Difference between revisions of "ספר הכללים במספר"

Problems of Various Types

Find a Number Problem - Subtraction of Fractions

1) If a man asks you: how large is the third from the quarter?

common denominator: say: 3 times 4 is 12.

The third is 4 dinar.

The quarter is 3 dinar.

We find that the third is greater than the quarter by a twelfth.

• If you are asked: how large is the quarter from the fifth?

common denominator: do as we did above and say: 4 times 5 is 20.

The quarter is 5-twentieths.

The fifth is 4-twentieths.

We find that the quarter is greater than the fifth by a twentieth.

• If you are asked: how large is the third from the fifth?

common denominator: say: 3 times 5 is 15.

The third is 5-fifteenths.

The fifth is 3-fifteenths.

We find that the third is greater than the fifth by two-fifteenths.

Find a Number Problem - Addition of Fractions

2) How much are a third, a quarter, a fifth, a sixth, and a seventh of one pašuṭ?

common denominator: you should divide the pašuṭ into the number in which you find all these fractions.

Follow this way and say: 3 times 4 is 12; 5 times 12 is 60; 6 times 60 is 360; 7 times 360 is 2520; in this number you find your question.

ה' פעמין י"ב הם ס‫'
ו' פעמי' ס' הם ש"ס
ז' פעמין ש"ס הם אלפי' ותק"כ ובזה השעור תמצא שאלתך

The third is 849 parts.

The quarter is 630 parts.

The fifth is 504 parts.

The sixth is 420 parts.

The seventh is 360 parts.

The total is 2754.

Subtract 2520 parts from it, which is one unit; you are left with 234 parts.

We find that a third, a quarter, a fifth, a sixth, and a seventh of one pašuṭ are one pašuṭ and 234 parts of one pašuṭ, when you divide it into 2520 parts, which is 117 parts of 1260.

שהוא קי"ז חלקי' מאלף ור"ס וכה"ל

Find a Quantity Problem - Whole from Parts Problem - Lance

3) Question: one-third of the lance is in the water, its quarter is in the ground, and 10 cubits of it are up above the water.

How many cubits is the whole lance?

כמה אמות כל הרומח

False Position: We look for a number that has a third and a quarter; it is 12.

The sum of its third and its quarter is 7.

We subtract it from 12; 5 remains.

Rule of Three:

We multiply the extremes; the result is 120.

We divide it by 5; the result is 24 and this is the length of the whole lance.

Check: Its third is 8; its quarter is 6; we subtract them from 24; 10 integers remain no more and no less.

How Much Problem - Money

4) Question: An amount of money - we sum its fifth, its seventh, and its ninth; they are 10.

How much is the amount of money?

כמה הממון

False Position - common denominator: We look for the denominator; it is 315 and the parts are 143.

When the part of 315 is a fifth, the result is 63.

When the part is a seventh, the result is 45.

When the part is a ninth, the result is 35.

Sum them up; the result is 143.

Rule of Three: We multiply 315 by 10; the result is 3150.

We divide it by 143; the result is 22 integers and 4 parts of 143 and this is the amount of money.

Check: Take a fifth of 22; it is 4 and 2 remain that are 2 times 143, which is 286.

Add to it 4 parts; it is 290.

Divide it by 5; the result is 58 parts of 143.

Do likewise with the ninth and the seventh; and the result is 10.

It has another method:

Consider the fractions you sum that are 10 as 143 and the original amount of money as 172.

Rule of Three: say: if 143 equals 10, how much is 172 equal to?

Say: 10 times 172 is 1720. Divide it by 143; the result is 12 integers and 4 parts of 143.

חלקם על קמ"ג יבאו י"ב שלמי' וד' חלקי' מקמ"ג

Add the 10 to it; it is 22 integers and 4 parts of 143 no more and no less.

This rule is also suitable for knowing the measure of the lance mentioned above.

Find a Quantity Problem - First from Last Problem – Money

5) We do the opposite: An amount of money - we subtract its fifth, its seventh, and its ninth from it and 10 remains.

We subtract 143, which are the fractions, from 315, which is the denominator; 172 remains.

Rule of Three: we do as follows: we multiply 10 by 315; the result is 3150. We divide it by 172; the result is 18 integers and 54 parts of 172.

חלקנום על קע"ב עלו י"ח שלמי' ונ"ד חלקי' מקע"ב

Check: we take a fifth, a seventh, and a ninth of this number; 10 integers remain.

How Much Problem – Money

6) Question: An amount of money – we add to it its half, its third, its fifth, and its sixth, and the total sum is 40.

How much is the money?

כמה היה הממון

False Position: we know that a half, a third, and a sixth are one integer. We consider [the amount of money] as one; so they are two plus a fifth.

Now, we have to divide 40 by 2 and a fifth and the result is the amount of money: we take 5 for each of the integers and add the fifth to them; they are [11]. We also multiply 40 by 5; it is 200. We divide it by 11; the result is 18 integers and 2 parts of 11.

הנה נקח לכל אחד מן השלמים ה' ונשים עמהם החמשית יהיו אלף
גם נכפול המ' על ה' עד שיהיו דרך אחד יהיו ר‫'
נחלקנו על י"א יעלו י"ח שלמים ועוד ב' חלקי' מי"א

How Much Problem – Money

7) Question: We take a fifth amount of money, its seventh, and its ninth.

How much is [their sum] in relation to the [original] amount of money?

כמה הוא מערך הממון

False Position - common denominator: we look for the common denominator; it is 315.

We sum [its] fifth, seventh, and ninth; they are 143.

We divide it by 35; it is 4-ninths and 3 parts of 35 remain, because 35 is the ninth, and 5 is a seventh of the ninth, so 3 parts are 3 fifths of a seventh of a ninth.

כי ל"ה הוא התשיעית וה' הוא שביעית התשיעית
לכן ג' חלקי' הם ג' חמשיות שבעית התשיעית

Pricing Problem - Find the Amount

8) Question: A man sells 13 measures for 23.

How many measures will he sell for 7 [pešiṭim]?

כמה מדות יתן בז' פשוטי‫'

Rule of Three: we say: 7 times 13 is 91. We divide it by 23; the result is 3 measures and 22 of 23 parts of one measure.

נחלקם על כ"ג בא ג' מדות וכ"ב מכ"ג חלקי' במדה אחת

Pricing Problem - Find the Price

• Now, we reverse the calculation and want to know how much will he get for 7 measures?

Rule of Three: we say: 7 times 23 is 161. We divide it by 17; it is 12 and 5 parts of 13 of one pašuṭ.

נחלקם על י"ז יהיו י"ב וה' חלקי' מי"ג בפשוט

Exchange Problem - Currencies

• If 3 of Treviso(?) are worth 4 of Cortona, how many of Cortona are 5 of Treviso(?) worth?

Rule of Three: of Cortona

חלקם על ג' בא ו' קורטוניס וב' שלשי' וכה"ל

Motion Problem – Pursuit

9) Question: A man sent a messenger to walk 29 miles a day.

After 10 days of walking, he sent another messenger to walk after him 37 miles a day.

When will he catch up with him?

אחר מהלך י' ימי' שלח רץ אחר אחריו שילך בכל יום ל"ז מילי‫'
מתי ישיגנו

Rule of Three: we multiply the miles he walks by 10 days; it is 290. We divide it by the difference between the two velocities, which is 8; it is 36 days and a quarter of a day.

נחלקם על היתרון שבין שני המהלכים שהוא ח' והנו ל"ו ימים ורבע יום

Motion Problem – Encounter

10) Question: Reuven left his city, walking to Shimon, to his city, on Sunday morning of the first of the month.

Shimon left his city on that same day, walking to Reuven's city.

The distance between the two cities is 100 miles.

Reuven is walking 19 miles a day and Shimon is walking 17 [miles a day].

When will they meet?

ובאותו יום עצמו יצא שמעון מעירו ללכת לעיר ראובן
והמרחק בין שני הערים ק' מלים
ומהלך ראובן ביום אחד י"ט מלים ומהלך שמעון י"ז
מתי יתחברו

Do as follows: sum up the two velocities; it is 36. Divide the 100 miles by it; it is 2 days and 28 parts of 36 of one day remain, which are 7-ninths of a day.

חלק עליו הק' מילים יהיו ב' ימים ישארו כ"ח חלקי' מל"ו ביום אחד שהם ז' תשעיו' יום

How Many Problem - Group of People

11) Question: A man passed by a group of people.

He said to them: hello one hundred people.

They answered him: we are not one hundred people, but all of us, and other like us, and half of us, and a quarter of us plus one will make 100

אמר להם שלו' לכם מאה איש
ענו לו אין אנו מאה רק אנו ואחרים כמונו ומחציתנו ורביעתנו עדיף נהיה מאה

False Position: we take one for their number and one "like us" - it is two; and "half of us" - it is two and a half. We add a quarter; it is [2 and] 3-quarters.

נוסיף רבעיתו יהיו ‫^[3]ג' רבעיות

Since we have quarter[s], we take 4 for each unit; they are 8. We add the 3-quarters to them; they are 11.

נחבר אליהם הג' רבעים יהיו י"א

Since they said that they are one hundred with him, their number with the additions is 99.

We convert them into quarters; they are 396.

We divide them by 11; they are 36 and this is their number.

Buy and Sell Problem

12) Question: A man bought 100 liṭra for 100 zehuvim.

He sold 50 [of them] at a liṭra and a quarter for one zahuv, and the other 50 at a liṭra minus a quarter for one zahuv.

We want to know: did he earn or lose?

אחרי כן מכר הנ' ליט' ורבע ליט' בזהוב והנ' ליט' מכר ליט' פחות רבע ליט' בזהוב
נבקש לדעת אם הרויח או הפסיד

We convert the first 50 into 200, because they are quarters, then divide them by 5, as he sold a liṭra and a quarter for one zahuv; they are 40 zehuvim.

נחלקם על ה' כי ליט' ורבע ליט' מכר בזהוב יהיו מ' זהובי‫'

We multiply the other 50 by 4 also; they are 200. We divide them by 3, as he sold 3-quarters for one zahuv; they are 66 zehuvim and two-thirds of a zahuv.

נחלקם על ג' כי ג' רבעים מכר בזהוב והנה יהיו ס"ו זהובים ושני שלישי זהוב

Add the 40 to them; the profit is 6 zehuvim and two-thirds of a zahuv.

Boiling Problem

13) Question: A man had 10 measures of apricot and he wanted to boil them so that only one third will remain.

He started to cook [them] until eight measures remained of them.

Then two measures overflow.

Now he wants to boil [the remainder] until it will be reduced [as planned for] the original [amount] of apricot

והנה החל לבשל עד שנשארו מהם ח' מדות
ונשפך מהם ב' מדות
והנה רצה לבשלם עד שיהיה כמשמש הראשון

You have three known numbers:

The first is a third of 10, which is 3 and a third.

The second is the 8 cooked measures.

The third is the 6 [measures] left over from what was overflowed.

Rule of Three: we multiply 6 by 3 and a third; it is 20. Divide it by 8; it is 2 and a half.

וחלקם על ח' יהיו ב' וחצי

Partnership Problem - For the Same Time

14) Question: Four people: one of them had 11 dinar, the second had 13 dinar, the third had 15 dinar, and the fourth had 17 dinar.

They earned 19 dinar.

How much should each of them take [from the profit]?

והרויחו י"ט דינרי‫'
כמה יקח כל אחד ואחד

False Position: we sum up all four amounts of their money; it is 56.

Rule of Three: as the ratio of each to 56, so each takes from 19.

We do as follows: we multiply 11 by 19; the result is 209. We divide [it] by 56; the result is 3 integers and 41 parts.

נחלק על נ"ו יעלו ג' שלמי' ומ"א חלקי‫'

We do the same with 13; the result is 247. We divide it by 56; the result is 4 integers and 23 parts.

חלקנום על נ"ו עלו ד' שלמי' וכ"ג חלקי‫'

We do the same with 15; it is 285. We divide it by 56; the result is 5 integers and 5 parts.

חלקנום על נ"ו עלו ה' שלמי' וה' חלקי‫'

We do the same with 17; the result is 323. We divide it by 56; the result is 5 integers and 43 parts.

חלקנום על נ"ו עלו ה' שלמי' ומ"ג חלקי‫'

Check: we sum up the integers and the fractions; the result is 19 integers, because the fractions are 56 parts.

Purchase Problem – Moneychanger

15) Question: The moneychanger has three [kinds of] coins.

One zahuv is worth three dinar of the first [kind of] coins; or four of the second [kind]; or six of the third [kind].

A man came and asked the moneychanger to give him from the three [kinds of] coins for one zahuv equally, so that the amount of the expensive will be equal to the amount of the inexpensive.

ובא אדם אחד ובקש למחליף שיתן לו מג' המטבעים בזהוב ויהיה המספר שוה מן היקרים כמו משאינם יקרים

False Position: look for a [number] that has a third, a quarter, and a sixth; it is 12.

The parts are 9 and this is a dinar.

We divide the denominator by it; it is 12-ninths and so he takes from each coin.

Another [way]: Say: what is the ratio of 12 to 9; it is the amount plus its third. We add 4 to 12; it is 16.

Payment Problem - Digging a Hole

16) Question: Reuven hired Shimon to dig for him in the ground 7 in length, 6 in width, 5 in depth, and he will pay him 11 pešiṭim, but he dug 6 in length, 5 in width, 4 in depth.

How much should be his payment?

והוא חפר ו' באורך ה' ברוחב ד' בעומק
כמה שכרו

We do as follows: 7 times 6 is 42. Multiply it by 5, which is the depth; it is 210.

We also multiply the second number, which is 6, by 5; it is 30. We multiply it by 4, which is the depth; it is 120.

Rule of Three: we multiply 11 times 120; it is 1320. Divide it by 210; the result is 6 integers and 60 remains, which is 2-sevenths of one pašuṭ.

חלקם על ר"י עלו ו' שלמי' ונשאר ס' שהם ב' שבעיות פשוט וכה"ל

Buy and Sell Problem

17) Question: A man bought three fifths of a liṭra for one pašuṭ, then he sold four sevenths of a liṭra for one pašuṭ and he earned one pašuṭ.

How much money did he have originally?

כמה היה ממונו

common denominator: look for the common denominator: multiply 5 by 7; it is 35.

Its 3-fifths are 21.

Its 4-sevenths are 20, so the original amount of money is 20.

It seems that he wants to say that 20 pešuṭim were spent.

18) Question: A man bought four sevenths of a liṭra for one pašuṭ, then he sold five ninth of a liṭra for one pašuṭ and he earned one pašuṭ.

How much money did he have originally?

כמה היה הממון

It is known that 4-sevenths is greater that 5-ninths.

common denominator: the common denominator is 63.

Its 5-ninths are 35.

Its 4-sevenths are 36.

Check: you can check this: as he buys 4-sevenths of a liṭra for one pašuṭ, and his money is 35, he has 20 liṭra.

Convert them into ninths; they are 180.

Divide this number by 5, because he sold 5-ninths for one pašuṭ; you receive 36.

• If it is said that he earned 2 pešiṭim:

Multiply them by 35 and they are the pešuṭim he spent.

• If it is said [that he earned] 3 pešiṭim:

He multiplies them by 35.

And so on for any number.

Exchange Problems

19) The Rule of Three: The rule of all calculations is in three categories: geometry; weights; arithmetic.

Therefore, we specify the appropriate rules for all three.

If one asks you for a calculation in one of the three categories: we multiply the thing we want to know by the one that is not of the same species, then divide it by the third.

נרבה אותו הדבר שנרצה לידע כנגד אותה שאיננה ממינה עצמה ממש
ותחלקנה על השלשי‫'

• Example: If 7 of Pisa are worth 9 of Cortona, how many of Cortona are 100 liṭra of Pisa worth?

You already know that the thing of Cortona is not of the same species, so we multiply 9 times 100; it is 900.

Divide it by 7; it is 128 liṭra and 11 dinar and this is the value of 100 liṭra of Pisa in Cortona.

Know that if 7 of Pisa equal 9 of Cortona: 7 dinar of Pisa are equal to 9 dinar of Cortona; 7 liṭra of Pisa are equal 9 liṭra of Cortona; 700 liṭra of Pisa are equal to 900 liṭra of Cortona.

20) Rule of Three: with fractions:

נדע באיזה המספר ימצאו אלו השבורים
ואחר שידעת באי זה המספר ימצאו אלו השבורים נרבה הב' חלקי' באותו המספר שימצאו בו אלו השבורים
אחרי כן נרבה אותו הדבר שנרצה לדעת כנגד אותו הדבר שאנינה ממינה עצמה
ונחלק אותה על השלישית וכה"ל

21) Rule of Three: with fraction:

נרבה הב' חלקי' באותו המספר דהיינו עם אותו השבור
אחרי כן נרבה אותו הדבר שנרצה לידע כנגד אותו הדבר שאנינה ממינה
ונחלק אותה על השלישית

• If 5¼ of Pisa are worth 7 of Cortona, how many of Cortona are 1000 liṭra of Pisa worth?
  

converting to quarters

לכן נרבה הב' חלקי' עם ד‫'

21 of Pisa = 28 of Cortona

reducing by 7

3 of Pisa = 4 of Cortona

Rule of Three:

1000 liṭra of Pisa = 1333 liṭra + 6 dinar + 8 pešuṭim of Cortona

ונחלק בג' שיבא אלף ושל"ג ליט' וו' דינרי' וח"פ וכן ישוו אלף ליט' מפיסני לקורטוט‫'

• If 7⅓ ounce of Pisa are worth 73 liṭra and 5 dinar of Cortona, how many of Cortona are 19 ounce of Pisa worth?

כמה ישוו הי"ט אונקיו' כסף לפי זה החשבון

converting to parts of 12

לכן יש לנו לרבות הב' חלקי' בי"ב

88 ounce = 879 liṭra

Rule of Three:

19 ounce = 189 liṭra + 15 dinar + 8²/₁₁ pešiṭim

חלקם בפ"ח יבא קפ"ט ליט' וט"ו די' וח"פ וב' חלקי' מי"א וכן ישוו הי"ט אונקיו' כסף באותו החשבון

Interest and Pricing Problems

22) Question: one kikkar yields 36 liṭra and 18 dinar a year.

How long will it take 65 liṭra to yield the same?

הס"ה ליט' בכמה זמן ירויחו כהם

Rule of Three: multiply the 100 liṭra by the 12 months of the year, say: 12 times 100 are 1200.

Divide them by 65; each part receives 18 months, 13 days and 11 parts of 13 parts of a day.

We find that 65 liṭra yield in 18 months, 13 days, and 11 parts of 13 of a day the same as one kikkar yields in one year.

23) If a man says: I lent some money at 3 pešuṭim a month for one liṭra.

How much will 60 liṭra make in 8 months?

כמה יבואו הס' ליט' ח' חדשים

Multiply the interest by the months and say: 3 times 8 are 24 [pešuṭim], which are 2 dinar.

Then, multiply these 2 dinar by 60 liṭra and say: 60 times 2 dinar are 120 dinar, which are 6 liṭra.

We find that 60 liṭra at 3 pešuṭim for one liṭra a month will make 6 liṭra in 8 months.

24) If a man asks you: I lent some money at one liṭra for so and so a month.

How many liṭra will yield one pašuṭ a day?

כמה ליט' ירויחו היום א"פ

We divide 30 liṭra by the amount of money that one liṭra makes a month.

• Example: I lent one liṭra for 2 pešuṭim and a half a month.

You want to know how many liṭra will yield one pašuṭ a day?

ותרצה לידע כמה ליט' ירויחו היום א"פ

We divide 30 liṭra by 2 and half; the result is 12 liṭra.

We find that 12 liṭra will make one pešuṭim a day.

25) If a man asks you: I lent one liṭra for so and so a month.

In how many days will one liṭra yield one pašuṭ?

Divide 30 days by the amount of money that one liṭra yields a month.

• Example: you lent [one liṭra] for 3 [a month].

You want to know in how many days will the liṭra yield one pašuṭ?

Divide 30 days into 3 parts; the result is 10 days.

So, the liṭra will yield one pašuṭ in 10 days.

26) If a man asks you: one kikkar yields so and so liṭra a year.

How many liṭra will yield one pašuṭ a day?

כמה ליט' ירויחו היום א"פ

Divide 150 liṭra by the number of liṭra that one kikkar yields a year.

• Example: one kikkar yields 12 liṭra a year.

You want to know how many liṭra will yield one pašuṭ a day?

ותרצה לידע כמה ליט' ירויחו היום א"פ

Divide 150 liṭra into 12 parts; the result is 12 and a half.

So, 12 liṭra and a half will yield one pašuṭ a day.

27) One kikkar yields so and so liṭra a year.

In how many days will one liṭra yield one pašuṭ?

בכמה ימים ירויח הליט' אח‫'

150÷(the liṭra earned a year)

• One kikkar yields 9 liṭra a year.

In how many days will one liṭra yield one pašuṭ?

days

הרי שבי"ו ימים וב' שלישי ירויח הליט' אח' וכה

28) So and so liṭra yield so and so [liṭra] in so and so [months].

In how many [months] will so and so liṭra yield the same?

Rule of Three:

• 50 liṭra yield 3 liṭra in 4 months.

In how many [months] will 75 liṭra yield the same?

הע"ה ליט' בכמה זמן ירויחו כהם

months

• 29) So and so liṭra yield so and so liṭra in so and so [months].

How many liṭra will yield the same in so and so [months]?

כמה ליט' בכל כך זמן ירויחו כהם

(months₂×liṭra)÷months₁

• 25 liṭra yield 40 dinar in 6 months.

How many liṭra will yield the same in 8 months?

כמה ליט' ירויחו בהם בח' חדשים

18 liṭra and 15 dinar

חלקם בח' חלקי' יבא י"ח ליט' וט"ו דינרי‫'
נמצא שי"ח ליט' וט"ו דינרי' ירויחו בח' חדשים מ' דינרי' כמו הכ"ה ליט' ו' חדשים

30) I lent one liṭra for so and so [pešuṭim] a month.

In how many [years] will so and so liṭra be doubled?

כל כך ליט' בכמה זמן יכפלו בלתי לשים ריוח אקרן

20÷(the pešuṭim earned a month)

• I lent one liṭra for 3 pešuṭim a month.

In how many [years] will 95 liṭra be doubled?

6 years and 8 months

31) One kikkar yields so and so liṭra a year.

In how many [years] will so and so liṭra be doubled?

כל כך ליט' בכמה זמן יכפלו בלתי לשים ריוח אקרן

100÷(the liṭra earned a year)

• One kikkar yields 8 liṭra a year.

In how many [years] will 45 liṭra be doubled?

המ"ה ליט' בכמה זמן יכפלו

12 years and 6 months

נמצא שבי"ב שנים וחצי יכפלו המ"ה ליט‫'

32) One kantar [measurement of weight] of lint is worth 17 liṭra‏ and 5 dinar.

One kantar is 165 liṭra.

How much is one liṭra worth?

כמה ישוה הליט‫'
והקנטרו' הוא קס"ה ליט‫'

pešuṭim

אחרי כן חלק אלו הה' ליט' וט"ו די' בה' חלקי' שיבא לכל חלק כ"ג דינרי‫'
עת' תחלק אלו הכ"ג דינרי' בי"א חלקי' שיבא לכל חלק בה"פ וא' חלק מי"א
נמצא שהליט' מן המוך שוה וא' חלק מי"א בפשוט

33) So and so liṭra yield one pašuṭ a day.

How many [pešuṭim] will one liṭra yield a month?

לאי זה חשבון יבא החדש הליט‫'

30÷(the liṭra that yield one pašuṭ a day)

• 12 liṭra yield one pašuṭ a day.

How many [pešuṭim] will one liṭra yield a month?

ותרצה לידע לאי זה חשבון יבא החדש הליט‫'

pešuṭim

נמצא שיצא החדש הליט' ב"פ וחצי וכה

34) So and so liṭra yield one pašuṭ a day.

How many liṭra will 100 liṭra yield a year?

כמה ליט' ירויחו ק' ליט' א' שנה

• 15 liṭra yield one pašuṭ a day.

How many liṭra will 100 liṭra yield a year?

כמה ירויחו ק' ליט' א' שנה‫'

liṭra for 100 liṭra a year

נמצא שק' ליט' ירויחו א' שנה י' ליט‫'

35) One kikkar yields 29 liṭra 13 dinar and 7 pešuṭim a year.

How many liṭra will yield the same in 7 months?

כמה ליט' ירויחו בהם ז' חדשי‫'

171 liṭra, 8 dinar, and pešuṭim

וחלקם בכל כך חלקי' כמו החדשי' שנרצה לידע דהיינו הז' חדשים
ואמור י"ב פעמי' ק' ליט' יבא אלף ור‫'
חלקם על ז' שיבא קע"א ליט' וח' דינרי' וחצי וו' חלקי' מז' בפשוט
נמצא שקע"א ליט' וח' די' וחצי וו' חלקי' מז' בפשוט ירויחו כ"ב בז' חדשי' כמו הככר א' שנה וכה

36) One kikkar yields so and so liṭra a year.

In how many [years] will one kikkar be doubled?

בכמה זמן יכפלו בלתי לשים ריוח אקרן

• One kikkar yields 14 liṭra a year.

In how many [years] will [one kikkar] be doubled?

7 years, 1 month and 21³/₇ days

נמצא שבז' שנים וא' חדש וכ"א ימים וג' חלקי' מז' ביום יכפלו

37) One liṭra yields 2½ pešiṭim a month.

In how many [years] will [so and so dinar] be doubled?

בכמה זמן יכפלו בלתי לשים רבית אקרן

• One liṭra yields 2½ pešiṭim a month.

In how many [years] will [20 dinar] be doubled?

years

נמצא שבח' שנים יכפלו וכה

38) One kikkar yields so and so liṭra a year.

How many liṭra will yield one pašuṭ a day?

כמה ליט' ירויחו היום א"פ

• One kikkar yields 13 liṭra a year.

How many liṭra will yield one pašuṭ a day?

11 liṭra, 10 dinar and 9³/₁₃pešuṭim

39) One liṭra [yields] so and so a month.

How much will 100 liṭra yield a day?

כמה ירויחו היום ק' ליט‫'

3⅓×(money earned)

• One liṭra [yields] 3 pešiṭim a month.

How much will 100 liṭra yield a day?

ותרצה לידע כמה ירויחו היום ק' ליט‫'

pešuṭim a day

נמצא שק' ליט' ירויחו היום י"פ וכה

40) One kikkar yields so and so pešuṭim a day.

How many [liṭra] will one kikkar yield a year?

כמה ירויח הככר השנה

• One kikkar yields 10 pešuṭim a day.

How much will it yield a year?

liṭra

נמצא שירויח הככר ט"ו ליט' השנה

41) One kikkar yields so and so pešuṭim a day.

How much will one liṭra yield a month?

כמה יבא החדש הליט‫'

• One kikkar yields 10 pešuṭim a day.

How much will one liṭra yield a month?

pešuṭim

42) One kikkar yields so and so liṭra a year.

How much will one liṭra yield a month?

כמה יבא החדש הליט‫'

• One kikkar yields 12 liṭra a year.

How much will one liṭra yield a month?

pešuṭim

43) You lend 100 liṭra for one month at 8 liṭra a year.

You want to know how much will be earned

100 pešuṭim = 8 dinar and 4 pešuṭim

1⅗ pešuṭim for a liṭra

the profit =

160 pešuṭim = 13 dinar and 4 pešuṭim

• At 7 liṭra a year

1⅖ pešuṭim for a liṭra

12 dinar minus 4 pešuṭim

• another method

You lend 100 liṭra for one month at 8 liṭra a year.

You want to know how much will be earned.

20 pešuṭim = 5 yadot a month

1 yadot = ⅓ dinar

40 yadot = 13 dinar and 4 pešuṭim

35 yadot = 12 dinar minus 4 pešuṭim

44) I lent 20 dinar for one year at 4 pešuṭim a month and at the end of the year I cashed some.

I lent [what remained] for a second year at 4 pešuṭim a month also and cashed some.

I lent [what remained] for a third year again at 4 pešuṭim a month and cashed some as I did in the first and the second year.

Then I had no pašuṭ left.

How much is the sum of all the repayments, and how much was each repayment?

הלותים שנה שניה לחשבון דפ"ה כמו כן ופרעתי מהם קצת
הלותים שנה שלשית לחשבון דפ"ה כמו כן ועשיתי הפרעון כמו שעשתי בשנה ראשונה ושנייה
ולא נשאר בידי אי פשוט
כמה עלו כל הפרעונות וכמה כל פרעון ופרעון לבדו

the three payments =

= 28 dinar + pešuṭim

each payment =

= 9 dinar and pešuṭim

if the payment in each year was 6 dinar:

• at the end of the third year remained:

dinar

• at the beginning of the third year remained:

dinar

אמור אם בסוף שנה שלישית נשארו ו' דינרי' הסר מהם השתות והנשאר הוא הקרן שנשאר בסוף שנה שנייה
כי לעולם בכל חשבון עולה בו ריוח לחשבון דפה"ה החומש מלג שהוא שתות מל' ולכן כשתסיר ממנו השתות ופרי הוא מעתה רשאי לעשות מהם ובהם ככל אות נפשו אך א' עליו פרוע בתנאי שיפרע לעולם כל השתות ישאר הקרן לבד
א"כ בראש שנה שלישית נשארו ה' די‫'

• at the end of the second year remained:

dinar

• at the beginning of the second year remained:

= 9 dinar and 2 pešuṭim

• at the end of the first year remained:

= 15 dinar and 2 pešuṭim

• at the beginning of the first year remained:

= 12 dinar and 7⅔ pešuṭim

Rule of Three:

if for 12 dinar and 7⅔ pešuṭim the payment is 6 dinar each year, how much will be the payment for 20 dinar?

Rule of Three:

• another method:

dinar

dinar

dinar

each payment:

= 9 dinar and pešuṭim

אחרי כן תסיר מן הנשאר החומש ותכפול המותר
יעלה בידך ט' די' וה"פ ופ"ה חלקי' מצ"א וכן יבא לכל פרעון ופרעון וכה"ל

45) You bought some fruit in trade at so and so liṭra for one kikkar and you want to know how much one liṭra is worth

• I bought 100 liṭra of pepper for 40 liṭra and you want to know how much one liṭra is worth

one liṭra of pepper for dinar

if each liṭra is worth pešuṭim

• If you bought one kikkar in trade for 40 liṭra and you want to know how much one liṭra is worth

one liṭra for dinar

46) How much will be the profit of 100 liṭra for one day at 8 liṭra a year?

Rule of Three: pešuṭim

Problems of Various Types

Rent Problem

47) If you rent a house at so and so liṭra a year and you want to know how much [should be paid] a day

take for every liṭra pešuṭim a day

• You rent a house at 20 liṭra a year

pešuṭim

Exchange Problems

48) 22 of Pisa are worth 25 of Bologna.

What is the exchange of the liṭra of Bologna to a liṭra of Pisa?

Rule of Three:

20 dinar and pešuṭim?

חלקם על כ"ב יבא כ' די' וח"פ ח' חלקי' מי"א בפשוט
וכן יתן חלוף הליט' מן הבוליי' לליט' מן הפיסני‫'

Proportions of Fractions

49) If one third equals a quarter, how much is one fifth equal?

False Position = commen denominator:

אמור ג' פעמי' ד' י"ב
ה' פעמי' י"ב הם ס‫'

Rule of Three:

חלקם בכ' חלקי' הרי שיבא ט' שהם ט' חלקי' מס‫'
שהם ג' חלקי' מכ' וכה"ל

Find a Quantity Problem - Whole from Parts – Tree

50) If one asks you: I have a tree - its half and its third are in the water; a third and a quarter of what is left are planted in the soil; and three spans remain.

How much is the length of the tree?

והשליש והרבע מן הנשאר הוא מכוסה תקוע בארץ
ונשאר ממנו מן האילן ג' זרתות
אשאלך כמה גבהות כל האילן

False Position = common denominator: Look for the denominator, which is 72, by multiplying 3 by 2, which is 6; 3 by 6 is 18; 4 by 18 is 72.

So, you find in 72 a half, a third, a third, and a quarter.

Extract a third and a half from 72; it is 60 and 12 remains.

For the part that is in the ground, extract a third and a quarter of 12; it is 7.

Now, sum 7 with 60; it is 67 and 5 remains.

Rule of Three: Do [as follows:] if 5 is 3, how much is 72?

72 times 3 is 216.

Divide it by 5, the result is 43⅕.

Hence, the length of the whole tree is 43⅕ spans.

Motion Problem - To and From - an Ant Climbing a Tower

51) A tower is 20 cubits tall.

An ant wants to climb up.

Every day it climbs up one third of a cubit and every night it goes down a quarter [of a cubit].

In how many days will it reach the top?

ונמלה אחת רוצה לעלות למעלה
ובכל יום עולה שליש אמה ובכל לילה יורדת רביע
בכמה ימי' עולה למעלה

each day it moves upwards by of a cubit

it climbs up one cubit in 12 days

it will reach the top in days

הרי שבר"מ ימים תעלה על המגדל

Find a Quantity Problem - First from Last Problem – Money

52) You have some money in your purse.

You take a third, a quarter, and a fifth of it and their sum is nine.

How much remains?

והוצאת מהם השליש והרביע והחומש והם ט‫'
כמה יהיו הנשארים

False Position = common denominator:

Rule of Three:

the money that was in the purse: pešiṭim

the money that remained:

Find a Quantity Problem - First from Last Problem – Money

53) You have some money in your purse.

You subtract a tenth from it and ten remain.

How much was the money at first?

והוצאתי מהם העשירית ונשאר י‫'
כמה היו בתחלה

False Position:

Rule of Three:

חלקם על ט' יהיו א' ותשיעי‫'

[the money that was in the purse:]

Check:

• If ten and a sixth remain

Rule of Three:

תאמ' אם אחת היא אחת ותשיעי' השתות כמה היא

[the money that was in the purse:]

Divide a Quantity – Money

54) If you want to divide 5 pešuṭim to a third and a quarter without a remainder

False Position - common denominator:

denominator:

• the portion of the one who has a third -

Rule of Three:

אמור ד' פעמי' ה' בעבור שהם הפ' יבא כ‫'
וחלק אותם לז' יבאו ב' וו' חלקי' מז‫'

• the portion of the one who has a quarter -

Rule of Three:

אמור ג' פעמי' ה' ט"ו
חלקם לז' יבא ב' וא' חלק מז‫'

Check:

Divide a Quantity – Money

55) If you want to divide 12 pešuṭim to a half, a third, and a quarter without a remainder

False Position - common denominator:

denominator:

• the portion of the one who has a half -

Rule of Three:

ואמור ו' פעמי' י"ב בעבור המעות שהם י"ב יבואו ע"ב
חלקם על י"ג יבואו ה' וז' חלקי' מי"ג

• the portion of the one who has a third -

Rule of Three:

אמור ד' י"ב הם מ"ח
חלקם בי"ג יבואו ג' וט' חלקי' מי"ג

• the portion of the one who has a quarter -

Rule of Three:

אמור ג' פעמי' י"ב הם ל"ו
חלקם בי"ג יבוא ב' וי' חלקי' מי"ג

Check:

Ordering Problem - Six Coins

56) A man has a worker for 30 pešuṭim for 30 days.

[The worker] wants to be paid each day, but the employer has only six silver coins that are worth a total of 30 pešuṭim.

He pays him every day with these six coins.

How much should [each of] the six coins [be worth] so that he will not need to exchange them?

ורוצה לקבל פרעונו בכל יום ואין לשוכר רק ו' מטבעות כסף ששוים ל"פ סך כולם
והוא פורע אותו בכל יום עם אלו הו' מטבעות
היאך יהיו אלו הו' מטבעות ולא יצטרך להחליפם

1; 2; 3; 4; 8; 12

Find a Number Problem - Ratio of Integers and Fractions

57) If three and one third equal four and a quarter, how much are five and one fifth equal?

הרי שעשית מה' וחומש כ"ו חמשיות ותזמן אותם כאשר תראה

הרי שעשית מד' ורביע י"ז רבעיות

ונחלק אותם אחרי כן לרביע וחומש ונאמ' ד' פעמי' ה' כ‫'
נחלק אותם תמ"ב יעלו כ"ב וב' חלקי' מכ' דהיינו א' עשירי‫'

converting to thirds:

נחלק אחרי כן ג' עשיריות בי' חלקי' יבואו' ג' חלקי' מק‫'
הרי שעולה החשבון ו' שלמים וו' עשירייות וג' חלקי' מק‫'
נקבץ אותם יחד ונעשה מו' עשיריות חלקי' מק' ויהיו ס' וג' חלקי' שהיו לנו הרי ס"ג
נמצא שיבא ו' שלמים וס"ג חלקי' ממאה

Multiple Quantities - Boys Selling Eggs

58) A man gave his sons eggs to sell: to one [of them] he gave 50 eggs, to the second 30, and to the third 10.

He said to them: go, sell the eggs equally, and bring me equal amounts of money

ואמ' להם לכו ומכרו הבצים בשוה וכולכם תביאו לי מעות בשוה

first they sold 7 eggs for one pašuṭ

• The first who had 50 eggs: 49 eggs for 7 pešuṭim

egg remains

• The second who had 30 eggs: 28 eggs for 4 pešuṭim

eggs remain

• The third who had 10 eggs: 7 eggs for 1 pašuṭ

eggs remain

then they sold one egg for 3 pešuṭim

• The first who had 1 egg left: 1 egg for 3 pešuṭim

he had a total of pešuṭim

• The second who had 2 eggs left: 2 eggs for 6 pešuṭim

he had a total of pešuṭim

• The third who had 3 eggs left: 3 eggs for 9 pešuṭim

he had a total of pešuṭim

they sold the eggs for the same price and received the same amount of money

Find a Quantity Problem - Whole from Parts – Fish

59) A fish - its head and tail were cut off.

The body weighs 10 liṭra.

The head weighs one third and a quarter of the whole fish.

The tail weighs a fifth and a sixth of the whole fish.

How much does the whole fish weigh?

והגוף שוקל י' ליט‫'
והראש היה שוקל השליש והרביע מכל הדג
והזנב שוקל החומש והשתות מכל הדג
כמה ישקול כולו

False Position: First say: a third, a quarter, a fifth, and a sixth are found in 60.

They are 20, 15, 12, and 10, which are 57 parts of 60.

We find that the head and tail that weigh together a third, a quarter, a fifth, and a sixth are 57 parts of 60, which is the whole.

The 3 parts that remain from 60 are the body that weighs 10 liṭra.

Rule of Three: So say: if 3 parts of 60, i.e. the body, weigh 10 liṭra, how much do the remaining 57 parts of 60, i.e. the third, the quarter, the fifth, and the sixth, which are the head and the tail weigh?

Say: 57 times 10 is 570.

Divide it by 3 parts; the result is 190 liṭra.

We find that the head and the tail weigh 190 liṭra.

The body weighs 10 liṭra, so the total is 200 liṭra.

If you wish to know how much does the head weigh alone:

You already know that the head weighs a third and a quarter; and the third and the quarter of 60 are 35 parts of 60.

Rule of Three: Say: if 3 parts of 60, i.e. the body, weigh 10 liṭra, how much do these 35 parts, i.e. the head that weighs a third and a quarter, weigh?

Say 35 times 10 [is] 350.

Divide it by 3; the result is 116⅔ and this is the weight of the head.

Do the same with the tail and you will get the matter.

Simultaneous Division - Money in a Purse

60) Two men had money in one purse.

One said to the other: hear me my brother, one third and a quarter of my money are equal to a sixth and a seventh of your money.

They checked and found this as they said.

How much money did each one of them have?

אמ' זה לזה שמעני אחי כך הם השליש והרביע ממעותי כמו השתות והשבע ממעותיך
והם בדקו ומצאו כן כאשר אמרו
אשאלך כמה מעות היו לכל אחד ואחד

the money of the one with the third and the quarter:

pešuṭim

Rule of Three:

the money of the one with the sixth and the seventh:

חלקם בי"ג יהיו כ"ב וח' חלקי' מי"ג וכך מעות היו לו

Check: pešuṭim

Partnership - for the same time

61) Three people wanted to form a partnership and contribute together 40 ounces of gold.

The first has gold that is worth 3 liṭra per ounce.

The second has gold that is worth 5 liṭra per ounce.

The third has gold that is worth 8 liṭra per ounce.

How much gold should each of them contribute so that the total will be 40 ounces and the share of each of them will be equal to the shares of his friends?

לאחד יש לו זהב ששוה ג' ליט' האונ‫'
והשני יש לו זהב ששוה ה' ליט' האונקי‫'
ולשלשי יש לו זהב ששוה ח' ליט' האונקי‫'
כמה זהב ישים כל אחד שיהיו מ' אונקיו' וכל אחד יהיה שוה השלו כמו של חברו

False Position - common denominator:

בקש המורה והוא ק"כ

denominator:

• the one whose gold is worth 3 liṭra contributes:

ounces of gold

אמור השלשי הוא מ' ותרבה אותו עם סך האונקי' שהם מ' ואמור מ' פעמי' מ' אלף ות"ר
חלקם בע"ט יבוא כ' וכ' חלקי' מע"ט
הרי שישים בחברותה כ' אונ' וכ' חלקי' מע"ט

liṭra

Rule of Three:

pešiṭim

חלקם בע"ט יבואו קפ"ב פשוטי' וכ"ב חלקי' מע"ט

his share is worth: 60 liṭra, 15 dinar, 2²²/₇₉ pešiṭim

the same for the others

Give and Take Problem - gatekeeper

62) A man passed through four towns.

In the first town, he doubled all his money and when leaving town the gatekeeper took from him 8 pešuṭim.

The same happened in the second and the third towns – in each town he doubled all his money and when leaving town the gatekeeper took from him 8 pešuṭim.

In the fourth town, he doubled again all the remaining money and paid 8 pešuṭim.

Then he had no pašuṭ left.

How much money did he have at first?

ובעיר ראשונה הכפיל כל מעותיו ובצאתו מן העיר לקח לו השוער ח"פ
וכן בעיר השנית והשלישית הכפיל כל מעותיו ובצאתו מן העיר לקח לו השוער ח"פ בכל עיר
וכן בעיר הרביעית הכפיל כל מעותיו הנשארים ופרע ח"פ ומ.. לא נשאר בידו פשוט
אשאלך כמה מעות היו לו קודם

If you wish to know this, since this man passed through four towns and in each town he paid 8 pešuṭim:

For the first town, the 8 pešuṭim should be divided into half; they are 4 pešuṭim.

For the second town, the 4 pešuṭim should be divided into half; they are 2 pešuṭim.

For the third town, the 2 pešuṭim should be divided into half; they are 1 pašuṭ.

Divide the pašuṭ into half; it is a half pašuṭ.

Then, sum up all together; they are 7½ pešuṭim and this was the original amount of money.

Shared Work Problem - Rivers Filling a Fountain

63) Four rivers are flowing towards a fountain.

The first fills it in a day.

The second fills it in two days.

The third [fills it] in 3 days.

The fourth [fills it] in 4 days.

If all are flowing together, how long will it take them to fill [the fountain]?

האחד ממלאו ביום אחד
והשני ממלאו בב' ימים
והשלשי בג' ימים
והרבעי בד' ימים
ותרצה לידע אם ירוצו בו כלם יחד בכמה זמן ימלאוהו

False Position:

denominator:

together they fill 25 fountains in 12 days:

• the one that fills it in 1 day - fills 12 fountains in 12 days

• the one that fills it in 2 days - fills 6 fountains in 12 days

• the one that fills it in 3 days - fills 4 fountains in 12 days

• the one that fills it in 4 days - fills 3 fountains in 12 days

together they fill 25 fountains in 12 days

Rule of Three:

together they fill one fountain in of a day

חלקם בכ"ה יבא י"ב חלקי' מכ"ה הרי שבי"ב חלקי' מכ"ה מתמלא ביום המעיין

• the one that fills it alone in 1 day, fills parts of the fountain in of a day

the same for the others

the number of fountains they fill in one day: fountains

Motion Problem – Pursuit

64) A man is walking ten miles a day.

Another man is walking one mile on the first day, two miles on the second day, three miles on the third day, and so on he adds in each day one more mile.

In how many days will [the total distance each of them walked] be equal?

[See the additional problems below]

ואדם אחר מהלך ביום אחד א' מיל וביום שני ב' וביום ג' ג' והוא מוסיף בכל יום מיל אחד
בכמה ימים יעמדו בשוה

days

• If the slower is walking 1, 3, 5, 7, and so on - each day he walks an odd number [of miles]

days

• If the slower is walking 2, 4, 6, 8, and so on - each day he walks an even number [of miles]

days

Mixture and Alligation Problem - Goldsmith making rings from several kinds of coins

65) You gave the goldsmith 10 ounces of gold to create rings from them.

Two ounces of them were of 14 carat per ounce.

Three ounces of them were of 16 carat per ounce.

Five ounces of them were of 18 carat per ounce.

The goldsmith mixed all the gold together.

You want to know how many carats per ounce all of them will be together

והב' אונקיו' מהם היה זהב מי"ד קראטי' האונקי‫'
והג' אונק' מהם היה זהב מי"ו קרטי' האונק‫'
והה' אונקיות מהם היה זהב מי"ח קרטי' האונק‫'
והצורף צירף כל הזהב ביחד
ותרצה לידע מכמה קראטי' יהיה האונק' כולם יחד

(the total sum of the carat)÷(the total sum of the ounces)

two ounces of 14 carat: carat

three ounces of 16 carat: carat

five ounces of 18 carat: carat

total sum: : carat

each ounce: carat

Divide a Quantity - Price of a Fish Paid by a Group of People

66) Three men bought one fish for nine pešuṭim.

One had a half; the second had a third; and the third had a ninth.

You want to know: how much was the share of each in the fish?

לאחד יש לו החצי ולשני יש לו השליש ולשלישי יש לו התשיעי
ותרצה לידע כמה חלק מן הדג יש לכל אחד

False Position:

denominator:

the one who has a half: of the fish

אמור החצי הוא ט' ותרבה אותו על הדג שהוא א' ואמור ט' פעמי' א' הוא ט‫'
חלקהו בי"ז יבא לחלקו ט' חלקי' מי"ז

the same for the others

Joint Purchase Problem - If You Give Me - two men - Amounts of Money

67) A man said to his friend: give me six pešuṭim and I will have as much as you have.

The other one said: if you give me six pešuṭim I will have twice as much as you have.

How much money did the first have and how much money did the second have?

והאחר אמ' אם תתן לי ו"פ יהיה לי פי שנים ממך
אשאלך כמה מעות היו לאחד וכמה מעות היו לאחר

the one who asked for 6 pešuṭim: pešuṭim.

the second: pešuṭim.

Interest and Discount Problem - find the fund

68) I produced two pešuṭim from one pašuṭ, I produced three [pešuṭim] from two [pešuṭim], four from three, five from four, and six from five, and I had a total of 400 pešuṭim.

How much was the money invested?

אשאלך כמה היה הקרן

Rule of Three:

א"כ צריך לראות כי ו' יצאו מא' א"כ מאיין מצים לצאת ת"פ

pešuṭim

Exchange Problems

69) 20 dinar of Ancona are worth 31½ pešuṭim of Bologna.

How many [of Bologna] are 37½ dinar of Ancona worth?

כמה יבא לל"ז די' וחצי מאנקונטני' מחלוף

Rule of Three:

חלק אותם לכ' חלקי' יבא נ"ט פשו' וא' חלק מכ' וא' חלק מפ‫'
דהיינו ה' חלקי' מפ' וכה"ל

• How many [of Bologna] are 91¼ dinar of Ancona worth?

Rule of Three:

וחלקם בכ' חלקי' ויהיו קמ"ג פשו' וג' רבעים וא' חלק מכ‫'
דהיינו קמ"ד פשו' פחות חומש וכה"ל

Interest and Discount Problem

70) A certain amount produced 17 liṭra in one year at 3 pešuṭim for one liṭra a month.

How much was the money invested?

כמה היה הקרן

Calculate and say: 23 liṭra at 3 pešuṭim for one liṭra a month are 20 liṭra of the fund and 3 liṭra of interest at the end of the year.

Rule of Three: Say: if 23 is 20, how much is 17?

Say: 17 times 20 is 340. Divide it by 23; it is 14 liṭra, plus 15 dinar and 15 parts of 23 of a [dinar].

חלקם על כ"ג יהיו י"ד ליט' וט"ו די' נוס' וט"ו חלקי' מכ"ג בפשוט וכה"ל

Pricing Problem - Find the Amount

71) If you are asked: One kikkar of leather for 21 liṭra and a half.

How much leather [can be bought] for 12 liṭra and a quarter?

False Position - denominator: Add 21 and a half with 21 and a half; they are 43 and this is the denominator.

Then, multiply 2 times 100; they are two hundred.

Rule of Three: Multiply 12 liṭra and a quarter by two hundred; they are 2450. Divide into 43 parts; they are 56 leathers and 42 parts of 43 of one unit.

וחלק במ"ג חלקים ויהיו נ"ו עורות ומ"ב חלקי' ממ"ג מא' עור וכה"ל

• If it is said: How much leather [can be bought] for 71 liṭra and a half?

Rule of Three: Multiply 71 liṭra and a half by 200; they are 143 kikkar. Divide by 43; they are 332 leathers and 24 parts of 43 of one unit.

וחלק במ"ג ויהיו של"ב עורות וכ"ד חלקי' ממ"ג מא' עור וכה"ל

Divide a Quantity - Sharing Food

72) Two men sat down to eat.

One had two loaves of bread and the second had three [loaves of bread].

A third came and ate with them.

The three ate the five loaves of bread and after they ate and drank, the third who came to eat with them gave five pešiṭim for the two.

How should they share the [five pešiṭim]?

לאחד יש לו ב' לחמים ולשני ג‫'
בא השלישי והיסב לאכול עמהם ואכלו כן שלשתם אלו הה' לחמים
לאחר שאכלו ושתו נתן השלשי שבא לאכול עמהם ה"פ לאותם השנים
היאך יחלקום

Say how many pieces of bread each ate: 1 loaf of bread and 2-thirds.

So, the one who had 2 loaves of bread and ate 1 loaf of bread and 2-thirds lost one third of a loaf.

The one who had 3 loaves of bread and ate 1 loaf of bread and 2-thirds lost one loaf of bread and a third, i.e. 4-thirds.

Therefore, the one who had 2 loaves of bread will receive 1 pašuṭ.

And the other will receive 4 pešiṭim.

Find a Number Problems - Multiplication of fractions

Fractions by Fractions

73) If a man asks you: How much are four fifths by four fifths?

Say: if he had asked for one fifth by one fifth.

You would have said: 5 times 5 is 25, so the result would have been one part of 25.

Now that he asks for 4-fifths by 4-fifths, you say: 4 times 4 is 16. We find that the result is 16 parts of 25.

74) If a man asks you: How much are seven ninths by four sevenths?

If he had asked for one ninth by one seventh.

You would have said: 7 times 9 is 63. We find that the result would have been one part of 63.

Now that he asks for 7-ninths by 4-sevenths, you say: 7 times 4 is 28. We find that the result is 28 parts of 63.

Integers and Fractions

• If a man asks you: How much are three integers and two fifths by two integers and four sevenths?

Say: 3 times 2 is 6 integers.

Say: 3 times 4-sevenths are 12 that are one integer and 5-sevenths, which is 25 parts of 35.

Say: 2 times 2-fifths are 4-fifths, which is 28 parts of 35.

Say: 2-fifths by 4-sevenths is 8 parts of 35.

Add them up; the result is 8 integers and 26 parts of 35.

75) If a man asks you: How much are one integer and five sevenths by one integer and four fifths?

Say: 1 time 1 is 1.

Say: 1 time 4-fifths is 28 parts of 35.

Say: 1 time 5-sevenths is 25 parts of 35.

Then, say: 4-fifths by 5-sevenths is 20 parts of 35.

Add them up; the result is 3 integers and 3 parts of 35.

Multiple Quantities - Coins that Worth an Amount of Money

76) Four coins are worth all in all 80 pešuṭim:

The first is equal to a half and a sixth of the second.

What is left from the second equals two thirds of the third.

The third is equal to a third, a quarter, and a sixth of the first.

The fourth is equal to four fifths of the third.

How much is each [of the coins] worth?

והראשון שוה החצי והשתות מן השני
הנשאר מן השני הם ב' שלישיות מן השלשי
והשלשי הוא שליש ורביע ושתות מן הראשון
והרבעי הוא ד' חמשים מהשלשי
ותרצה לידע כמה שוה כל אחד ואחד

Follow this way:

Say for the first that is equal to a half and a sixth of the second:

Say: a half and a sixth are found in 6:

The half is 3; the sixth is 1; so they are 4 parts of 6, which are 2-thirds.

Therefore, for the one that equals a half and a sixth of the second, it is as if it is said: the first is equal to 2-thirds of the second.

For the third that is equal to a third, a quarter, and a sixth of the first:

Say: a third, a quarter, and a sixth are found in 60:

The third is 20; the quarter is 15; the sixth is 10; so they are 45.

Therefore, for the third that is equal to a third, a quarter, and a sixth of the first, which are 45 parts of 60, it is as if it is said: 3-quarters of the first.

For the fourth that is equal to 4-fifths of the third - it is clear.

Therefore, this is the order: the first equals 2-thirds of the second; the third equals 3-quarters of the first; and the fourth equals 4-fifths of the third.

והשלשי שוה ג' רבעיות מן הראשון
והרבעי שוה ד' חמשים מהשלשי

False Position: First say: the fractions mentioned in this calculation, which are a half, a third, a quarter, a fifth, and a sixth, are all found in 60.

Hence, the second is 60.

The first that is equal to 2-thirds of the second, which is 60, is 40, because 40 is 2-thirds of 60.

Then, the third that is equal to 3-quarters of the first, which is 40, is 30, because 30 is 3-quarters of 40.

The fourth that is equal to 4-fifths of the third, which is 30, is 24, because 24 is 4-fifths of 30.

For what is left of the second that is 2-thirds of third, follow this way:

Say: how much is left of the second, which is 60, after you subtract the first from it, which is 40? 20 remain.

So, 20, which is what is left of the second, is 2-thirds of the third, which is 30, because 20 is 2-thirds of 30.

Denominator: So, the first is 40; the second is 60; the third - 30; and the fourth - 24. Sum them together; the result is 154 and this is the denominator.

Rule of Three: If you wish to know how much the first, which is 40, is worth, say: 40 times 80, because they are worth together 80; it is 3200. Divide it by 154; the result is 20 and 120 parts of 154.

אמור מ' פעמ' פ' בעבור שכלם שוים פ' יהיו ג' אלפי' ומאתיים
חלקם בקנ"ד יבאו כ' וק"כ חלקי' מקנ"ד

Do the same for the others and you will get the same answer.

Find a Number Problems - Sums

77) 1, 2, 3, 4, 5, 6, 7, 8.

[The number of items] can be divided in half [= last term is even ]:

Add 1 to the [last number] and multiply it by its half.

Say: 4 times 9 is 36; so their sum is 36.

78) 1, 2, 3, 4, 5, 6, 7.

[The number of items] cannot be divided in half [= last term is odd ]:

Take the greater half and multiply it by the last number.

Say: 4 times 7 is 28.

79) 1, 3, 5, 7, 9 - that are odds.

Multiply the greater half by itself.

Say: the greater half of 9 is 5. Say 5 times 5 is 25 and this is their sum.

80) 2, 4, 6, 8 - that are evens.

Multiply their half by the number that follows it.

Say: half 8 is 4. Say: 4 times 5 is 20 and this is their sum.

Find a Number Problem - Multiplication of integers and fractions

81) If a man asks you: How much are one integer and a quarter by one integer and a fifth?

First, say: 1 time 1 is 1.

Then, say: one time a quarter and one times a fifth are 9-twentieths.

Say: a quarter by a fifth is 1 part of 20.

We find that the total is 1 and 10-twentieths, i.e. 1 and a half.

Interest and Discount Problems

Find the Earned Interest

82) If a man asks you: I lent some money at so and so pešuṭim for a liṭra a month, how much is the profit of 100 liṭra for a year?

כמה יעלה ריוח בק' ליט' א' שנה

Multiply the money that the liṭra yields per month by 5.

• Example: I lent 100 liṭra for one [year] at 7 pešuṭim for one liṭra a month.

Say: 5 times 7 is 35.

We find that 100 liṭra yield 35 liṭra a year.

Compound Interest

83) The one who wants to combine several loans made at different times and convert them to one time:

He should sum up all the monthly interests and divide [the sum] by the amount of money, or peraḥim, or any currency [specified] for those loans.

• Example: I lent 100 peraḥim of gold for a month, 100 peraḥim for two months, 100 peraḥim for three months, 100 peraḥim for four months, and 100 peraḥim for five months

Do as follows: one, two, three, four, five - which is [the number of] months - is 15.

Divide it by the amount of money, which is 5; the quotient is 3.

So, all are set for 3 months,

The same for every currency and every number.

Find the Earned Interest

84) The one who wants to know for one kikkar at so and so [a year], how much will be [the interest] a month?

Rule of Three: multiply the interest of the kikkar by the amount of money you want, then divide by five and the resulting quotient is [the interest] per month for the amount you want.

• Example: At 7 liṭra for one kikkar a year: how much will 30 liṭra yield a month or two months or more?

Multiply 30 by 7; it is 210.

Divide by five; each part is 42.

So, [the interest is] 42 pešuṭim a month.

Then, multiply by [the number of] months you want and you will get the answer.

Multiple Quantities Problem - Five Weights

85) The butcher sells meat but he has only five weights summed up together to 121 liṭra.

How much should each of the weights weigh, so that the butcher will be able to weigh the meat with these weights be it light or heavy?

כמה ישקול כל משקל לעצמו שיוכל הקצב לשקול הבשר באלו המשקלות למי רב ולמי מעט

So, if someone wants to buy 121 liṭra of meat, he puts all the weights against the meat.

If he wants to buy 50 liṭra of meat, or more or less, he puts as many weights in the scales against the meat, so that they will weigh as much as the meat he wants to buy.

If there is no balance, he changes the weights.

Sometimes he must put one, or two, or three weights with the meat against one weight to fulfill the buyer's wish.

Now, I ask you: How much does each weight weigh by itself?

One weighs 1 liṭra.

The second weighs 3 liṭra.

The third - 9 liṭra.

The fourth - 27 liṭra.

The fifth - 81 liṭra.

Add them together; they are 121 liṭra, no less and no more.

With these weights the butcher can sell as he wishes, to some a lot and to some a little.

Guessing - coins

86) Three different coins [gold, silver, and copper, given] to three different people: give the first one pašuṭ, the second two [pešuṭim], and the third four [pešuṭim].

The one who has the largest coin should multiply his share [= the number of the pešuṭim he has] by 4, the second by 3, and the one who has the smallest coin should double his share once.

Then they should sum up [the products] and cast out the sevens from the result

ובעל המטבע הגדול יכפול חלקו ד' פעמ' והשני ג' פעמים ‫^[9]ובעל המטבע הקטן יכפול חלקו פעם אחד
ויקבצום וישלכום ז"ז

1 → copper, gold, silver.

2 → silver, copper, gold.

3 → copper, silver, gold.

4 → gold, silver, copper.

5 → silver, gold, copper.

6 → gold, copper, silver.

Payment Problem - three workers, three different daily wages, the same actual payment

87) A man hired three brothers – Reuven, Shimon, and Levi – to do his work for 20 days from morning until evening, any one of them in turns so that the work will not cease.

If Reuven works all the days he will pay him 5 zehuvim; if Shimon – 4, if Levi – 3.

They all worked during the 20 days and he was sitting and watching over them: how many hours and parts of hours each of them is working a day.

Finally, he paid each of them an equal share.

How much is the share of each of them?

והנה אם עבד ראובן כל הימים יתן לו ה' זהובים ואם שמעון ד' ואם לוי ג‫'
והנה בין כלם עבדו הכ' שנים ימים והוא יושב ושומר עליהם כמה שעות ביום עבד כל אחד מהם וכמה חלקי שעות
והנה באחרונה נתן לכל אחד מהם חלק שוה
נרצה לדעת כמה החלק של כל אחד

Know that Reuven is paid one zahuv for 4 days.

Shimon [is paid one zahuv] for 5 days.

Levi [is paid one zahuv] for 6 days and 2-thirds of a day.

The total is 15 days and 2-thirds.

days

We divide 20 by this number; the result is one integer; 4 integers and a third remain.

We convert all into thirds because of the third:

We convert the 20 days into 60-thirds; the 15 and 2-thirds into 47-thirds; and the 4 and a third into 13.

So, each one was paid one zahuv and 13 pešuṭim.

Now, we check how much each must work, so that they complete the 20 days.

We start with Levi, who has to work 6 days and 2-thirds for the zahuv he is paid.

We convert them into thirds; they are 20.

We want to know how much he must work for the 13 pešuṭim he received:

We multiply 13 by 20; it is 260.

We divide it by 47; the result is 5 and 25 parts remain.

We add 5 to 20; they are 25-thirds and 25 parts of 47.

We divide the thirds by 3; the result is 8 integers and one remains.

We consider it as 4 hours, which are a third of a day.

We multiply 25 by 4; the result is 100.

We divide it by 47; the result is 2 hours and 6 parts remain.

So, Levi worked 8 days, 6 hours and 6 parts [of 47].

We want to know how much Shimon worked, who has to work 5 days, which is 15-thirds, for the zahuv he is paid.

We want to know how much he must work for the 13 pešuṭim he received:

We multiply 13 by 15; the result is 195.

We divide it by 47; the result is 4 and 7 parts remain.

We add 4 to 15, because they are thirds; they are 19-thirds.

We divide them by 3; they are 6 whole days.

We consider the one that remains as 4 hours.

We also multiply 7 by 4; it is 28.

So, Shimon worked 6 days, 4 hours and 28 parts [of 47].

We want to know how much Reuven must work: he must work 4 days for the one zahuv, which are 12 thirds.

We extract the ratio for the 13 pešuṭim he received:

We multiply 13 by 12; the result is 156.

We divide it by 47; the result is 3 and 15 parts remain.

We add 3 to the 12 we have; they are 15-thirds.

So, he worked 5 days.

We also multiply the 15 parts by 4; the result is 60.

We divide it by 47; the result is one hour and 13 parts of an hour remain in which Reuven worked.

Check: When you add up these parts, the result is one hour, no more and no less.

When you add this hour to the mentioned hours, they are 12 hours, which is one day.

When you add this day to the mentioned days, they are 20 days, no more and no less.

Divide a Quantity Problem - Proportional Division – Inheritance

88) Jacob died.

Reuven issued a deed with two witnesses, according to which his father Jacob has given him all the property he had and instructed so in case of death.

His son Shimon issued a deed as well according to which half of his property should be granted to him.

His son Levi also issued a deed according to which a third of his property should be given to him.

His son Yehudah too issued a deed according to which a quarter of his property should be granted to him.

All of them are writing this in Jerusalem in the same day, the same time, the same hour

והוציא ראובן שטר בשני עדים כשרים שנתן לו יעקב אביו כל ‫^[10]הממון שהיה לו וציוה כן מחמת מיתה
גם הוציא שמעון בנו שטר שיינתן לו חצי ממונו
גם הוציא לוי שטר שינתן לו שליש ממונו
גם הוציא יהודה שטר שינתן לו רביע ממונו
וכולם ביום אחד וזמן אחד ושעה אחת ברושלים שכותבים בו שעות

• The sages of Israel divide it according to the request of each.

• The Gentile sages - according to the ratio of each share.

• The arithmeticians consider the property as a whole, so when you add to it its half, its third, and its quarter, the total is two and half a sixth.

False Position: We suppose the whole is 60, which has all the mentioned parts; their sum is 125.

Another false position: Or, we suppose the whole is 12, and the [sum of the] mentioned fractions is 13.

The result, for whichever you take, will be the same.

Now, we look how much Reuven will take according to the value of his share: we extract the ratio from 60, as he ask for the whole property. We suppose the property is 10 dinar that are 120 pešuṭim.

This is the diagram of the share that Reuven takes:

0 60 120 125

Rule of Three: We multiply the extremes; the result is 7200. We divide it by 125; the result is 57 pešuṭim and 75 parts and this is Reuven's share.

חלקנום על קכ"ה עלו נ"ז פשו' וע"ה חלקי' וזה חלק ראובן

This is the diagram of Shimon's share:

0 30 120 125

Rule of Three: We multiply the extremes and divide according to the rule.

This is the diagram of Levi's share:

0 20 120 125

Rule of Three: We multiply and divide according to the rule.

This is the diagram of Yehudah's share:

0 15 120 125

Rule of Three: We multiply and divide according to the rule.

• Another shorter method [according to the Gentile sages]:

Shimon takes a half of Reuven's share.

Levi takes a third of Reuven's share.

Yehudah takes a quarter of Reuven's share.

When you sum up all these fractions and the integers, they are 120 pešuṭim that are 10 dinar.

• According to the method of the sages of Israel:

The three older brothers say to Yehudah: you are only appealing about 30 pešuṭim and the appeal of each of us is the same about them, so take 7 and a half, which is a quarter and leave us.

Each of the three brothers takes the same also.

Then, Reuven says to Levi: you are only appealing about 40 pešuṭim and you have already taken your share from the 30 the four of us shared, so take a third of 10, which is a quarter of 40 and leave us.

קח אתה שלישית י' שהוא רבעית מ' ולך מעמנו

Hence, Levi's share is 10 pešuṭim and 5-sixths, because the half from the 7 and a half he already took is 3-sixths and the third from the 10 are 2-sixths, so they are 5-sixths

Reuven also says to Shimon: you are only appealing about one half, which is 60; all the other half is mine; and you have already taken your share from the 40, so take half of the appeal that remains between you and me, and leave me.

So, Shimon's share is 20 pešuṭim and 5-sixths.

Reuven's share is 80 pešuṭim and 5-sixths.

When you sum up all these shares, they are 10 dinar.

Geometrical Problems

Triangulation Problem - Two towers

89) two towers, one is 15 cubits tall and the other is 12 cubits tall.

There are two pigeons there, [one pigeon at the top of each tower], and between them a spring.

[The pigeons] arrive to the spring at the same moment.

How far are the towers from each other and where is the place of the spring?

ושם ב' יונים ומעיין אחד ביניהם ומגיעים ברגע אחד במעיין
כמה מרוחקים המגדלים זה מזה ואנה מקום המעיין

Suppose the distance between them is the same as the measure of both heights, i.e. 12 plus 15, which is 27.

Divide 27 into 12 and 15.

Give the 15 to the tower that is 12 cubits high.

Give the 12 to the tower that is 15 cubits high.

The place of the spring is 12-15 cubits from the bases [of the towers].

Rectangle

90) Question: a rectangle, its diagonal together with its one side are 18, and its other side is 6.

How much are its area, its diagonal and its side that is summed with the diagonal?

כמה הוא ריבועו ואלכסונו וצלעו המנויה עם האלכסון

[this figure does not appear in the text]

The one who answers this question squares the known side, which is 6; its square is 36.

He divides it by the sum of the diagonal and the second side, which is 18; the quotient is 2.

He adds 2 to 18; it is 20.

So, he knows that half 20 is the diagonal.

The remainder from 18 is the second side; it is 8.

Their product, 8 by 6, is the area.

Quadrangle

91) A quadrangle whose sides are not equal - its length is not equal to its breadth, the one length is not equal to the other length, and the one breadth is also not equal to the other breadth.

Add up the lengths, divide [the sum] in half and keep its half.

Add up the breadth with the breadth, and keep its half.

Multiply half the sum of the two lengths by half the sum of the two breadths; its area is as their product.

[The author does not note that this formula is not accurate in the general case]

• Example: its one length is 4, and its other length is 6 its one breadth is 5, and its other breadth is 3

Add the length to the length and say: 6 plus 4 is 10. Take its half; it is 5. Keep it.

Then add the breadth to the breadth and say: 3 plus 5 is 8. Take its half; it is 4.

Multiply it by 5, which is half the sum of the two lengths; say: 4 by 5 is 20. So, its area is 20.

Triangulation Problem - Tree leaning against a wall

92) A tree 30 cubits high was standing straight next to a wall.

Then it was withdrawn from the bottom [of the wall] by 10 cubits [so that its top is leaning against the wall].

By how much was the top of the tree brought down from its original place?

נרצה לידע מזה כמה הורד ראש העץ ממקום שהיה

[this figure does not appear in the text]

You know that from the top of the tree to the ground and from the wall to the base of the tree is a right angle.

Pythagorean Theorem:

The hypotenuse is the tree, which is 30 cubits. Its square is 900.

Then, square the line of the ground, which is 10; its square is 100.

Subtract it from 900; 800 remains.

Extract the root and it is the length of the wall; it is slightly more than 28 and a quarter.

Subtract the length of the wall from 30 when the tree was upright; a little less than 1 and 3-quarters remain and this is [the length of the part of] the tree that was brought down.

Sometimes the result is exact.

93) A tree 10 cubits high was standing straight next to a wall.

Then it was withdrawn from the bottom [of the wall] by 6 cubits [so that its top is leaning against the wall].

By how much was the top of the tree brought down from its original place?

כמה הורד ראש העץ ממקום שהיה

Pythagorean Theorem:

Do as we did above: square the length of the tree, it is 100.

Then, square the 6 cubits that were brought down; it is 36.

Subtract it from 100; 64 remains.

Extract the root, meaning 8, and it is the length of the wall.

Subtract the length of the wall from 10 when the tree was upright; you are left with [the length of the part of] the tree that was brought down, meaning 2 cubits.

Now, the result is exact.

Transformation Problems

94) A square, its area is 38 and a half and you want to create a circle from it, so that its area is the same as the area of the square.

Multiply 38 and a half by 12 and 4-sevenths; the result is 484.

Extract the root of 484; it is 22 and it is the perimeter of the circle, whose area is the same as the area of the square, whose area is 38 and a half.

95) A circle, its area is 100, and you want to create a square from it.

Extract the root of 100; it is 10 and it is the size of each of its sides.

Volume

Sphere

96) The volume of the sphere.

As the geometricians stated, it is that you square the diameter of the sphere, then multiply this square 3 times and a seventh; you receive the surface of the sphere.

Multiply it by a sixth of the diameter; you receive the volume of the sphere.

• Example: a sphere whose diameter is 7 cubits

The square of the diameter is 49 cubits.

When you multiply this number 3 times and a seventh, it is 154 and this is the surface of the sphere.

Multiply it by a sixth of the diameter; it is 180 minus a third and this is the volume of the sphere.

Segment of a Sphere

From this calculation you can deduce the volume of the segments of a sphere.

• Example: a water pool - its body is rounded from the inside, its upper surface is a circular, its breadth [= diameter] is 7 cubits, and its depth is 3 cubits and a half.

You know that this pool is a hemisphere. Multiply its depth by the breadth of its upper surface, which is the diameter of the sphere, then multiply the product you received 3 times and a seventh, as you have done with the sphere; the result is 77 and this is the surface of the hemisphere.

Multiply it by a sixth of the diameter; it is 90 minus a sixth and it is the volume of the hemispherical pool.

If the depth is less than half the breadth [= less than half the diameter], know that the pool is less than a hemisphere.

• Example: a spherical solid [spherical cap] – its depth is 2 cubits and the breadth of the circle [= the diameter] of its base is a root of 40 which is approximately 6 and a third.

You know that this pool is less than a hemisphere, because its depth is less than half the breadth of its upper surface.

If you extract its diameter as you have learned, you find it 7.

Multiply 2, which is the depth of the pool, by 7, which is the diameter of the sphere; it is 14.