the unique facets of startup fundraising in the Philippines. There are similarities to the Silicon Valley; there are similarities to the processes and experiences of fundraising elsewhere in ASEAN, but overall: startup fundraising in the Philippines is unique.
# Startups in the Philippines/A Survival Guide for Startups in the Philippines/Startup Stack Philippine Edition
## Intro

For the time being, we'll organize the section, list-style, in alphabetical order.
# Starting and Running a Wiki Website/Links

External links:
* MediaWiki User's Guide - MediaWiki is the software underlying Wikibooks and Wikipedia
* At the original WikiWikiWeb (contains a great deal of useful information)
** Wiki Choicetree
** Wiki Engines
** Choosing a wiki
* Wikiversity: Wiki science
* Articles
** Which Open Source Wiki Works For You? (by Shlomi Fish)
** Getting Your Wiki Adopted
** A Tale of Two Wikis: Techniques for building, managing and promoting collaborative communities
* Other sources
** Wiki That! A blog of wiki case studies, scenarios, advice, and resources for the non-technical wiki user.
** Free Wiki - Wiki Info : Wiki Demos : Wiki Screenshots : Wiki Links : Wiki Feeds
** MeatballWiki
** hive-wiki, center-wiki for wiki-hives
** the Wiki Choice Wizard helps you narrow down which wiki engines or hosts will work for you.