Difference between revisions of "ספר דיני ממונות"

Prologue

Chapter One: Addition

• If a person says: sum three numbers, the first is 223, the second 342 and the third 422.

• As if you say: sum 203 with 402 and with 809.

• If you wish to sum numbers: in the first of which there are zeros in the units and the tens and 2 in the hundreds; in the second there is a zero in the units, 4 in the tens and 2 in the hundreds; in the third there is 3 in the units, zero in the tens and 4 in the hundreds.

• If you wish to sum 4 numbers: in the first of which there are 4 thousands, 2 hundreds and 4; in the second there are 2 hundreds and 12; in the third there is 12; and in the fourth 30.

Addition of Coins

Chapter Two: Subtraction

• Example: we wish to subtract 123 from 245.

• Another example: we wish to subtract 207 from 349.

• Another example: we wish to subtract 253 from 304.

• Another example: we wish to subtract 1403 from 2040.

• Example: we wish to subtract 135 from 246.

• Example: we wish to subtract 382 from 468.

• Example: we wish to subtract 3456 from 4321.

• Example: we wish to subtract 2304 from 5060.

Subtraction of Coins

Chapter Three: Discusses the Multiplication in all Possible Ways Found in Integers

Type one

• Example: we wish to multiply 23 by 2[4].

• Example: we wish to multiply 234 by 24.

• Example: we wish to multiply 222 by 333.

• Example: we wish to multiply 230 by 324.

• Example: we wish to multiply 240 by 368.

• Example: we wish to multiply 403 by 230.

Type two

• Example: we wish to multiply 24 by 25.

• Example: we wish to multiply 123 by 456.

• Example: we wish to multiply 246 by 140.

Type three

• Example: we wish to multiply 23 by 17.

• Example: we wish to multiply 242 by 144.

• Example: we wish to multiply 240 by 170.

Type four

• Example: we wish to multiply 122 by 232.

• Example: we wish to multiply 470 by 580.

Type five

• Example: we wish to multiply 27 by 32.

• Another example: we wish to multiply 231 by 342.

Type six: Gelosia

• Example: we wish to multiply 25 by 25.

• Example: we wish to multiply 212 by 132.

Type seven: multiplying by tens

• Example: we wish to multiply 33 by 10.

• Example: we wish to multiply 45 by 20.

• Example: we wish to multiply 15 by 40.

Type eight: subtraction and multiplication

• Example: we wish to multiply 4 minus 2 by 8 minus 3.

• Another example: we wish to multiply 7 minus 2 by 8 minus 4.

Chapter Four: Division

• Example: we wish to divide 36 by 48.

• Example: we wish to divide 11 by 18.

• Another example: we wish to divide 23 by 36.

• Example: we wish to divide 37 by 48.

• Example: we wish to divide 11 by 19.

• Example: we wish to divide 5 into 17 parts.

• Example: we wish to divide 245 by 34.

• Example: we wish to divide 12345 by 234.

• Example: we wish to divide 20503 by 304.

• Example: we wish to divide 4321 by 23.

• Another example: we wish to divide 70213 by 136.

• Example: we wish to divide two hundred by 50.

Chapter Five: Addition of Fractions

Chapter Six: Subtraction of Fractions

Chapter Seven: Multiplication of Fractions

Chapter Eight: Division of Fractions

Chapter Nine: Ratios

ואם השבר הוא למטה ראוי לך לכפול כל השלמים מלמטה נגד כל המטבעים וגם ראוי לקחת מכל המטבעים כפי השבר אם חצי מחציתו ואם שליש שלישיתו ואם ב' שלישיות ב' שלישיות

Chapter Ten: Discusses the Extraction of Roots

• The first part is the extraction of roots of integers

• The second part is the extraction of roots of fractions

• The third part is the extraction of roots of integers and fractions

Part One: Extracting the Roots of Integers

Know that 1 is a compound root and square.

• Know that the squares that are in third rank are analogous to the first rank and so are their roots, only that their roots should be raised by one rank, i.e. their roots are tens.

גם שרשיהם אבל צריך שתעלה שרשיהם מדרגה אחת ר"ל כי שרשיהם יהיו עשרות

• The roots and the squares of the fourth rank are analogous to the second rank, but there is no need to raise their roots by one rank, they stay in their rank.

ואין צורך להעלות שרשיהם מדרגה אחת אלא יעמדו במקומם

• The roots and the squares of the fifth rank are analogous to the first rank, but the root should be raised to the third rank, so their roots are hundreds.

אלא שצריך שתעלה השרש עד המדרגה השלישית ושרשיהם יהיו מאיות

• The roots and the squares of the sixth rank are analogous to the second rank, but the analogous root should be raised to the hundreds.

וצריך שתעלה השרש הנמשל אל המאיות

• The squares and the roots of the seventh rank are analogous to the first rank, but the root should be thousands.

אלא צריך שיהיה השרש אלפים

• The eighth rank is analogous to the second [rank], but the root should be also thousands.

וצריך שיהיה השרש ג"כ אלפים

• The ninth rank is analogous to the first rank, but the root is tens of thousands.

אלא השרש יהיה עשרות אלפים

• The tenth rank is analogous to the second [rank] and the roots are tens of thousands.

• Example: we wish to know the root of 980.

• Another example: we wish to know the root of 49[3]2.

• Another example: we wish to know the root of one hundred and fifty thousand.

Part Two: Extracting the Roots of Fractions

• I.e. if you are told: how much is the root of a quarter?

• Example: we wish to know the root of one quarter.

• Another example: we wish to know the root of one third.

• Another example: we wish to know the root of 3 fifths.

• Another example: we wish to know how much is the root of 5 eighths.

• Another example: we wish to know the root of five.

• Another example: we wish to know the root of six.

• Another example: we wish to know the root of 8.

• Example: we wish to know the root of 7.

• Another example: we wish to know the root of 12.

Part Three: Extracting the Roots of Fractions and integers

• Example: we wish to know the root of 3 and a half.

• Another example: we wish to know the root of 4 and a fifth.

• Another way: we wish to know the root of 6 and a fifth.

• Another example: we wish to know the root of 9 and 3 quarters and a quarter of a quarter.

Chapter Eleven: Checking Methods

Chapter Twelve: Word Problems

Purchase Problems – Unequal Amount

1) Question: a man wants to buy a canna from four kinds of clothes, one canna from each types - red, green, black and white, and the total amount is 17 peraḥim.

He said that the canna of the black [cloth] costs 12 carlini and two parts of 15 more than [the canna of] the white [cloth].

The canna of the green [cloth] costs 15 carlini and two sevenths more than the canna of the black [cloth].

The canna of the red [cloth] costs 23 carlini and two fifths more than [the canna of] the green [cloth].

We wish to know how much is the price of each, so that when all the prices are summed together the sum is 17 peraḥim.

ואמר כי הקנה מהשחור שוה יותר מהלבן י"ב קרליני וב' חלקים מט"ו
והקנה מהירוק שוה יותר מקנה השחור ט"ו קרליני וב' חלקים מז‫'
ואמר שהקנה מהאדום שוה יותר מהירוק כ"ג קרליני וב' חמישיות
נרצה לדעת כמה ערך כל א' וא' באופן שכשנחבר ערך כלם יעלו י"ז פרחים

The answer: Take 12 and 2 parts of 15, sum them with 15 and 2-sevenths and with 23 and 2-fifths, then add once more 12 and 2 parts of 15, with 15 and 2-sevenths and with 23 and 2-fifths; the result is 9 peraḥim, 44 cavalli and 4-sevenths.

Subtract all this from 17 peraḥim; the remainder is 7 peraḥim, 9 carlini, 75 cavalli and 3-sevenths.

Divide it into 4 parts; the result is 17 carlini, 102 cavalli and 6-sevenths and this is the price of the white cloth that is the cheapest.

Then, add it to 12 carlini and 2 parts of 15; the result is 31 carlini, 118 cavalli and 6-sevenths and this is the price of the black [cloth].

Add 15 cavalli and 2-sevenths to the price of the black [cloth]; the result is 47 carlini, 39 cavalli and one-seventh and this is [the price] of the green [cloth].

Add 23 carlini and 2-fifths to the price of the green [cloth]; the total result is 70 carlini, 87 cavalli and one-seventh. We define it as is the price of the red [cloth].

When we sum up all, the result is 17 peraḥim no more and no less.

Partnership Problems - for different times

2) Question: two men formed a partnership.

One contributed 100 minyanim, stayed in the partnership for 6 months, and earned 40 minyanim.

The other contributed unknown amount of money, stayed in the partnership for 7 months, and earned 90 minyanim.

How much was his money?

הא' שם ק' מנינים ועמדו בשותפות ו' חדשים והרויחו מ' מנינים
והאחר שם ממון נעלם ועמדו בשותפות ז' חדשים והרויחו צ' מנינים
כמה היה ממונו

Answer: you should multiply the amount of money that the second earned, which is 90, by 10; the result is 9 thousand.

Divide it by the profit of the first, which is 40; the result is 225.

So, the amount of money of the second for 6 months was 225 and he stayed in the partnership for 7 months.

Rule of Three: We say: if 7 is equal to 225, how much is 6 equal to?

The result is 192 and 6-sevenths and it is the amount of money of the second.

Interest and Discount Problems - Find the time

3) Question: a man earned 5 pešuṭim for 4 minyanim in 3 months. In how many months would 6 minyanim yield 7 [pešuṭim]?

ו' מנינים בכמה חדשים ירויחו ז‫'

Answer: multiply 3 by 4; it is 12.

Then, divide 12 by 6; the result is two.

Hence, 6 minyanim yield 5 [pešuṭim] in 2 months.

Rule of Three: Our profit was 7, so we do as follows: if 5 minyanim are equal to 2 months, how many months are 7 minyanim equal to?

Multiply two by 7; the result is 14.

Divide 14 by 5; the result is 2 months and 4-fifths of a month.

So, they yield 7 minyanim in 2 months and 4-fifths of a month.

Find a Quantity

How Much - Money

4) Question: we summed the third and the quarter of an unknown amount of money and the result is 24.

How much is the amount of money?

כמה הממון

False Position: Answer: take the common denominator, which is 12, from the multiplication of 3 by 4.

Then, take its third and quarter; they are 7.

Rule of Three: As the ratio of 7 to 12, so is the ratio of 24 to our unknown number.

We multiply 12 by 24; the result is 288.

We divide it by 7; you get 41 and a seventh and this is the amount of the money.

Check: You can check it by taking a third of 41 and a seventh; it is 13 integers and 5-sevenths.

Take also a quarter of 41 and a seventh; you get 10 and 2-sevenths.

Sum them with the 13; they are 24 no more and no less.

5) Question: we took the third and the fifth of an amount of money plus a third of what remains and they are 12.

How much is the total amount of money?

כמה היה כל הממון

False Position: The answer: take 1 as an example and extract its third and fifth; they are 8 parts and 15.

Take the remaining third, i.e. the third of 8 parts of 15 and sum them; the total sum is 62 parts of 90.

Rule of Three: Then, you should multiply 12 by 62 and divide the product by 90; you get 17 integers and 13 parts of 31 and this is the whole amount of money.

Check: To check it, take a third of this number; it is 5 integers and 25 parts of 31.

Take also a fifth of the number; it is 3 integers and 15 parts of 31.

Sum them together; the result is 9 integers and 9 parts of 31.

Subtract this number from 17 integers and 13 parts of 31; the remainder is 8 integers and 4 parts of 31.

Divide it by 3 and add the quotient to 9 integers and 9 parts of 31; the total result is 12.

First from Last - Money

6) Question: if a man says to you: we took a third and a quarter from an amount of money and 14 still remain, how much was the total amount of money?

False Position: The answer: one should multiply 3 by 4; it is 12.

Then, subtract the third and the quarter from it; 5 still remains.

Rule of Three: The ratio of this 5 to 12 is as the ratio of 14 to the whole amount of money.

We relate as follows: if 5 is equal to 12, how much is 14 equal to?

You get 33 and three-fifths.

Check: To check it, take a third of 33 and three-fifths; it is 11 and a fifth.

Then, take a quarter of 33 and 3-fifths; it is 8 and 2-fifths.

Add it to 11 and a fifth; the sum is 19 and three-fifths.

Subtract it from 33 and 3-fifths; 14 remains no more and no less.

How Much - Money

7) Question: we added to an amount of money its quarter and its fifth and they are 12.

How much was the amount of money in the beginning?

כמה היה הממון בתחלה

False Position: The answer: seek the common denominator, which is 20, from the multiplication of [five by four].

Take its quarter, which is 5, and its fifth, which is 4, and add them to the denominator; it is 29.

Rule of Three: Then, take the denominator, which is 20, multiply it by 12 and divide the product by 29; you get 8 and 8 parts and so was the amount of money originally.

Check: To check it, take a quarter of the amount of money, which is 8 [and 8 parts]; the result is 2 integers and 2 parts of 29.

Take a fifth of the amount of money; the result is one integer and 3 remain.

We convert each of them into 29 parts and add the 8 parts to them; they are 95.

Divide them by 5; it is 19.

Sum up all the numbers; they are 12.

8) Question: if a man says to you: you took a fifth and a sixth of an amount of money plus 4 pešuṭim and the result is 24, how much is the total amount of money?

False Position: The answer: multiply 5 by 6; it is 30.

Take its fifth and its sixth and sum them up; it is 11.

Then, subtract 4, which is the addition, from 24; twenty remains.

Rule of Three: the ratio of 11 to 30 is the same as the ratio of twenty to the total amount of money.

Multiply 20 by 30; the result is 600

Divide it by 11; the result is 54 integers and 6 parts of 11.

Check: To check it, take a fifth of 54; it is ten integers and ten parts of 11.

Then, take a sixth of 54; it is 9 integers and one part of 11.

Sum them up; they are twenty integers.

Add 4 to them; they are 24 integers.

First from Last - Money

9) Question: we subtracted from an amount of money its third and its fifth, then we take 2 from what remained and 4 still remain.

False Position: The answer: look for the common denominator by multiplying 3 by 5; it is 15.

Then, we subtract the third and the fifth; 7 remains.

Since it is said that after the third and the fifth are subtracted, 2 is also subtracted from the remaining amount, we add it to 4; it is 6.

Rule of Three: we take the 7 that remains from the 15, after the third and the fifth are subtracted, and say: if 7 is equal to 19, how much is 6 equal to?

You receive 12 and 6-sevenths.

Check: To check it, take a third of 12 [and 6-sevenths]; it is 4 and 2-sevenths.

Take a fifth of 12 and 6-sevenths; it is 2 and 4-sevenths.

Subtract all this from 12 and 6-sevenths; the remainder is 6.

Subtract 2 from this 6; 4 remains.

How Much - Money

10) Question: we took a third of an amount of money plus 5 pešuṭim and its quarter minus 2 pešuṭim and the result is 8 pešuṭim.

The answer: you should know that 5 is additive and 2 is subtractive, so you should subtract 2 from 5; 3 remains. Then, subtract these 3 from 8 pešuṭim.

It is as if one says: We took a third and a quarter of an amount of money and the result is 5 pešuṭim.

Rule of Three: you should multiply 5 by 12 and divide it by 7; the result is 8 integers and 4-sevenths.

Check: You can check it by that you take a third of 8 [and 4-sevenths]; it is 2 and 6-sevenths.

Add 5 to it, as it is said "its third plus 5"; it is 7 integers and 6-sevenths.

Take a quarter of the total amount, which is 8 and 4-sevenths; the result is 2 integers and one part of 7.

Subtract 2 from it, as it is said "its quarter minus two"; one seventh remains.

Add it to 7 integers and 6-sevenths; it is 8 integers no more and no less.

11) Question: if an inquirer says: we summed a quarter of a munber plus 2 with its fifth minus 3 and the result is 6.

The answer: you already know that the addition is 2 and the subtraction is 3 and this is the opposite of the previous question. So, we do the opposite, that is instead of subtracting we add: we subtract 2 from 3; the remainder is 1. Then, we add the 1 to 6; it is 7.

It is as if it is said: we take a quarter and a fifth of an amount of money and the result is 7, how much is the amount of money?

Rule of Three: we relate as follows: we multiply 7 by 20; it is 140.

We divide it by 9; we receive 15 integers and 5 parts of 9 and this is the amount of money.

Check: You can know it by that you take a quarter of 15 [and 5-ninths]; it is 3 integers and 8 parts of 9.

Then, take a fifth of 15 [and 5-ninths]; it is 3 and one part of 9.

Add 2 to it and subtract 2; 6 remains.

12) Question: we took a half and a third of an amount of money then we took the square of the result and it is the initial amount.

How much is the amount of money?

False Position: The answer: take the half and the third from 6.

It is said "from 6" because it has a half and a third; they are 5.

Take its square; it is 25.

Take also the square of the common denominator, which is 6; it is 36.

Divide 36 by 25; you receive 1 integer and [1]1 parts of 25 and this is the amount of money.

Check: Check it: take a half of 1 and 11 parts of 25; it is 18 parts of 25.

Then, take a third of 1 and 11 parts of 25; it is 12 [parts of 25].

Add it to 18 [parts of 25]; it is 1 and a fifth.

Take its square; the result is 1 integer and 11 parts of 25, as it was at first.

13) Question: we took a half and a third of an amount of money then we took a square of the result and it is equal to the amount of money plus its third.

False Position: The answer: do as follows: take the common denominator, i.e. the number that consists of a half and a third; it is 6.

Then, take the half and the third; it is 5.

Take its square; it is 25.

Take the square of the common denominator, which is 36, and add its third to it; it is 48.

Divide 48 by 25; you receive 1 and 23 parts of 25 and this is the amount of money.

Check: To check it, take a half of this number, which is 24 parts of 25.

Then, take its third, which is 16.

Sum them up; it is 1 and 3-fifths.

Take its square; it is 2 integers and 14 parts of 25, which is the same as the amount of money plus its third.

14) Question: we took a half and a third of an amount of money, then from their sum we took its quarter plus a fifth of the remainder from the half and the third and it is equal to 12.

How much is the amount of money?

False Position: The answer: find a number that has a half, a third, a quarter, and a fifth; it is 60.

We take from 60 its third and its half; it is 50.

10 still remains. We take a fifth of the remaining 10; it is 2 integers.

We take a quarter of fifty; it is 12 and a half.

We add it to the 2 we have; it is 14 and a half.

Rule of Three: we multiply 12, which is the known number, by the common denominator, which is 60; the result is 720.

Divide it by 14 and a half; you receive 49 and 19 parts of 29.

Check: To check it, take a half of the 49 integers and their parts; you get 24 and 24 parts of 29.

Then, take a third of the 49 and its parts; you get 16 and 16 parts of 29.

Sum them up; it is 41 and 11 parts of 29.

Then, take a quarter of this number, as it is said "and a quarter of the sum of a half and a third"; it is 10 integers and 10 parts of 29.

We subtract 41 integers and 11 parts, which is a half and a third, from the amount of money, which is 49 integers and 19 parts of 29; the remainder is 8 and 8 parts of 29.

Take its fifth, as it is said "plus a fifth of the remainder"; it is 1 integer and 19 parts of 29.

Add it to the 10 integers and 10 parts of 29; it is 12.

15) Question: we summed half of an amount of money with a third of what remained and a quarter of the remainder after subtracting the third and the total sum is equal to 14.

How much is the amount of money?

כמה הממון

The answer: a number should be found that has a half, a third, and a quarter; you find it is 12.

Subtract its half from it, which is 6; the remainder is 6.

Take from the remaining 6 its third, which is 2; the remainder is 4.

Take a quarter of the 4, which is 1; the remainder is 3.

False Position: all the fractions are 9.

Rule of Three: multiply 14, which is the known number, by the common denominator, which is 12, then divide the product by 9; you receive 18 and 2-thirds and this is the [required] number.

Check: To check it, take a half of 18 and 2-thirds, which is 9 and a third.

Take a third of the remainder; it is 3 and a ninth.

Take a quarter of the remainder; it is 1 and 5-ninths.

Sum up all; it is 14 integers.

Multiple Quantities Problem - Two Amounts of Money

16) Question: we summed a quarter of an amount of money with a fifth of another different amount of money and the sum is 8.

How much is each amount?

כמה סך כל א' וא‫'

The answer: you can solve it in different ways [→ it is an indeterminate problem].

It is that you define any number to be a quarter of the first amount and you subtract this number from 8; then; you define what remains from the 8 as the fifth of the second amount. This way you find them.

• Suppose, for example that 3 is a quarter of the first amount.

Then, the first amount is 12, because 3 is a quarter of 12.

Subtract 3 from 8; the remainder is 5. Define the 5 as a fifth of the second number.

So, the second amount is 25.

• Or, if you wish, define 2 as the quarter of the first amount.

Then, the first amount is 8.

Subtract 2 from 8; the remainder is 6. Define the 6 as a fifth of the second amount.

So, the second amount is 30.

• Or, if you wish, define 4 as the quarter of the first number.

Then, the first number is 16.

Subtract 4 from 8; the remainder is 4. Define it as a fifth of the second number.

So, the second number is 20.

And so on endlessly - define any number you wish as a quarter of the first amount, subtract it from 8, and define what remains as a fifth of the second number.

Sums

• sum of natural numbers

17) Question: if you want to sum from 1 to 14, i.e. 1 with 2, with 3, and so on until 14.

1 should be added to 14; it is 15.

Then, multiply 15 by 14; it is 210.

Divide it by 2; you receive 105 and this is the required.

• even number of terms \ last term is even

18) Question: if the number [of terms] is even take half this number and multiply it by the whole number, then add the result to half the number.

• Example: we want to know [the sum] from 1 to 16.

A half of 16 should be taken, which is 8.

Multiply it by 16; the result is 128.

Add a half of 16 to it also, which is 8; it is 136 and this is the required.

• odd number of terms \ last term is odd

19) Question: if [the number of terms] is odd the larger part should be taken and multiplied by the whole number and the result is the required.

• Example: we want to know the sum from 1 to 25.

The greater part should be taken from 25, which is 13.

Multiply 13 by 25; the result is 325 and this is the required.

Or, if you wish, divide 25 by 2; it is 12 and a half.

Multiply it by 25; it is 312 and a half.

Then, add a half of 25 to it, which is 12 and a half; it is 325.

• the first term is not 1

20) Question: sum from 1 to 10 without the [first] three.

1 to 10 should be summed, which is 55 as you know.

Then, sum from 1 to 3; which is 6.

Subtract 6 from 55; the remainder is 49 and this is the required.

• the number of terms \ the last term is unknown

21) Question: we sum up all the numbers until an unknown number and the sum is 120.

This number should be divided by 2; it is 60; then, multiplied by 4; it is 240.

Now, you should extract the preceding root, i.e. that is closest to 240; you find it is 15 and this is the required.

I.e., when you sum from 1 to 15, the result is 120.

• sum of odds

22) Question: sum all the odds from 1 to 9.

1 should be added to the number mentioned, the resulting sum should be divided by 2, then the square of the quotient should be taken and this is the required.

We add 1 to 9; it is 10.

Then, we divide it by 2; it is 5.

We take its square; it is 25 and this is the sum of all the odd numbers from 1 to 9.

• sum of evens

23) Question: if you want to know the sum of all the evens from 2 to 12.

The half of 12 should be taken; it is 6; then its square, which is 36 should be [added]; it is 42.

• sum of odds: number of terms \ last term is unknown

24) Question: we sum up all the odds until an unknown numbers and the sum is 25.

The root of 25 should be extracted, then multiply it by 2, and subtract 1 from the product.

The root of 25 is 5. Double it; it is 10. Subtract 1 from it; 9 remains and this is the required.

• sum of evens: number of terms \ last term is unknown

25) Question: we summed up all the evens from 1 to an unknown number and the sum is 42.

The preceding root should be extracted. It is known that the preceding root is 6. Multiply it by 2; it is 12 and this is the required.

• sum of squares

26) Question: if you want to sum up all the squares [of the numbers] from 1 to 4.

All the numbers from 1 to 4 should be summed; they are ten

Then, take 2-thirds of 4 plus a third; it is 3.

Multiply 3 by 10; it is 30.

27) Question: if you want to sum up all the squares [of the numbers] from 1 to 25

Sum up all the numbers; they are 15.

Take 2-thirds of 5 plus a third; it is 3 and 2-thirds.

Multiply it by 15; it is 55.

• sum of squares: first term is not 1

28) Question: sum up [the squares] from the square of 4 to the square of 8.

All the squares [of the numbers] from 1 to 8 should be summed; they are 204.

Then, sum up the squares [of the numbers] from 1 to 3

Subtract 14 from 204; the remainder is 190 and this is the required.

• sum of the squares of even numbers

29) Question: if you want to sum up all the squares of the even numbers from 2 to 8.

All the even numbers from 2 to 8 should be summed; they are twenty.

Then, take 2-thirds of 8 plus 2-thirds; it is 6 integers.

Multiply it by 20; it is 120 and this is the required.

• sum of the squares of odd numbers

30) Question: sum up the squares of the odd numbers from 1 to 7

Sum up all the odd numbers until 7; they are 16.

Subtract a third of 16 plus 2 parts of 12; or subtract from 15 it third.

או תפיל מט"ו שלישיתם

[Subtract 15 from] 16; 1 remains.

Then, subtract from this 1 its half and add it to 2-thirds of 15; it is 10 and a half.

Add 1 to 7; it is 8.

Multiply is by 10 and a half; you receive 84.

• sum of cubes

31) Question: sum up from the cube of 1 to the cube of 7

Know that the cubes are 1 time 1, 1 time 2, 1 time 3, 1 time 4, and so on until 7; then take 2 and multiply it by all the numbers from 1 to 7; then multiply 3 by 1 to 7; then 4, and so on until the end of the numbers.

ואח"כ תקח הב' ותכה אותו עם כל המספרים מא' עד ז‫'
ואח"כ תכה הג' מא' עד ז‫'
וכן הד' וכן עד כלות כל המספרים

If you want to know the total sum, you should sum up all the [numbers] from 1 to 7; they are 28.

Then, take the square of 28, which is 784, and this is the sum of all the cubes from 1 to 7.

• square of sum of natural numbers: number of terms \ last term is unknown

32) Question: sum up from the cube of 5 to the cube of 12.

Sum up all the cube numbers from 1 to 12; they are 6084.

Then, sum up all the cube numbers from 1 to 4; they are 100.

Subtract it from 6084; the remainder is 5984 and this is the required.

• sum of cubes: number of terms \ last term is unknown

33) Question: we summed up all the cubes from 1 to an unknown number and the sum is 225.

How much is the unknown number?

כמה הוא המספר הנעלם

The root of the total sum should be extracted, i.e. the root of 225; its root of 15.