Difference between revisions of "ספר דיני ממונות"

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|Since the divine intention and the will of God is to justify us with His holy Torah, given to us as a possession by Moses, the trusted of His house, to whom He spoke ''mouth to mouth, in a vision and not in riddles'' [Numbers 12, 8] and whom He singled out from all the prophets, as the sages said: ''All the prophets looked through a mirror that does not shine brightly, but Moses our master looked through a brightly shining mirror'' [Talmud, Yevamot 49b].

|Since the divine intention and the will of God is to justify us with His holy Torah, given to us as a possession by Moses, the trusted of His house, to whom He spoke ''mouth to mouth, in a vision and not in riddles'' [Numbers 12, 8] and whom He singled out from all the prophets, as the sages said: ''All the prophets looked through a mirror that does not shine brightly, but Moses our master looked through a brightly shining mirror'' [Talmud, Yevamot 49b].

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|This is from God's love for His people, because God does not do anything except for the sake of Israel, as Rashi wrote in his commentary of the Torah that as long as Israel were in the desert, condemned by God, blessed be He, Moses was like one of them, and He did not speak to him with bright countenance.

|This is from God's love for His people, because God does not do anything except for the sake of Israel, as Rashi wrote in his commentary of the Torah that as long as Israel were in the desert, condemned by God, blessed be He, Moses was like one of them, and He did not speak to him with bright countenance.

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|In giving His Torah to His people of Israel, God aimed at two reasons, in which all the affairs of this world and the next are included

|In giving His Torah to His people of Israel, God aimed at two reasons, in which all the affairs of this world and the next are included

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|As the cabbalist sage R. Menahem from Recanati [1310-1250] wrote and said that the one who eats abominations and creeping creatures and species of sick torn-to-pieces animals causes himself strange diseases and transgresses [the commandment] ''sanctify yourselves and you shall become holy'' [Leviticus 11, 44]. Furthermore, he causes himself beastliness, rudeness of the brain and the confusion of good thought and shall not ''see the beauty of the God'' [Psalms 27,4] in the abode of the righteous with his companions.

|As the cabbalist sage R. Menahem from Recanati [1310-1250] wrote and said that the one who eats abominations and creeping creatures and species of sick torn-to-pieces animals causes himself strange diseases and transgresses [the commandment] ''sanctify yourselves and you shall become holy'' [Leviticus 11, 44]. Furthermore, he causes himself beastliness, rudeness of the brain and the confusion of good thought and shall not ''see the beauty of the God'' [Psalms 27,4] in the abode of the righteous with his companions.

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|Moreover, many commandments occur in the Torah, by which we direct ourselves toward the political and subtle [commandments] concerning matters of acquiring assets, such as the laws in Portion Mishpaṭim [Exodus 21:1-24:18], which constitute the principle and the foundation of the whole Torah.

|Moreover, many commandments occur in the Torah, by which we direct ourselves toward the political and subtle [commandments] concerning matters of acquiring assets, such as the laws in Portion Mishpaṭim [Exodus 21:1-24:18], which constitute the principle and the foundation of the whole Torah.

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|For instance, Reuven owes Shimon 5 ma‘ot, and he comes before the judge for verdict, if the judge would not know what is a ma‘ah is and how many peshuṭim or peruṭot it is, he would not be able to rule the verdict and verify it completely.

|For instance, Reuven owes Shimon 5 ma‘ot, and he comes before the judge for verdict, if the judge would not know what is a ma‘ah is and how many peshuṭim or peruṭot it is, he would not be able to rule the verdict and verify it completely.

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|Also in Bechukotai portion regarding the sale of the lands it is said: ''He shall reckon with his buyer'' [Leviticus 25, 50] ''and return the surplus to the man to whom he sold it'' [Leviticus 25, 27]

|Also in Bechukotai portion regarding the sale of the lands it is said: ''He shall reckon with his buyer'' [Leviticus 25, 50] ''and return the surplus to the man to whom he sold it'' [Leviticus 25, 27]

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|If the judge does not know to divide fairly and correctly as his duty imposed upon him, he will be found breaking the law Thou shalt not wrest the judgment [Exodus 23, 6].

|If the judge does not know to divide fairly and correctly as his duty imposed upon him, he will be found breaking the law Thou shalt not wrest the judgment [Exodus 23, 6].

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|R. [Ya’ako] says: ''there is no reward [for performance] of commandments in this world'' [Chullin 142a], because we find the length of days for ''honoring one's father and mother'' [Mishnah Peah 1] and for sending away the mother, as a man said to his son: "Go up to the attic and bring me chicks!" and he went up to fulfill his father's commandment and sent the mother away from the children and took the children and while he was going down, the ladder broke and he fell and died. Where is the goodness of this one? Where is the length of days of this one? Rather, “that it may be well with you” means in the world where all is good; and “that your days may be long” means in the world that is entirely long [Chullin 142a].

|R. [Ya’ako] says: ''there is no reward [for performance] of commandments in this world'' [Chullin 142a], because we find the length of days for ''honoring one's father and mother'' [Mishnah Peah 1] and for sending away the mother, as a man said to his son: "Go up to the attic and bring me chicks!" and he went up to fulfill his father's commandment and sent the mother away from the children and took the children and while he was going down, the ladder broke and he fell and died. Where is the goodness of this one? Where is the length of days of this one? Rather, “that it may be well with you” means in the world where all is good; and “that your days may be long” means in the world that is entirely long [Chullin 142a].

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|However the first reason forced me to write a short treatise, in which to include all the matters of number in brief.

|However the first reason forced me to write a short treatise, in which to include all the matters of number in brief.

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::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.

::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.

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::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.

::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.

::<math>\scriptstyle{\color{blue}{8-\left(5-2\right)=8-3=5}}</math>

::<math>\scriptstyle{\color{blue}{8-\left(5-2\right)=8-3=5}}</math>

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=== <span style=color:Green>Multiple Quantities Problem - Two Amounts of Money</span> ===

=== <span style=color:Green>Multiple Quantities Problem - Two Amounts of Money</span> ===

::The answer: we have to divide the 300 by 20, which is what the first walks [a day]; the result is 15. So, he walks 300 miles in 15 days.

::The answer: we have to divide the 300 by 20, which is what the first walks [a day]; the result is 15. So, he walks 300 miles in 15 days.

::<math>\scriptstyle{\color{blue}{\frac{300}{20}=15}}</math>

::<math>\scriptstyle{\color{blue}{\frac{300}{20}=15}}</math>

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::He buys one liṭra from each for 3 dinar and 4 pešuṭim.

::He buys one liṭra from each for 3 dinar and 4 pešuṭim.

::<math>\scriptstyle{\color{blue}{\frac{7}{12}+\frac{8}{12}+\frac{10}{12}+\frac{15}{12}=3+\frac{4}{12}}}</math>

::<math>\scriptstyle{\color{blue}{\frac{7}{12}+\frac{8}{12}+\frac{10}{12}+\frac{15}{12}=3+\frac{4}{12}}}</math>

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:{{#annot:men and beasts|612|IawN}}55) Question: Reuven hired Shimon to do his work 9 days for 10 liṭra. The work is to hire for him each day 13 men, each of whom should lead seven beasts, each of which should carry 15 measure and walk 6 parsa. He hired for him for 8 days 17 men, each of whom led 6 beasts, each of which carried 11 measures and walked 7 parsa.  

:{{#annot:men and beasts|612|IawN}}55) Question: Reuven hired Shimon to do his work 9 days for 10 liṭra. The work is to hire for him each day 13 men, each of whom should lead seven beasts, each of which should carry 15 measure and walk 6 parsa. He hired for him for 8 days 17 men, each of whom led 6 beasts, each of which carried 11 measures and walked 7 parsa.  

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:Then multiply the result also by the measure of the length.

:Then multiply the result also by the measure of the length.

:Divide the product by 144 and the result are the measures that the container contains

:Divide the product by 144 and the result are the measures that the container contains

דע כמה אצבעות מבוהן היד יש בארכו מבפנים ובעמקו באמצע ג"כ בעומק ראשו מפני שהוא צנה<br>

דע כמה אצבעות מבוהן היד יש בארכו מבפנים ובעמקו באמצע ג"כ בעומק ראשו מפני שהוא צנה<br>

וקח שעור חצי האצבעות שמצאת בעמקו באמצעו ובראשו ותכפלם בעצמם<br>

וקח שעור חצי האצבעות שמצאת בעמקו באמצעו ובראשו ותכפלם בעצמם<br>

והעולה תחלקהו על קמ"ד והיוצא הוא המדות שיכיל הכלי{{#annotend:j94W}}

והעולה תחלקהו על קמ"ד והיוצא הוא המדות שיכיל הכלי{{#annotend:j94W}}

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:*{{#annot:container|2582|GaHs}}Example: we want to know: how much will be contained in a container in which there are 50 thumbs of the size of my thumb from the inside, there are 28 thumbs in the depth of its center and 20 in the depth of its top which is narrower

:*{{#annot:container|2582|GaHs}}Example: we want to know: how much will be contained in a container in which there are 50 thumbs of the size of my thumb from the inside, there are 28 thumbs in the depth of its center and 20 in the depth of its top which is narrower

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::Divide it by 144; the result is two hundred and these are the measures that the container contains, according to the measure that it is now customary to measure the wine here, in the city of Caronia.

::Divide it by 144; the result is two hundred and these are the measures that the container contains, according to the measure that it is now customary to measure the wine here, in the city of Caronia.

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:Take its half and multiply it by itself, then divide by it half the [number of the] measures that the container will hold when it is full

:Take its half and multiply it by itself, then divide by it half the [number of the] measures that the container will hold when it is full

:The result is the [number of the] measures that are missing in the container.

:The result is the [number of the] measures that are missing in the container.

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:*{{#annot:container|2582|0X2x}}For example: we have a container that contains 200 small measures and in its depth in the middle there are 28 thumbs.

:*{{#annot:container|2582|0X2x}}For example: we have a container that contains 200 small measures and in its depth in the middle there are 28 thumbs.

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:Divide by it the number of the measures that you want the container to contain multiplied by 144.

:Divide by it the number of the measures that you want the container to contain multiplied by 144.

:Extract the root of the result and this is the volume that should be in the depth of the container, in its quarter which is between the middle and the top.

:Extract the root of the result and this is the volume that should be in the depth of the container, in its quarter which is between the middle and the top.

דע תחלה אורך הלוחות שתרצה לעשותו והוצא מהם הנצרך לשייר לעשות מקום לישיבת תחתיות הכלי וקח הנשאר מאורך הלוח ושמרהו וחלק עליהם שעור המדות שרצונך שיכיל הכלי מוכפלים על קמ"ד ומהיוצא קח שרשו והוא השעור שצריך להיות בעומק הכלי ברביעיתו שהוא בין האמצע ובין הראש{{#annotend:G2Z7}}

דע תחלה אורך הלוחות שתרצה לעשותו והוצא מהם הנצרך לשייר לעשות מקום לישיבת תחתיות הכלי וקח הנשאר מאורך הלוח ושמרהו וחלק עליהם שעור המדות שרצונך שיכיל הכלי מוכפלים על קמ"ד ומהיוצא קח שרשו והוא השעור שצריך להיות בעומק הכלי ברביעיתו שהוא בין האמצע ובין הראש{{#annotend:G2Z7}}

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:*{{#annot:container|2582|pVZR}}For example: we want to make a container that would hold two hundred measures and we have to make them out of boards that are 54 thumbs long.

:*{{#annot:container|2582|pVZR}}For example: we want to make a container that would hold two hundred measures and we have to make them out of boards that are 54 thumbs long.

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::Subtract 4 from them for the lower base of the container; 50 remain.

::Subtract 4 from them for the lower base of the container; 50 remain.

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::In this example, the resulting root is 24. Multiply it by 3 and a seventh; it is 75 and 3-sevenths. So, make the the width of all the boards at the quarter of the length of the board equal to the 75 and 3-sevenths mentioned and make the container as you wish.

::In this example, the resulting root is 24. Multiply it by 3 and a seventh; it is 75 and 3-sevenths. So, make the the width of all the boards at the quarter of the length of the board equal to the 75 and 3-sevenths mentioned and make the container as you wish.

::<math>\scriptstyle{\color{blue}{24\sdot\left(3+\frac{1}{7}\right)=75+\frac{3}{7}}}</math>

::<math>\scriptstyle{\color{blue}{24\sdot\left(3+\frac{1}{7}\right)=75+\frac{3}{7}}}</math>

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:The third said to the fourth: if you give me a quarter of your money with what I have left after giving the third the result will be ten and you will be left with ten.

:The third said to the fourth: if you give me a quarter of your money with what I have left after giving the third the result will be ten and you will be left with ten.

:How much was the money?

:How much was the money?

אמר הא' לב' אם תתן לי חצי ממונך עם שלי יעלה עשרה<br>

אמר הא' לב' אם תתן לי חצי ממונך עם שלי יעלה עשרה<br>

אמר הב' לג' אם תתן לי שליש ממונך עם חצי הממון שהיה לי קודם שנתתי לחבירי מחציתו יעלה עשרה ר"ל עם מה שנשאר לי<br>

אמר הב' לג' אם תתן לי שליש ממונך עם חצי הממון שהיה לי קודם שנתתי לחבירי מחציתו יעלה עשרה ר"ל עם מה שנשאר לי<br>

::We return, then, to the second. We already know that he received 6 and 2-thirds from his friend. So, he needs 3 and a third to complete the amount.

::We return, then, to the second. We already know that he received 6 and 2-thirds from his friend. So, he needs 3 and a third to complete the amount.

::<math>\scriptstyle{\color{blue}{a=10-\left(6+\frac{2}{3}\right)=3+\frac{1}{3}}}</math>

::<math>\scriptstyle{\color{blue}{a=10-\left(6+\frac{2}{3}\right)=3+\frac{1}{3}}}</math>

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:The third said to the first and the second: if you give me a fifth of your money with all of my money the result will be 100.

:The third said to the first and the second: if you give me a fifth of your money with all of my money the result will be 100.

:How much did each have?

:How much did each have?

אמ' הא' לב' ולג' אם תתנו לי שליש ממונכם עם כל ממוני יעלה ק&#x202B;'<br>

אמ' הא' לב' ולג' אם תתנו לי שליש ממונכם עם כל ממוני יעלה ק&#x202B;'<br>

השיב הב' לג' ולא' אם תתנו לי רביעית ממונכם עם כל ממוני יעלו ק&#x202B;'<br>

השיב הב' לג' ולא' אם תתנו לי רביעית ממונכם עם כל ממוני יעלו ק&#x202B;'<br>

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::The answer: do as follows: take a number that has a third, a quarter, and a fifth; it is 60.

::The answer: do as follows: take a number that has a third, a quarter, and a fifth; it is 60.

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::<math>\scriptstyle{\color{blue}{\left(62+\frac{1}{2}\right):\left(42+\frac{1}{2}\right)=100:b}}</math>

::<math>\scriptstyle{\color{blue}{\left(62+\frac{1}{2}\right):\left(42+\frac{1}{2}\right)=100:b}}</math>

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:The fourth said: give a fifth of your money with all my money and the result will be 100.

:The fourth said: give a fifth of your money with all my money and the result will be 100.

:How much did each have?

:How much did each have?

אמר א' אני אתן כל ממוני ותנו אתם חצי ממונכם ויהיו ק&#x202B;'<br>

אמר א' אני אתן כל ממוני ותנו אתם חצי ממונכם ויהיו ק&#x202B;'<br>

השיב הב' לג' אני אתן כל ממוני ותנו אתם שליש ממונכם ויעלה ק&#x202B;'<br>

השיב הב' לג' אני אתן כל ממוני ותנו אתם שליש ממונכם ויעלה ק&#x202B;'<br>

:Then place them on the table one by one, from top to bottom and from bottom to top, and so on, until you have placed all five rows on the table.

:Then place them on the table one by one, from top to bottom and from bottom to top, and so on, until you have placed all five rows on the table.

:Then ask each them in which row is the card he touched.

:Then ask each them in which row is the card he touched.

ואמור לה' בני אדם שכל א' יגע א' בשורה א' ר"ל שבשורה שיגע הא' לא יגע הב&#x202B;'<br>

ואמור לה' בני אדם שכל א' יגע א' בשורה א' ר"ל שבשורה שיגע הא' לא יגע הב&#x202B;'<br>

אח"כ הניחם על השלחן אחת אחת ממעלה למטה וממטה למעלה וכן עד שהנחת כל הה' שורות על השלחן<br>

אח"כ הניחם על השלחן אחת אחת ממעלה למטה וממטה למעלה וכן עד שהנחת כל הה' שורות על השלחן<br>

ואח"כ שאל להם באיזו שורה השטר שנגע וכן לכל א' וא&#x202B;'{{#annotend:dqeR}}

ואח"כ שאל להם באיזו שורה השטר שנגע וכן לכל א' וא&#x202B;'{{#annotend:dqeR}}

:You count all the points and you are 245 short to 312.

:You count all the points and you are 245 short to 312.

:Turn over the cards a second time and count the missing five of whichever kind they are i.e. whether of horses or cups or sticks or swords and you will receive the required.

:Turn over the cards a second time and count the missing five of whichever kind they are i.e. whether of horses or cups or sticks or swords and you will receive the required.

ידוע תדע כי כל הנקדות מצחוק שלם משטרות עולים שי"ב וכל כך שישוה הפרש עשרה ורוכב הסוס י"א והמלך י"ב<br>

ידוע תדע כי כל הנקדות מצחוק שלם משטרות עולים שי"ב וכל כך שישוה הפרש עשרה ורוכב הסוס י"א והמלך י"ב<br>

ותמנה כל הנקדות ויחסר לך עד תשלום שי"ב רמ'ה&#x202B;'<br>

ותמנה כל הנקדות ויחסר לך עד תשלום שי"ב רמ'ה&#x202B;'<br>

:The third said to the first: if you give me a quarter of your money plus 20 with all of my money we will buy the horse that is worth 100.

:The third said to the first: if you give me a quarter of your money plus 20 with all of my money we will buy the horse that is worth 100.

:How much did each have?

:How much did each have?

ואמר הא' לב' תן לי חצי ממונך עם כל ממוני ונקנה סוס הס&#x202B;'<br>

ואמר הא' לב' תן לי חצי ממונך עם כל ממוני ונקנה סוס הס&#x202B;'<br>

אמר הב' לג' אם תתן לי שליש ממונך עם כל שלי נקנה סוס הפ&#x202B;'<br>

אמר הב' לג' אם תתן לי שליש ממונך עם כל שלי נקנה סוס הפ&#x202B;'<br>

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::The answer is as follows:

::The answer is as follows:

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::Multiply 48 by 50; it is 2400.

::Multiply 48 by 50; it is 2400.

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:The third asks for a quarter of their money minus 4.

:The third asks for a quarter of their money minus 4.

:How much did each have?

:How much did each have?

א' שואל לב' הנשארים מחצית ממונם פחות ב&#x202B;'<br>

א' שואל לב' הנשארים מחצית ממונם פחות ב&#x202B;'<br>

והאחר שואל שליש ממונם פחות ג&#x202B;'<br>

והאחר שואל שליש ממונם פחות ג&#x202B;'<br>

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::The answer: find a number that includes a half, a third, and a quarter; it is 12.

::The answer: find a number that includes a half, a third, and a quarter; it is 12.

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:How much did each have?

:How much did each have?

:<math>\scriptstyle\begin{cases}\scriptstyle a+\left[\frac{1}{2}\sdot\left(b+c\right)\right]+2=22+\frac{1}{2}\\\scriptstyle b+\left[\frac{1}{3}\sdot\left(a+c\right)\right]+3=22+\frac{1}{2}\\\scriptstyle c+\left[\frac{1}{4}\sdot\left(a+b\right)\right]+4=22+\frac{1}{2}\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle a+\left[\frac{1}{2}\sdot\left(b+c\right)\right]+2=22+\frac{1}{2}\\\scriptstyle b+\left[\frac{1}{3}\sdot\left(a+c\right)\right]+3=22+\frac{1}{2}\\\scriptstyle c+\left[\frac{1}{4}\sdot\left(a+b\right)\right]+4=22+\frac{1}{2}\end{cases}</math>

אמר הא' לב' אם תתנו לי חצי ממונכם עם תוספת שנים ועם שלי יהיו כ"ב וחצי<br>

אמר הא' לב' אם תתנו לי חצי ממונכם עם תוספת שנים ועם שלי יהיו כ"ב וחצי<br>

אמר הב' לא' ולג' אם תתנו שליש ממונכם עם תוספת ג' ועם שלי יהיו כ"ב וחצי<br>

אמר הב' לא' ולג' אם תתנו שליש ממונכם עם תוספת ג' ועם שלי יהיו כ"ב וחצי<br>

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::The answer: look for a number that includes a half, a third, and a quarter; it is 12.

::The answer: look for a number that includes a half, a third, and a quarter; it is 12.

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::Subtract also 21 from it; 12 and a half remains and this is the share of the one who asks for a third of the money of his friends plus 3.

::Subtract also 21 from it; 12 and a half remains and this is the share of the one who asks for a third of the money of his friends plus 3.

::<math>\scriptstyle{\color{blue}{b=\left(33+\frac{1}{2}\right)-21=12+\frac{1}{2}}}</math>

::<math>\scriptstyle{\color{blue}{b=\left(33+\frac{1}{2}\right)-21=12+\frac{1}{2}}}</math>

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:We ask: if we want to drop a rope from the top of the wall to the hole which is 30 cubits far from the bottom of the wall.

:We ask: if we want to drop a rope from the top of the wall to the hole which is 30 cubits far from the bottom of the wall.

:How long should it be?  

:How long should it be?  

נשאל אם נרצה להוריד חבל אחד מראש החומה עד סוף הגומא הרחוקה מרגל החומה ל' אמות<br>

נשאל אם נרצה להוריד חבל אחד מראש החומה עד סוף הגומא הרחוקה מרגל החומה ל' אמות<br>

כמה יהיה ארכה{{#annotend:x5y8}}

כמה יהיה ארכה{{#annotend:x5y8}}

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::The answer: we should multiply the [height of the] wall, which is 50, by itself; it is 2500.

::The answer: we should multiply the [height of the] wall, which is 50, by itself; it is 2500.

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:{{#annot:equilateral triangle|679|ZIN2}}79) Question: a triangle such as this - if the length of the height is 30, how much is the length of each side of the triangle?  

:{{#annot:equilateral triangle|679|ZIN2}}79) Question: a triangle such as this - if the length of the height is 30, how much is the length of each side of the triangle?  

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::The answer: you should multiply the length of the line [= the height], which is 30, by 30; it is 900.

::The answer: you should multiply the length of the line [= the height], which is 30, by 30; it is 900.

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:We wish to build a bridge from the top of one tower to the other.

:We wish to build a bridge from the top of one tower to the other.

:How long would it be?

:How long would it be?

כמה יהיה ארכו{{#annotend:fq24}}

כמה יהיה ארכו{{#annotend:fq24}}

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::The answer: we should multiply the distance between the two towers, which is 20, by itself; it is 400.

::The answer: we should multiply the distance between the two towers, which is 20, by itself; it is 400.

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:The thickness of the wall from each of its corners is 2 cubits and a quarter.

:The thickness of the wall from each of its corners is 2 cubits and a quarter.

:We wish to know: how much is the perimeter of the tower?

:We wish to know: how much is the perimeter of the tower?

נרצה כמה הקף המגדל{{#annotend:ekBH}}

נרצה כמה הקף המגדל{{#annotend:ekBH}}

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::The answer: you should multiply 4, since there are four walls, by 2 and a quarter; the result is 9.

::The answer: you should multiply 4, since there are four walls, by 2 and a quarter; the result is 9.

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:{{#annot:square|679|ZZDk}}82) Question: here is a square such as this, whose inner line is 10 cubits long, i.e. its diagonal.

:{{#annot:square|679|ZZDk}}82) Question: here is a square such as this, whose inner line is 10 cubits long, i.e. its diagonal.

:How much is the length of each side of the square?  

:How much is the length of each side of the square?  

כמה אורך כל קו וקו מהמרובע{{#annotend:ZZDk}}

כמה אורך כל קו וקו מהמרובע{{#annotend:ZZDk}}

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::The answer: you should multiply ten by itself; it is one hundred.

::The answer: you should multiply ten by itself; it is one hundred.

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:{{#annot:square|683|X7Nc}}83) Question: a square whose four sides are 10 long.

:{{#annot:square|683|X7Nc}}83) Question: a square whose four sides are 10 long.

:We wish to know: how much is the length of its diagonal?

:We wish to know: how much is the length of its diagonal?

נרצה לדעת כמה אורך הקו ההולכת באלכסון{{#annotend:X7Nc}}

נרצה לדעת כמה אורך הקו ההולכת באלכסון{{#annotend:X7Nc}}

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:The tree was broken and its top touched the edge of the stream, i.e. 10 cubits far from the foot of the tree.

:The tree was broken and its top touched the edge of the stream, i.e. 10 cubits far from the foot of the tree.

:We ask: at what point did the tree break?

:We ask: at what point did the tree break?

נשאל באיזה מקום נשבר האילן{{#annotend:36n4}}

נשאל באיזה מקום נשבר האילן{{#annotend:36n4}}

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::The answer: you should multiply the length of the tree by itself, i.e. 20 by 20; it is 400.

::The answer: you should multiply the length of the tree by itself, i.e. 20 by 20; it is 400.

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:We want to create a triangle inside it, so that the length of each of its three sides will be 20.

:We want to create a triangle inside it, so that the length of each of its three sides will be 20.

:How much is the perimeter of the quadrangular?

:How much is the perimeter of the quadrangular?

כמה יקיף כל המרובע{{#annotend:Z82J}}

כמה יקיף כל המרובע{{#annotend:Z82J}}

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:{{#annot:equilateral triangle|685|DSFq}}86) Question: here is a triangle, the length of each of its sides is 35.

:{{#annot:equilateral triangle|685|DSFq}}86) Question: here is a triangle, the length of each of its sides is 35.

:How long is its height?

:How long is its height?

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:{{#annot:square to circle|2583|sy3G}}87) Question: if you want to create a square from a circle

:{{#annot:square to circle|2583|sy3G}}87) Question: if you want to create a square from a circle

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:{{#annot:right-angled triangle|679|T2On}}88) Question: here are two intersecting lines, one is 30 long and the second is 40 long.

:{{#annot:right-angled triangle|679|T2On}}88) Question: here are two intersecting lines, one is 30 long and the second is 40 long.

:If we wish to create a triangle from these two lines, how long should the third line be?

:If we wish to create a triangle from these two lines, how long should the third line be?

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=== <span style=color:Green>Encounter Problem - Two Men</span> ===

=== <span style=color:Green>Encounter Problem - Two Men</span> ===

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:{{#annot:two men|658|dgk5}}89) Question: A man walks from Naples to Rome in 5 days and on the same day and at the same time another man walks from Rome to Naples in 7 days.

:{{#annot:two men|658|dgk5}}89) Question: A man walks from Naples to Rome in 5 days and on the same day and at the same time another man walks from Rome to Naples in 7 days.

:We would like to know in how many days they will meet each other?

:We would like to know in how many days they will meet each other?

נרצה לדעת בכמה ימים יפגשו זה את זה{{#annotend:dgk5}}

נרצה לדעת בכמה ימים יפגשו זה את זה{{#annotend:dgk5}}

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=== <span style=color:Green>Gaging Problem - Barrel</span> ===

=== <span style=color:Green>Gaging Problem - Barrel</span> ===

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:{{#annot:barrel|2582|ZrkJ}}90) Question: one barrel of twenty boards contains twenty measures of water.

:{{#annot:barrel|2582|ZrkJ}}90) Question: one barrel of twenty boards contains twenty measures of water.

:If we remove two boards, how much will it contain?

:If we remove two boards, how much will it contain?

אם נחסר ב' קרשים כמה יכיל{{#annotend:ZrkJ}}

אם נחסר ב' קרשים כמה יכיל{{#annotend:ZrkJ}}

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:{{#annot:three men|2644|qbn3}}91) Question: three men found a purse with some dinar.

:{{#annot:three men|2644|qbn3}}91) Question: three men found a purse with some dinar.

:The second answered: if you give me the purse I will have two times as much dinar as both of you.

:The second answered: if you give me the purse I will have two times as much dinar as both of you.

:The third said: if you give it to me I will have three times as much money as both of you.

:The third said: if you give it to me I will have three times as much money as both of you.

:How much money was in the purse and how much did each have?

:How much money was in the purse and how much did each have?

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::The answer: look for a number that has a half and a third; it is 12.

::The answer: look for a number that has a half and a third; it is 12.

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:I ask: how much did each have and how much is the price of the horse?

:I ask: how much did each have and how much is the price of the horse?

:<math>\scriptstyle\begin{cases}\scriptstyle a_1+\left(\frac{1}{3}a_2+\frac{1}{4}a_2\right)=N \\\scriptstyle a_2+\left(\frac{1}{4}a_1+\frac{1}{5}a_1\right)=N\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle a_1+\left(\frac{1}{3}a_2+\frac{1}{4}a_2\right)=N \\\scriptstyle a_2+\left(\frac{1}{4}a_1+\frac{1}{5}a_1\right)=N\end{cases}</math>

אמר הא' לב' אם תתן לי שליש ממונך ורביעיתו עם כל ממוני אקנה הסוס<br>

אמר הא' לב' אם תתן לי שליש ממונך ורביעיתו עם כל ממוני אקנה הסוס<br>

אמר הב' לא' אם תתן לי רביעית ממונך וחמישיתו עם כל ממוני אקנה הסוס<br>

אמר הב' לא' אם תתן לי רביעית ממונך וחמישיתו עם כל ממוני אקנה הסוס<br>

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::The answer: find a number that has a third and a quarter for the first; you find it is 12.

::The answer: find a number that has a third and a quarter for the first; you find it is 12.

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=== <span style=color:Green>Geometrical Problems</span> ===

=== <span style=color:Green>Geometrical Problems</span> ===

:{{#annot:square city|680|xXVE}}93) Question: Here is a city, the walls surrounding the whole city are 3 miles [long].

:{{#annot:square city|680|xXVE}}93) Question: Here is a city, the walls surrounding the whole city are 3 miles [long].

:We wish to know how much a city with ten walls is greater than the other.

:We wish to know how much a city with ten walls is greater than the other.

נרצה לדעת עיר שחומותיה עשרה כמה היא יותר גדולה מהאחרת{{#annotend:xXVE}}

נרצה לדעת עיר שחומותיה עשרה כמה היא יותר גדולה מהאחרת{{#annotend:xXVE}}

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::Divide one hundred by 9; you receive 11 and a ninth and this is the number of times, by which the one of ten [miles] is greater than the one of 3 [miles].

::Divide one hundred by 9; you receive 11 and a ninth and this is the number of times, by which the one of ten [miles] is greater than the one of 3 [miles].

::<math>\scriptstyle{\color{blue}{\frac{10^2}{3^2}=\frac{100}{9}=11+\frac{1}{9}}}</math>

::<math>\scriptstyle{\color{blue}{\frac{10^2}{3^2}=\frac{100}{9}=11+\frac{1}{9}}}</math>

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:{{#annot:circle|682|XJvI}}94) Question: here is a circle whose diameter is 7.

:{{#annot:circle|682|XJvI}}94) Question: here is a circle whose diameter is 7.

:We wish to know how much is the perimeter of the circle?

:We wish to know how much is the perimeter of the circle?

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:The diameter of the circle is 7.

:The diameter of the circle is 7.

:How much is the length of each side of the square?

:How much is the length of each side of the square?

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::You should multiply 7 by itself; it is 49.

::You should multiply 7 by itself; it is 49.

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::Divide it by 2; it is 24 and a half.

::Divide it by 2; it is 24 and a half.

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::Extract its root; it is approximately 5 and this is the length of each side of the square.

::Extract its root; it is approximately 5 and this is the length of each side of the square.

::<math>\scriptstyle{\color{blue}{\sqrt{\frac{7^2}{2}}=\sqrt{\frac{49}{2}}=\sqrt{24+\frac{1}{2}}\approx5}}</math>

::<math>\scriptstyle{\color{blue}{\sqrt{\frac{7^2}{2}}=\sqrt{\frac{49}{2}}=\sqrt{24+\frac{1}{2}}\approx5}}</math>

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==== <span style=color:Green>Find the Side - Inscribed Triangle</span> ====

==== <span style=color:Green>Find the Side - Inscribed Triangle</span> ====

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:{{#annot:inscribed triangle|679|obt9}}95) Question: here is a circle whose diameter is 16.

:{{#annot:inscribed triangle|679|obt9}}95) Question: here is a circle whose diameter is 16.

:If you want to create a triangle inside it, how much will be its perimeter and how much will be the length of each side of the triangle?  

:If you want to create a triangle inside it, how much will be its perimeter and how much will be the length of each side of the triangle?  

אם תרצה לעשות משולש בתוכו כמה יהיה הקפו וכמה יהיה אורך כל קו וקו מהמשולש{{#annotend:obt9}}

אם תרצה לעשות משולש בתוכו כמה יהיה הקפו וכמה יהיה אורך כל קו וקו מהמשולש{{#annotend:obt9}}

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:{{#annot:circle|683|MVIQ}}96) Question: here is a circle whose perimeter is 30.

:{{#annot:circle|683|MVIQ}}96) Question: here is a circle whose perimeter is 30.

:How much is the length of the diameter?  

:How much is the length of the diameter?  

כמה אורך הבריח המחלקת{{#annotend:MVIQ}}

כמה אורך הבריח המחלקת{{#annotend:MVIQ}}

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:{{#annot:rectangle|683,680|ikHf}}97) Question: here is a quadrangular - the length of its two long sides is 40 and the breadth of its two short sides is 30.

:{{#annot:rectangle|683,680|ikHf}}97) Question: here is a quadrangular - the length of its two long sides is 40 and the breadth of its two short sides is 30.

:We wish to know: how long is the diagonal and how much is the area of the whole quadrangular?

:We wish to know: how long is the diagonal and how much is the area of the whole quadrangular?

נרצה לדעת כמה אורך הבריח האלכסונית וריקות כל המרובע{{#annotend:ikHf}}

נרצה לדעת כמה אורך הבריח האלכסונית וריקות כל המרובע{{#annotend:ikHf}}

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:{{#annot:wall|685|WhQX}}98) Question: Here is a city whose walls are of unknown height.

:{{#annot:wall|685|WhQX}}98) Question: Here is a city whose walls are of unknown height.

:If you would like to know their height

:If you would like to know their height

אם תרצה לדעת גבהם{{#annotend:WhQX}}

אם תרצה לדעת גבהם{{#annotend:WhQX}}

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::If it is not as high, move away or get closer, while lying on the ground, until it seems to you that it is of the same height.

::If it is not as high, move away or get closer, while lying on the ground, until it seems to you that it is of the same height.

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=== <span style=color:Green>Find the Price - Barrel</span> ===

=== <span style=color:Green>Find the Price - Barrel</span> ===

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:{{#annot:barrel|629|yvCf}}99) Question: there is a barrel full of oil here that is 7 cubits wide from the foot below and gets shorter and shorter until the upper base that is a cubit, which is worth 8 minyanim.

:{{#annot:barrel|629|yvCf}}99) Question: there is a barrel full of oil here that is 7 cubits wide from the foot below and gets shorter and shorter until the upper base that is a cubit, which is worth 8 minyanim.

:How much is a barrel that is 7 cubits wide at the bottom and the top base is also 7 cubits wide?

:How much is a barrel that is 7 cubits wide at the bottom and the top base is also 7 cubits wide?

כמה ישוה חבית שרחבה למטה ז' אמות גם הראש העליון רחבו ז' אמות{{#annotend:yvCf}}

כמה ישוה חבית שרחבה למטה ז' אמות גם הראש העליון רחבו ז' אמות{{#annotend:yvCf}}

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=== <span style=color:Green>Find the Side - Rectangle Formed by Walls</span> ===

=== <span style=color:Green>Find the Side - Rectangle Formed by Walls</span> ===

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:The distance plus the height of one of the walls is 11.

:The distance plus the height of one of the walls is 11.

:How high is the wall and how far are they from each other?

:How high is the wall and how far are they from each other?

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:We know that the height of the wall exceeds the distance between the two walls by 2 cubits.

:We know that the height of the wall exceeds the distance between the two walls by 2 cubits.

:We would like to know how high each one of the walls is and how far apart they are.

:We would like to know how high each one of the walls is and how far apart they are.

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::The way is as follows:  

::The way is as follows:  

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:We know that the distance between its two ends is 25, for example.

:We know that the distance between its two ends is 25, for example.

:We ask: by how much did it break?

:We ask: by how much did it break?

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=== <span style=color:Green>Multiple Quantities - Selling Apples</span> ===

=== <span style=color:Green>Multiple Quantities - Selling Apples</span> ===

:At what price did they sell?

:At what price did they sell?

:<math>\scriptstyle10a_1=30a_2=50a_3</math>

:<math>\scriptstyle10a_1=30a_2=50a_3</math>

לא' נתן י' ולאחר נתן ל' ולאחר נתן נ&#x202B;'<br>

לא' נתן י' ולאחר נתן ל' ולאחר נתן נ&#x202B;'<br>

ואמר כל כך ישא לי ממון הא' כמו האחר לא פחות ולא יתר ובערך שימכור הא' ימכרו האחרים<br>

ואמר כל כך ישא לי ממון הא' כמו האחר לא פחות ולא יתר ובערך שימכור הא' ימכרו האחרים<br>

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=== <span style=color:Green>Gaging Problem - Wine and Water</span> ===

=== <span style=color:Green>Gaging Problem - Wine and Water</span> ===

:{{#annot:wine and water|2582|rzGP}}104) Question: Here is a barrel that contains [10] kabin, half water and half wine.

:{{#annot:wine and water|2582|rzGP}}104) Question: Here is a barrel that contains [10] kabin, half water and half wine.

:We take out one kab, and pour one kab of water in it.

:We take out one kab, and pour one kab of water in it.

:We take another kab out a third time, and fill it with water.

:We take another kab out a third time, and fill it with water.

:We take another kab out a fourth time, and fill it with water.

:We take another kab out a fourth time, and fill it with water.

:We would like to know how many kabim of wine are left in the barrel after the fourth time.

:We would like to know how many kabim of wine are left in the barrel after the fourth time.

ונוציא קב אחד ואח"כ נשים בו קב מים<br>

ונוציא קב אחד ואח"כ נשים בו קב מים<br>

עוד נוציא קב פעם ג' ונמלאנה מים<br>

עוד נוציא קב פעם ג' ונמלאנה מים<br>

עוד נוציא קב פעם ד' ונמלאנה מים<br>

עוד נוציא קב פעם ד' ונמלאנה מים<br>

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::The way is as follows:

::The way is as follows:

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:Its shadow is 100 cubits and this shadow reaches another smaller tower that is ten cubits high and its [shadow] is 25.

:Its shadow is 100 cubits and this shadow reaches another smaller tower that is ten cubits high and its [shadow] is 25.

:We would like to know how much is the unknown height of the tower.

:We would like to know how much is the unknown height of the tower.

נרצה לדעת כמה גובה המגדל הנעלם{{#annotend:foOl}}

נרצה לדעת כמה גובה המגדל הנעלם{{#annotend:foOl}}

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:The length of the rope is 120 cubits.

:The length of the rope is 120 cubits.

:We would like to know how high the wall is.

:We would like to know how high the wall is.

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:{{#annot:³√10|439|WS2D}}107) Question: if you want to extract the cubic root of ten.

:{{#annot:³√10|439|WS2D}}107) Question: if you want to extract the cubic root of ten.

:<math>\scriptstyle\sqrt[3]{10}</math>

:<math>\scriptstyle\sqrt[3]{10}</math>

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:{{#annot:perfect number|618|IXt8}}108) Question: if you want to find a perfect number

:{{#annot:perfect number|618|IXt8}}108) Question: if you want to find a perfect number

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:{{#annot:perfect number|618|x3Is}}109) If you want to find the second [perfect number]

:{{#annot:perfect number|618|x3Is}}109) If you want to find the second [perfect number]

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:{{#annot:perfect number|618|CaJp}}110) If you want to find the third [perfect number]

:{{#annot:perfect number|618|CaJp}}110) If you want to find the third [perfect number]

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::To find the fifth, take 64 and do as mentioned.

::To find the fifth, take 64 and do as mentioned.

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:The sixth is 2048.

:The sixth is 2048.

:We wish to know how much is the second?

:We wish to know how much is the second?

דמיון זה יש לנו ו' {{#annot:term|994,2263|jjwV}}מספרים יחסיים{{#annotend:jjwV}} ר"ל שיחס הו' אל הה' כיחס הה' אל הד' וכיחס הד' אל הג' וכיחס הג' אל הב' וכיחס הב' אל הא&#x202B;'<br>

דמיון זה יש לנו ו' {{#annot:term|994,2263|jjwV}}מספרים יחסיים{{#annotend:jjwV}} ר"ל שיחס הו' אל הה' כיחס הה' אל הד' וכיחס הד' אל הג' וכיחס הג' אל הב' וכיחס הב' אל הא&#x202B;'<br>

והמספר הא' ב&#x202B;'<br>

והמספר הא' ב&#x202B;'<br>

:*{{#annot:six proportional numbers|618|Wrir}}For example: we have six proportional numbers, the first is 2 and the sixth is 64.

:*{{#annot:six proportional numbers|618|Wrir}}For example: we have six proportional numbers, the first is 2 and the sixth is 64.

::We wish to know which is the second.

::We wish to know which is the second.

נרצה לדעת מהו השני{{#annotend:Wrir}}

נרצה לדעת מהו השני{{#annotend:Wrir}}

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:{{#annot:a²=40+41\b|618|aiR7}}112) Question: find a number such that when you multiply it by itself the result is 40 and 41 parts of a fraction.

:{{#annot:a²=40+41\b|618|aiR7}}112) Question: find a number such that when you multiply it by itself the result is 40 and 41 parts of a fraction.

:<math>\scriptstyle a^2=40+\frac{41}{b}</math>

:<math>\scriptstyle a^2=40+\frac{41}{b}</math>

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::<span style=color:Green>'''Check:'''</span> because, if you multiply 6 and 3-eighths by 6 and 3-eighths, it is 40 integers and 41 parts of 64.

::<span style=color:Green>'''Check:'''</span> because, if you multiply 6 and 3-eighths by 6 and 3-eighths, it is 40 integers and 41 parts of 64.

::<math>\scriptstyle{\color{blue}{\left(6+\frac{3}{8}\right)^2=40+\frac{41}{64}}}</math>

::<math>\scriptstyle{\color{blue}{\left(6+\frac{3}{8}\right)^2=40+\frac{41}{64}}}</math>

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:{{#annot:rectangle|683|TX4D}}113) Question: A quadrilateral, the length of its two lines of longitude is 8 and the length of its two lines of latitude is 6.

:{{#annot:rectangle|683|TX4D}}113) Question: A quadrilateral, the length of its two lines of longitude is 8 and the length of its two lines of latitude is 6.

:We would like to know the length of the diagonals of the quadrilateral from end to end, i.e. the ones that go diagonally.

:We would like to know the length of the diagonals of the quadrilateral from end to end, i.e. the ones that go diagonally.

נרצה לדעת כמה אורך כל קו וקו מהמרובע המבריחות מן הקצה אל הקצה ר"ל ההולכות באלכסון{{#annotend:TX4D}}

נרצה לדעת כמה אורך כל קו וקו מהמרובע המבריחות מן הקצה אל הקצה ר"ל ההולכות באלכסון{{#annotend:TX4D}}

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::Take half the length, which is 4, and half [the width], which is 3, and multiply them by each other; it is 12 and this is the area of each of the four quadrilaterals. This is its figure:

::Take half the length, which is 4, and half [the width], which is 3, and multiply them by each other; it is 12 and this is the area of each of the four quadrilaterals. This is its figure:

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=== <span style=color:Green>Find the Height - Well</span> ===

=== <span style=color:Green>Find the Height - Well</span> ===

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:{{#annot:well|685|Snqi}}114) Question: A man wants to know the depth of a well that has 12 measures of water in it; each of the four sides of the well is 5 cubits long, and if he throws a stone into the well, each of its four sides is 4 cubits long and its height is twenty, the water reaches the surface of the well

:{{#annot:well|685|Snqi}}114) Question: A man wants to know the depth of a well that has 12 measures of water in it; each of the four sides of the well is 5 cubits long, and if he throws a stone into the well, each of its four sides is 4 cubits long and its height is twenty, the water reaches the surface of the well

:We would like to know how deep the well is.

:We would like to know how deep the well is.

נרצה לדעת כמה עמקו של בור{{#annotend:Snqi}}

נרצה לדעת כמה עמקו של בור{{#annotend:Snqi}}

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::Add it to the 12 measures that are in the well; it is 24 and 4-fifths and this is the depth of the whole well.

::Add it to the 12 measures that are in the well; it is 24 and 4-fifths and this is the depth of the whole well.

::<math>\scriptstyle{\color{blue}{\left(12+\frac{4}{5}\right)+12=24+\frac{4}{5}}}</math>

::<math>\scriptstyle{\color{blue}{\left(12+\frac{4}{5}\right)+12=24+\frac{4}{5}}}</math>

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:{{#annot:circle to square|2583|NFMC}}115) Question: what square can be made from a circle whose diameter is 7?

:{{#annot:circle to square|2583|NFMC}}115) Question: what square can be made from a circle whose diameter is 7?

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:{{#annot:circle to squares|2583|Ozqb}}117) Question: How many squares of [one cubit] can be made from a circle whose circumference is 22.

:{{#annot:circle to squares|2583|Ozqb}}117) Question: How many squares of [one cubit] can be made from a circle whose circumference is 22.

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:We want to make its length 12.

:We want to make its length 12.

:How much will its perimeter be?

:How much will its perimeter be?

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:The second said: if I had 1 I would have half as much as the both of you.

:The second said: if I had 1 I would have half as much as the both of you.

:The third said: if I had 1 I would have third as much as the both of you.

:The third said: if I had 1 I would have third as much as the both of you.

אמר הא' אם היה לי א' היה לי כמו שניכם<br>

אמר הא' אם היה לי א' היה לי כמו שניכם<br>

אמר השני אם היה לי אחד היה לי חצי שלכם<br>

אמר השני אם היה לי אחד היה לי חצי שלכם<br>

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::The answer: look for the common denominator; it is 6.

::The answer: look for the common denominator; it is 6.

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:The third said: I will give a quarter of what I have and you will give all you have and we will buy the fish.

:The third said: I will give a quarter of what I have and you will give all you have and we will buy the fish.

:We ask: how much did each have and how much is the price of the fish?

:We ask: how much did each have and how much is the price of the fish?

אמר הראשון אני אתן החצי ממה שיש לי ממעות ואתם תתנו כל מה שיש לכם<br>

אמר הראשון אני אתן החצי ממה שיש לי ממעות ואתם תתנו כל מה שיש לכם<br>

אמר השני אני אתן השליש ממה שיש לי ואתם תתנו כל מה שלכם<br>

אמר השני אני אתן השליש ממה שיש לי ואתם תתנו כל מה שלכם<br>

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::The answer: look for a number that has a half, a third, and a quarter; you find it is 12.

::The answer: look for a number that has a half, a third, and a quarter; you find it is 12.

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:The third said: I will give a quarter of my money and you will give all of what you have.

:The third said: I will give a quarter of my money and you will give all of what you have.

:We wish to know how much did each have?

:We wish to know how much did each have?

אמר הא' אני אתן חצי ממוני ותנו אתם כל אשר לכם<br>

אמר הא' אני אתן חצי ממוני ותנו אתם כל אשר לכם<br>

אמר השני אני אתן שליש ממוני ותנו אתם כל אשר לכם<br>

אמר השני אני אתן שליש ממוני ותנו אתם כל אשר לכם<br>

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

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

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::Then, say: 12 is 3-quarters of which number? You find it is 16.

::Then, say: 12 is 3-quarters of which number? You find it is 16.

::<math>\scriptstyle{\color{blue}{\frac{3}{4}b_3=12\longrightarrow b_3=16}}</math>

::<math>\scriptstyle{\color{blue}{\frac{3}{4}b_3=12\longrightarrow b_3=16}}</math>

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::You know that if the price of the [rooster] was unknown, we have already found the answer to the question, because the price of the rooster was 46 as stated above.

::You know that if the price of the [rooster] was unknown, we have already found the answer to the question, because the price of the rooster was 46 as stated above.

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::<math>\scriptstyle{\color{blue}{46:24=11:a_1}}</math>

::<math>\scriptstyle{\color{blue}{46:24=11:a_1}}</math>

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=== <span style=color:Green>Partnership Problem – Three Partners</span> ===

=== <span style=color:Green>Partnership Problem – Three Partners</span> ===

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:The third contributed 10 canna of cloth and he ought to take 32 dinar from the profit.

:The third contributed 10 canna of cloth and he ought to take 32 dinar from the profit.

:We ask how much are the 10 canna worth and how much did the one who earned 17 dinar contribute?

:We ask how much are the 10 canna worth and how much did the one who earned 17 dinar contribute?

הא' שם י"ו מנינים וראוי לו לקחת מהריוח כ"ה דינרין<br>

הא' שם י"ו מנינים וראוי לו לקחת מהריוח כ"ה דינרין<br>

והשני שם כל כך דינרין באופן שראוי לו לקחת מן הריוח י"ז דינרין<br>

והשני שם כל כך דינרין באופן שראוי לו לקחת מן הריוח י"ז דינרין<br>

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::Answer: you already know that the one who contributed 16 dinar should take 25.

::Answer: you already know that the one who contributed 16 dinar should take 25.

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:{{#annot:ring|667|dQMu}}123) Question: if you want to know to which person your friend has given a ring, on which finger is it, and on which finger joint.

:{{#annot:ring|667|dQMu}}123) Question: if you want to know to which person your friend has given a ring, on which finger is it, and on which finger joint.

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::Example: I gave the ring to the fourth man on the third finger on the second joint.

::Example: I gave the ring to the fourth man on the third finger on the second joint.

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::2 [units] - the second joint.

::2 [units] - the second joint.

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!<span style=color:Green>Guessing a chosen number</span>

!<span style=color:Green>Guessing a chosen number</span>

:Then sum up all [the results] and cast out all the hundreds and for each hundred subtract 5.

:Then sum up all [the results] and cast out all the hundreds and for each hundred subtract 5.

:The remainder will be the requested [number].

:The remainder will be the requested [number].

תאמר לו אם הולך ג"ג ואם הולך לא תקח כלום ואם אינו הולך קח בעד כל א' שישאר ע&#x202B;'<br>

תאמר לו אם הולך ג"ג ואם הולך לא תקח כלום ואם אינו הולך קח בעד כל א' שישאר ע&#x202B;'<br>

ומן החמישיות יש לך לקחת כ"א<br>

ומן החמישיות יש לך לקחת כ"א<br>

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::Example: I choose the number 4.

::Example: I choose the number 4.

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:Then tell him to subtract 4 from it.

:Then tell him to subtract 4 from it.

:This is your rule: take 1 for each 4 you subtract and this is the required.  

:This is your rule: take 1 for each 4 you subtract and this is the required.  

תאמר לו שיכפלהו<br>

תאמר לו שיכפלהו<br>

ואח"כ תאמר לו שיכפלהו פעם ב&#x202B;'<br>

ואח"כ תאמר לו שיכפלהו פעם ב&#x202B;'<br>

:Take 1 form every ten that remains and this is the required.

:Take 1 form every ten that remains and this is the required.

:<math>\scriptstyle5\sdot\left(2a+5\right)+10-35</math>

:<math>\scriptstyle5\sdot\left(2a+5\right)+10-35</math>

תאמר לו שיכפלהו ואח"כ יוסיף עליו ה&#x202B;'<br>

תאמר לו שיכפלהו ואח"כ יוסיף עליו ה&#x202B;'<br>

ואחר יכפול אותו ה' פעמי' ואחר יוסיף בו עשרה<br>

ואחר יכפול אותו ה' פעמי' ואחר יוסיף בו עשרה<br>

:This is the procedure to do it: I cast the Ishmaelites out by nines, I cast the lots, I throw to sea, through the strength of the waves I throw them to it by tricks.

:This is the procedure to do it: I cast the Ishmaelites out by nines, I cast the lots, I throw to sea, through the strength of the waves I throw them to it by tricks.

:Indeed, it is appropriate to start with the Jews and count them by initials.

:Indeed, it is appropriate to start with the Jews and count them by initials.

אמר יהודי אחד בואו כלכם בכאן והושיבם היהודים וההגרים באופן כי כשהיה מונה אותם לעולם היה כלה כל התשיעיות אל ההגרים והוכיח כי בעבורם בא הצער בים<br>

אמר יהודי אחד בואו כלכם בכאן והושיבם היהודים וההגרים באופן כי כשהיה מונה אותם לעולם היה כלה כל התשיעיות אל ההגרים והוכיח כי בעבורם בא הצער בים<br>

ובזה האופן השליך ט"ו הגרים בים והיהודים נשארו<br>

ובזה האופן השליך ט"ו הגרים בים והיהודים נשארו<br>

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=== <span style=color:Green>Guess - Stone</span> ===

=== <span style=color:Green>Guess - Stone</span> ===

:After the third reversal, ask him in which row it is.

:After the third reversal, ask him in which row it is.

:This is the rule in your hands: the third term is the one you are asking for.

:This is the rule in your hands: the third term is the one you are asking for.

קח י"ו אבנים ועשה ב' שורות מח' אבנים ותאמר לאיש שיגע אבן אחת מאלו השורות<br>

קח י"ו אבנים ועשה ב' שורות מח' אבנים ותאמר לאיש שיגע אבן אחת מאלו השורות<br>

אח"כ שאל לו באיזו שורה היא האבן ואם היא בשמאל תהפך כל השורות בשמאל ואם בימין כן תעשה<br>

אח"כ שאל לו באיזו שורה היא האבן ואם היא בשמאל תהפך כל השורות בשמאל ואם בימין כן תעשה<br>

אח"כ שאל באיזו שורה וכן תהפך ג' פעמים<br>

אח"כ שאל באיזו שורה וכן תהפך ג' פעמים<br>

ואחר ההפוך הג' תשאל לו באיזו שורה היא<br>

ואחר ההפוך הג' תשאל לו באיזו שורה היא<br>

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:After he sold them he looked at his hand and said: I did not do well, because if I had given 1 less for a dinar, I would have had 4 times more coins.

:After he sold them he looked at his hand and said: I did not do well, because if I had given 1 less for a dinar, I would have had 4 times more coins.

:Tell him how many eggs were sold and how much was the money earned from the eggs.

:Tell him how many eggs were sold and how much was the money earned from the eggs.

תאמ' לו כמה היו הביצות הנמכרות וכמה הוא הממון שיצאו מהביצות

תאמ' לו כמה היו הביצות הנמכרות וכמה הוא הממון שיצאו מהביצות

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::Answer: do as follows: since he said that he would have had 4 times more coins, multiply 4 by itself; the result is 16.

::Answer: do as follows: since he said that he would have had 4 times more coins, multiply 4 by itself; the result is 16.

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=== <span style=color:Green>Too Much and Too Little Problem - Money</span> ===

=== <span style=color:Green>Too Much and Too Little Problem - Money</span> ===

:But he knew that if he gave 12 to each person, they would have more than 52 and if he gave 16 to each person, he would have less than 60.

:But he knew that if he gave 12 to each person, they would have more than 52 and if he gave 16 to each person, he would have less than 60.

:I would like to know how much money there was and how many people there were?

:I would like to know how much money there was and how many people there were?

אבל ידע שאם נתן י"ב לכל איש יהיו יותר מה'ב' ואם יתן י"ו לאיש יחסר מס&#x202B;'<br>

אבל ידע שאם נתן י"ב לכל איש יהיו יותר מה'ב' ואם יתן י"ו לאיש יחסר מס&#x202B;'<br>

הנה היתי רוצה לדעת כמה ממון היה לזה וכמה אנשים היו{{#annotend:v4xY}}

הנה היתי רוצה לדעת כמה ממון היה לזה וכמה אנשים היו{{#annotend:v4xY}}

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::The answer: the missing money should be added to the surplus money, i.e. 52 with 60; the result is 112.

::The answer: the missing money should be added to the surplus money, i.e. 52 with 60; the result is 112.

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=== <span style=color:Green>Multiple Quantities Problem - Three Men - Money</span> ===

=== <span style=color:Green>Multiple Quantities Problem - Three Men - Money</span> ===

:The two without the third have 30.

:The two without the third have 30.

:We say: how much money does each have?

:We say: how much money does each have?

נאמר כמה ממון יש לכל א' וא&#x202B;'{{#annotend:5Y4D}}

נאמר כמה ממון יש לכל א' וא&#x202B;'{{#annotend:5Y4D}}

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:{{#annot:male and female|647|l10Y}}132) Question: Reuven reached his deathbed and made a will this way: from a hundred ducat that I would leave - if my wife, who is pregnant, gives birth to a male, she will have one-third of it and the rest will be for her son, and if she gives birth to a female, the third will be for her daughter and the rest for her.

:{{#annot:male and female|647|l10Y}}132) Question: Reuven reached his deathbed and made a will this way: from a hundred ducat that I would leave - if my wife, who is pregnant, gives birth to a male, she will have one-third of it and the rest will be for her son, and if she gives birth to a female, the third will be for her daughter and the rest for her.

:When he died, his wife gave birth to twins, a male and female.

:When he died, his wife gave birth to twins, a male and female.

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::Answer: it is clear that the son should have double the mother and the mother double the daughter.

::Answer: it is clear that the son should have double the mother and the mother double the daughter.

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::How much should each receive?

::How much should each receive?

|style="text-align:right;"|ותעשה ככה הנה בכאן ג' הא' שם א' והב' ב' והג' ד' והרויחו ק&#x202B;'<br>

|style="text-align:right;"|ותעשה ככה הנה בכאן ג' הא' שם א' והב' ב' והג' ד' והרויחו ק&#x202B;'<br>

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:{{#annot:7a/9=45|618|d4nh}}133) Question: find me a number such that if you multiply it by 7 and divide by 9, your result will be 45.

:{{#annot:7a/9=45|618|d4nh}}133) Question: find me a number such that if you multiply it by 7 and divide by 9, your result will be 45.

:<math>\scriptstyle \frac{a\sdot7}{9}=45</math>

:<math>\scriptstyle \frac{a\sdot7}{9}=45</math>

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:{{#annot:(5/a)·8=23|618|km4W}}134) Question: find a number such that if you divide it by 5 and multiply by 8 the result will be 23.

:{{#annot:(5/a)·8=23|618|km4W}}134) Question: find a number such that if you divide it by 5 and multiply by 8 the result will be 23.

:<math>\scriptstyle \frac{a}{5}\sdot8=23</math>

:<math>\scriptstyle \frac{a}{5}\sdot8=23</math>

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:{{#annot:a-⅓a-¼a=29|618|3Zd3}}135) Question: find me a number such that if you subtract its third and its quarter from it the remainder is 29.

:{{#annot:a-⅓a-¼a=29|618|3Zd3}}135) Question: find me a number such that if you subtract its third and its quarter from it the remainder is 29.

:<math>\scriptstyle a-\left(\frac{1}{3}a\right)-\left(\frac{1}{4}a\right)=29</math>

:<math>\scriptstyle a-\left(\frac{1}{3}a\right)-\left(\frac{1}{4}a\right)=29</math>

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:{{#annot:a+⅓a+⅖a=81|618|khzV}}136) Question: find me a number such that if you add to it its third and its two fifths it makes [81].

:{{#annot:a+⅓a+⅖a=81|618|khzV}}136) Question: find me a number such that if you add to it its third and its two fifths it makes [81].

:<math>\scriptstyle a+\left(\frac{1}{3}a\right)+\left(\frac{2}{5}a\right)=81</math>

:<math>\scriptstyle a+\left(\frac{1}{3}a\right)+\left(\frac{2}{5}a\right)=81</math>

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::The third of 15 is 5 and its two-fifths are 6. Add them to 15; the result is 26.

::The third of 15 is 5 and its two-fifths are 6. Add them to 15; the result is 26.

::<math>\scriptstyle{\color{blue}{15+\left(\frac{1}{3}\sdot15\right)+\left(\frac{2}{5}\sdot15\right)=15+5+6=26}}</math>

::<math>\scriptstyle{\color{blue}{15+\left(\frac{1}{3}\sdot15\right)+\left(\frac{2}{5}\sdot15\right)=15+5+6=26}}</math>

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=== <span style=color:Green>Divide a Number Problems</span> ===

=== <span style=color:Green>Divide a Number Problems</span> ===

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:{{#annot:a+b=30, 4a=3b|619|ahVb}}137) Question: if you want to divide 30 into two parts such that if you multiply the one by 4 it becomes the same as the other part if you multiply it by 3.

:{{#annot:a+b=30, 4a=3b|619|ahVb}}137) Question: if you want to divide 30 into two parts such that if you multiply the one by 4 it becomes the same as the other part if you multiply it by 3.

:<math>\scriptstyle\begin{cases}\scriptstyle a+b=30 \\\scriptstyle 4a=3b\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle a+b=30 \\\scriptstyle 4a=3b\end{cases}</math>

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:{{#annot:a+b+c=18, a-⅕a=b+⅓b=2½c|619|Slvf}}138) Question: if you want to produce three parts from 18 so that when you subtract a fifth from one part and add to the second [part] its third and multiply the third [part] by 2 and a half, all will be the same.

:{{#annot:a+b+c=18, a-⅕a=b+⅓b=2½c|619|Slvf}}138) Question: if you want to produce three parts from 18 so that when you subtract a fifth from one part and add to the second [part] its third and multiply the third [part] by 2 and a half, all will be the same.

:<math>\scriptstyle\begin{cases}\scriptstyle a+b+c=18\\\scriptstyle a-\frac{1}{5}a=b+\frac{1}{3}b=c\sdot\left(2+\frac{1}{2}\right)\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle a+b+c=18\\\scriptstyle a-\frac{1}{5}a=b+\frac{1}{3}b=c\sdot\left(2+\frac{1}{2}\right)\end{cases}</math>

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::Divide it by 9 and 3-fifths; the result is 5 and 5-eighths and this is the second part.

::Divide it by 9 and 3-fifths; the result is 5 and 5-eighths and this is the second part.

::<math>\scriptstyle{\color{blue}{\frac{3\sdot18}{9+\frac{3}{5}}=\frac{54}{9+\frac{3}{5}}=9+\frac{5}{8}}}</math>

::<math>\scriptstyle{\color{blue}{\frac{3\sdot18}{9+\frac{3}{5}}=\frac{54}{9+\frac{3}{5}}=9+\frac{5}{8}}}</math>

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::<math>\scriptstyle{\color{blue}{\left(5+\frac{5}{8}\right)+\frac{1}{3}\sdot\left(5+\frac{5}{8}\right)=7+\frac{1}{2}}}</math>

::<math>\scriptstyle{\color{blue}{\left(5+\frac{5}{8}\right)+\frac{1}{3}\sdot\left(5+\frac{5}{8}\right)=7+\frac{1}{2}}}</math>

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:{{#annot:a+b=20, b=4a|619|IrC7}}139) Question: if you want to produce two parts from 20 so that one part will be 4 times of the other part.

:{{#annot:a+b=20, b=4a|619|IrC7}}139) Question: if you want to produce two parts from 20 so that one part will be 4 times of the other part.

:<math>\scriptstyle\begin{cases}\scriptstyle a+b=20\\\scriptstyle b=4a\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle a+b=20\\\scriptstyle b=4a\end{cases}</math>

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:{{#annot:a+b=10, a/b=100|619|kZMG}}140) Question: if you want to produce two parts from 10 so that when you divide the greater part by the smaller the result will by a hundred.

:{{#annot:a+b=10, a/b=100|619|kZMG}}140) Question: if you want to produce two parts from 10 so that when you divide the greater part by the smaller the result will by a hundred.

:<math>\scriptstyle\begin{cases}\scriptstyle a+b=10\\\scriptstyle\frac{a}{b}=100\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle a+b=10\\\scriptstyle\frac{a}{b}=100\end{cases}</math>

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=== <span style=color:Green>Shared Work Problem - Cloth</span> ===

=== <span style=color:Green>Shared Work Problem - Cloth</span> ===

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:The first took it upon himself to do it in 3 weeks and the second in 4 weeks.

:The first took it upon himself to do it in 3 weeks and the second in 4 weeks.

:If they both do their work, how long will they do it?

:If they both do their work, how long will they do it?

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::<math>\scriptstyle{\color{blue}{\frac{12}{4}=3}}</math>   

::<math>\scriptstyle{\color{blue}{\frac{12}{4}=3}}</math>   

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::Together both do 7 tasks, but we want one task.

::Together both do 7 tasks, but we want one task.

::<math>\scriptstyle{\color{blue}{4+3=7}}</math>

::<math>\scriptstyle{\color{blue}{4+3=7}}</math>

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:{{#annot:a+b+c=30, 7a=b/5=c/10|619|AhlV}}142) Question: if you want to produce three parts from 30 so that when you multiply one [part] by 7 and divide the second [part] by 5 and the third [part] by 10 all the quotients and products will be the same amount.

:{{#annot:a+b+c=30, 7a=b/5=c/10|619|AhlV}}142) Question: if you want to produce three parts from 30 so that when you multiply one [part] by 7 and divide the second [part] by 5 and the third [part] by 10 all the quotients and products will be the same amount.

:<math>\scriptstyle\begin{cases}\scriptstyle a+b+c=30\\\scriptstyle7a=\frac{b}{5}=\frac{c}{10}\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle a+b+c=30\\\scriptstyle7a=\frac{b}{5}=\frac{c}{10}\end{cases}</math>

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::<span style=color:Green>'''Rule of Three:'''</span> You should say regarding to the first: if 1 results from [10]6, how much will result from 30?

::<span style=color:Green>'''Rule of Three:'''</span> You should say regarding to the first: if 1 results from [10]6, how much will result from 30?

::<math>\scriptstyle{\color{blue}{106:1=30:X}}</math>

::<math>\scriptstyle{\color{blue}{106:1=30:X}}</math>

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:{{#annot:a+b=11, ½a+¼a=⅓b+⅕b|619|3zr2}}143) Question: if you want to produce two parts from 11 such that a half and a quarter of one part will be the same as a third and a fifth of the other part.

:{{#annot:a+b=11, ½a+¼a=⅓b+⅕b|619|3zr2}}143) Question: if you want to produce two parts from 11 such that a half and a quarter of one part will be the same as a third and a fifth of the other part.

:<math>\scriptstyle\begin{cases}\scriptstyle a+b=11\\\scriptstyle\frac{1}{2}a+\frac{1}{4}a=\frac{1}{3}b+\frac{1}{5}b\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle a+b=11\\\scriptstyle\frac{1}{2}a+\frac{1}{4}a=\frac{1}{3}b+\frac{1}{5}b\end{cases}</math>

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:144) Question: if you wish to divide a number by a root or a root by a number.

:144) Question: if you wish to divide a number by a root or a root by a number.

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::<math>\scriptstyle{\color{blue}{12\div\sqrt{4}=\frac{\sqrt{144}}{\sqrt{4}}=\sqrt{36}}}</math>

::<math>\scriptstyle{\color{blue}{12\div\sqrt{4}=\frac{\sqrt{144}}{\sqrt{4}}=\sqrt{36}}}</math>

|style="text-align:right;"|ואח"כ תחלק שרש מקמ"ד עם שרש ד' ויבא לך שרש מל"ו

|style="text-align:right;"|ואח"כ תחלק שרש מקמ"ד עם שרש ד' ויבא לך שרש מל"ו

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:{{#annot:√9+√4|738|CvCx}}145) Question: if you want to add a root of 9 to [a root] of 4.

:{{#annot:√9+√4|738|CvCx}}145) Question: if you want to add a root of 9 to [a root] of 4.

:<math>\scriptstyle\sqrt{9}+\sqrt{4}</math>

:<math>\scriptstyle\sqrt{9}+\sqrt{4}</math>

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:146) Question: another method:

:146) Question: another method:

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=== <span style=color:Green>Subtraction of Roots</span> ===

=== <span style=color:Green>Subtraction of Roots</span> ===

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:{{#annot:√16-√9|739|Otry}}147) Question: if you want to subtract a root of 9 from a root of 16.

:{{#annot:√16-√9|739|Otry}}147) Question: if you want to subtract a root of 9 from a root of 16.

:<math>\scriptstyle\sqrt{16}-\sqrt{9}</math>

:<math>\scriptstyle\sqrt{16}-\sqrt{9}</math>

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:148) Question: another method:

:148) Question: another method:

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:{{#annot:a²+19=b²|618|DmeO}}149) Question: if you want to find a square number such that if you add to it 19 it will be a square.

:{{#annot:a²+19=b²|618|DmeO}}149) Question: if you want to find a square number such that if you add to it 19 it will be a square.

:<math>\scriptstyle a^2+19=b^2</math>

:<math>\scriptstyle a^2+19=b^2</math>

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::Do as follows: subtract 1 from 19; 18 remains.

::Do as follows: subtract 1 from 19; 18 remains.

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:{{#annot:a²-15=b²|618|eiGo}}150) Question: if you want to find a square number such that if you subtract 15 from it, it remains a square.

:{{#annot:a²-15=b²|618|eiGo}}150) Question: if you want to find a square number such that if you subtract 15 from it, it remains a square.

:<math>\scriptstyle a^2-15=b^2</math>

:<math>\scriptstyle a^2-15=b^2</math>

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:{{#annot:a+4=n², a-4=m²|618|bQqS}}151) Question: if you want to find a number such that if you add to it 4 it will be a square and if you subtract [from it 4] it will also be a square.

:{{#annot:a+4=n², a-4=m²|618|bQqS}}151) Question: if you want to find a number such that if you add to it 4 it will be a square and if you subtract [from it 4] it will also be a square.

:<math>\scriptstyle\begin{cases}\scriptstyle a+4=n^2\\\scriptstyle a-4=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle a+4=n^2\\\scriptstyle a-4=m^2\end{cases}</math>

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:152) Question: you can find it faster:

:152) Question: you can find it faster:

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:<math>\scriptstyle\sqrt{7}\sdot\sqrt{8}</math>

:<math>\scriptstyle\sqrt{7}\sdot\sqrt{8}</math>

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:{{#annot:√7×8|737|m2AG}}154) Question: if you want to multiply the root of 7 by 8.

:{{#annot:√7×8|737|m2AG}}154) Question: if you want to multiply the root of 7 by 8.

:<math>\scriptstyle\sqrt{7}\sdot8</math>

:<math>\scriptstyle\sqrt{7}\sdot8</math>

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:{{#annot:3/a=5+7/b|618|iyBa}}155) Question: if you want to find a number such that if you divide 3 by it, the result is 5 and 7 remain.

:{{#annot:3/a=5+7/b|618|iyBa}}155) Question: if you want to find a number such that if you divide 3 by it, the result is 5 and 7 remain.

:<math>\scriptstyle\frac{3}{a}=5+\frac{7}{b}</math>

:<math>\scriptstyle\frac{3}{a}=5+\frac{7}{b}</math>

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::Multiply it by 5; the result is 50.

::Multiply it by 5; the result is 50.

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:{{#annot:√a+√(a-√a)=3|618|e6ph}}156) Question: if you want to find a number such that if you subtract its root and add it to the root of the remainder it makes 3.

:{{#annot:√a+√(a-√a)=3|618|e6ph}}156) Question: if you want to find a number such that if you subtract its root and add it to the root of the remainder it makes 3.

:<math>\scriptstyle\sqrt{a}+\sqrt{a-\sqrt{a}}=3</math>

:<math>\scriptstyle\sqrt{a}+\sqrt{a-\sqrt{a}}=3</math>

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::It is as if you say: 9 times 9 is 81. Then, take the root of 9, which is 3, and multiply it by 81; the result is 243. So, the number 243 is a number whose [..] root is 9.

::It is as if you say: 9 times 9 is 81. Then, take the root of 9, which is 3, and multiply it by 81; the result is 243. So, the number 243 is a number whose [..] root is 9.

::<math>\scriptstyle{\color{blue}{9^2\sdot\sqrt{9}=81\sdot3=243}}</math>

::<math>\scriptstyle{\color{blue}{9^2\sdot\sqrt{9}=81\sdot3=243}}</math>

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:{{#annot:a³+b³+c³=27|619|Iw3S}}157) Question: if you want to produce three parts from 27 so that each of the parts has a cubic root<br>

:{{#annot:a³+b³+c³=27|619|Iw3S}}157) Question: if you want to produce three parts from 27 so that each of the parts has a cubic root<br>

:<math>\scriptstyle a^3+b^3+c^3=27</math>

:<math>\scriptstyle a^3+b^3+c^3=27</math>

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:{{#annot:a²-5a=n², a²+7a=m²|618|3YCC}}158) Question: if you want to find a square number such that if you subtract its five roots from it the remainder is a square and if you add it to its seven roots it will be a square.

:{{#annot:a²-5a=n², a²+7a=m²|618|3YCC}}158) Question: if you want to find a square number such that if you subtract its five roots from it the remainder is a square and if you add it to its seven roots it will be a square.

:<math>\scriptstyle\begin{cases}\scriptstyle a^2-5a=n^2\\\scriptstyle7a+a^2=m^2\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle a^2-5a=n^2\\\scriptstyle7a+a^2=m^2\end{cases}</math>

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::Because, its five roots are [45]. Subtract [it from 81]; 36 remains and it is a square number.

::Because, its five roots are [45]. Subtract [it from 81]; 36 remains and it is a square number.

::<math>\scriptstyle{\color{blue}{a^2-5a=81-{\color{red}{45}}=36}}</math>

::<math>\scriptstyle{\color{blue}{a^2-5a=81-{\color{red}{45}}=36}}</math>

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:{{#annot:⅖a²=a|618|LWXU}}159) Quaestion: if you want to find a number such that its two fifths are its root.

:{{#annot:⅖a²=a|618|LWXU}}159) Quaestion: if you want to find a number such that its two fifths are its root.

:<math>\scriptstyle\frac{2}{5}a^2=a</math>

:<math>\scriptstyle\frac{2}{5}a^2=a</math>

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:Their food costs 11 pešuṭim.

:Their food costs 11 pešuṭim.

:How much should each of them pay?

:How much should each of them pay?

הפרש אמר שרוצה לפרוע ב' פעמים יותר מהסוחר ויותר דינר א&#x202B;'<br>

הפרש אמר שרוצה לפרוע ב' פעמים יותר מהסוחר ויותר דינר א&#x202B;'<br>

והסוחר אמר שרוצה לפרוע ג' רביעיות יותר מהאורח ויותר ב' דינרין<br>

והסוחר אמר שרוצה לפרוע ג' רביעיות יותר מהאורח ויותר ב' דינרין<br>

כמה ראוי לפרוע לכל א' מהם{{#annotend:s6HV}}

כמה ראוי לפרוע לכל א' מהם{{#annotend:s6HV}}

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::Answer: take 1 for the guest that [pays] the least of all.

::Answer: take 1 for the guest that [pays] the least of all.

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::Then, take 1 for the poor man, 1 and 3-quarters for the merchant minus the two, and 3 and a half for the horseman, which is double the money of the merchant; the total is 6 and a quarter.

::Then, take 1 for the poor man, 1 and 3-quarters for the merchant minus the two, and 3 and a half for the horseman, which is double the money of the merchant; the total is 6 and a quarter.

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:161) Question: if you wish to divide a number into two parts, so that the sum of one part or more of the one plus one part or more of the other equals the first.

:161) Question: if you wish to divide a number into two parts, so that the sum of one part or more of the one plus one part or more of the other equals the first.

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:The second walks a mile on the first day, on the second 2, on the third 3 miles and so on every day he adds one mile.

:The second walks a mile on the first day, on the second 2, on the third 3 miles and so on every day he adds one mile.

:In how many days will he reach the first one, so that they both arrive together in one place?

:In how many days will he reach the first one, so that they both arrive together in one place?

א"כ בכמה ימים יגיע אל הראשון כדי שיגיעו שניהם יחד במקום אחד{{#annotend:cZiZ}}

א"כ בכמה ימים יגיע אל הראשון כדי שיגיעו שניהם יחד במקום אחד{{#annotend:cZiZ}}

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::The way: you should double the [number of] miles of the one who walks at a constant speed, i.e. the one who walks 30 miles; it is 60.

::The way: you should double the [number of] miles of the one who walks at a constant speed, i.e. the one who walks 30 miles; it is 60.

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:{{#annot:two men|657|33N2}}163) Question: a man walks 30 miles and the other 14 miles on the first day, and every day he adds a mile.

:{{#annot:two men|657|33N2}}163) Question: a man walks 30 miles and the other 14 miles on the first day, and every day he adds a mile.

:When will he catch up with the first one?

:When will he catch up with the first one?

מתי ישיג הראשון{{#annotend:33N2}}

מתי ישיג הראשון{{#annotend:33N2}}

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::It should be known that you should subtract the miles [that the second walks] on the first day from twice the miles [that the first walks] on the first day, i.e. we subtract 14 from 60; 46 remains, so he reaches him in 46 days.

::It should be known that you should subtract the miles [that the second walks] on the first day from twice the miles [that the first walks] on the first day, i.e. we subtract 14 from 60; 46 remains, so he reaches him in 46 days.

::<math>\scriptstyle{\color{blue}{\left(2\sdot30\right)-14=60-14=46}}</math>

::<math>\scriptstyle{\color{blue}{\left(2\sdot30\right)-14=60-14=46}}</math>

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:{{#annot:two men|657|8dlz}}164) Question: if you are told: on the first day he walks 2, on the second 4, on the third 6, and so on, every day he adds two miles?  

:{{#annot:two men|657|8dlz}}164) Question: if you are told: on the first day he walks 2, on the second 4, on the third 6, and so on, every day he adds two miles?  

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::1 mile should be subtract from the [number of miles of the one who walks at a constant speed] and the remainder is the number of days in which the one reaches the other.  

::1 mile should be subtract from the [number of miles of the one who walks at a constant speed] and the remainder is the number of days in which the one reaches the other.  

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=== <span style=color:Green>Encounter Problem - Two Couriers</span> ===

=== <span style=color:Green>Encounter Problem - Two Couriers</span> ===

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:Both of them will start moving together at the same time.

:Both of them will start moving together at the same time.

:Within how many days will they meet?

:Within how many days will they meet?

בכמה ימים יפגעו זה את זה{{#annotend:WiQX}}