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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::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}}

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

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:We know that a man subtracted double the sum of the two sides from the product of one height by the other.

:We know that a man subtracted double the sum of the two sides from the product of one height by the other.

:I ask how high is each of them?

:I ask how high is each of them?

והנה ידענו שאדם חסר מהכאת האחת בחבירו כפל מקובץ שני הצלעים<br>

והנה ידענו שאדם חסר מהכאת האחת בחבירו כפל מקובץ שני הצלעים<br>

אשאל כמה היה גובה כל א' מהם{{#annotend:kriv}}

אשאל כמה היה גובה כל א' מהם{{#annotend:kriv}}

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::The answer: suppose the remainder is 21, for instance, i.e. what remains from the product of the height of one tower by the other, when double the sum of the [sides of] the two towers is subtracted from it.   

::The answer: suppose the remainder is 21, for instance, i.e. what remains from the product of the height of one tower by the other, when double the sum of the [sides of] the two towers is subtracted from it.   

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:I ask how much money is in both purses [together]?

:I ask how much money is in both purses [together]?

:<math>\scriptstyle\frac{1}{2}X+\frac{1}{3}X+13=X</math>

:<math>\scriptstyle\frac{1}{2}X+\frac{1}{3}X+13=X</math>

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::The answer: take the common denominator, which is 6; its third and its half is 5, which is 1 less than 6, by which you divide.

::The answer: take the common denominator, which is 6; its third and its half is 5, which is 1 less than 6, by which you divide.

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:{{#annot:a(mod)3=a(mod5)=a(mod7)=1|618|lQns}}168) Question: find me a number, such that if you divide it by 3, by 5, and by 7, 1 remains.

:{{#annot:a(mod)3=a(mod5)=a(mod7)=1|618|lQns}}168) Question: find me a number, such that if you divide it by 3, by 5, and by 7, 1 remains.

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::The answer: take 70 for the 1 that remains above 3, when we divide it by 3; take 21 for the 1 that remains when we divide by 5; take 15 for the 1 that remains when we divide by 7. Sum them; it is [...] 106.

::The answer: take 70 for the 1 that remains above 3, when we divide it by 3; take 21 for the 1 that remains when we divide by 5; take 15 for the 1 that remains when we divide by 7. Sum them; it is [...] 106.

::<math>\scriptstyle{\color{blue}{70+21+15=106}}</math>

::<math>\scriptstyle{\color{blue}{70+21+15=106}}</math>

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:Do it this way:

:Do it this way:

:For instance, if it is said that when we divide it by 3, 2 remains; if its is divided by 5, 4 remains; and if it is divided by 7, 6 remains.

:For instance, if it is said that when we divide it by 3, 2 remains; if its is divided by 5, 4 remains; and if it is divided by 7, 6 remains.

עשה בדרך הזה<br>

עשה בדרך הזה<br>

עד"מ ואם אמר כשנחלקהו על ג' ישאר ב' ואם יחולק על ה' ישאר ד' ואם יחולק על ז' ישאר ו&#x202B;'{{#annotend:hx8d}}

עד"מ ואם אמר כשנחלקהו על ג' ישאר ב' ואם יחולק על ה' ישאר ד' ואם יחולק על ז' ישאר ו&#x202B;'{{#annotend:hx8d}}

:The total is 100 minyanim.

:The total is 100 minyanim.

:We ask how much is in each?

:We ask how much is in each?

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::The answer: we suppose that in the first purse there is 1 dinar, in the second 2, and in the third 3. The total is 6.

::The answer: we suppose that in the first purse there is 1 dinar, in the second 2, and in the third 3. The total is 6.

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:The second said: If you give me a third and a quarter of your money, with what I have, we will buy the horse.

:The second said: If you give me a third and a quarter of your money, with what I have, we will buy the horse.

:I ask: how much does each have and how much is the horse worth?

:I ask: how much does each have and how much is the horse worth?

אמר הראשון לשני אם תתן לי החצי ושליש ממעותיך עם מה שיש לי נקנה הסוס<br>

אמר הראשון לשני אם תתן לי החצי ושליש ממעותיך עם מה שיש לי נקנה הסוס<br>

אמר הב' אם תתן לי שלישית ורביעית ממעותיך עם מה שיש לי נקנה הסוס<br>

אמר הב' אם תתן לי שלישית ורביעית ממעותיך עם מה שיש לי נקנה הסוס<br>

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::Answer: look for the common denominator; it is 12.

::Answer: look for the common denominator; it is 12.

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::If you want to know the price of the horse, take a half and a third of 5; it is [4] and a sixth.

::If you want to know the price of the horse, take a half and a third of 5; it is [4] and a sixth.

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:Suppose we receive 128 and a quarter after adding to the square of the tree its four times and we wish to know the height of the tree.

:Suppose we receive 128 and a quarter after adding to the square of the tree its four times and we wish to know the height of the tree.

:<math>\scriptstyle4a+a^2=128+\frac{1}{4}</math>

:<math>\scriptstyle4a+a^2=128+\frac{1}{4}</math>

כמה גבהות האילן<br>

כמה גבהות האילן<br>

ונרצה לדעת גבהות האילן{{#annotend:57wS}}

ונרצה לדעת גבהות האילן{{#annotend:57wS}}

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:{{#annot:workers - house|2645|dD6m}}174) Question: a man built his house. He came to pay wages to the workers and said to them: I have an amount of money such that if I give 3 dinar to each one, I will have 15 dinar left and if I give 3 dinar and a half to each one, I will be short of ten dinar.

:{{#annot:workers - house|2645|dD6m}}174) Question: a man built his house. He came to pay wages to the workers and said to them: I have an amount of money such that if I give 3 dinar to each one, I will have 15 dinar left and if I give 3 dinar and a half to each one, I will be short of ten dinar.

:I ask: how many are the workers and how much is his money?

:I ask: how many are the workers and how much is his money?

אשאל כמה היו הפועלים וכמה הם מעותיו{{#annotend:dD6m}}

אשאל כמה היו הפועלים וכמה הם מעותיו{{#annotend:dD6m}}

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::The answer: subtract 3 dinar from 3 and a half; half a dinar remains, by which you divide.

::The answer: subtract 3 dinar from 3 and a half; half a dinar remains, by which you divide.

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:{{#annot:three men - money|652|WUTT}}175) three men have money, we do not know how much, only that the first has half the second, the second has half the third and together they all have 12.

:{{#annot:three men - money|652|WUTT}}175) three men have money, we do not know how much, only that the first has half the second, the second has half the third and together they all have 12.

:We ask: how much does each have?

:We ask: how much does each have?

נשאל כמה לכל א' וא&#x202B;'{{#annotend:WUTT}}

נשאל כמה לכל א' וא&#x202B;'{{#annotend:WUTT}}

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::The answer: the first has 1 pašuṭ, the second has 2 pešuṭim, and the third has 4.

::The answer: the first has 1 pašuṭ, the second has 2 pešuṭim, and the third has 4.

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:He worked for a known [number of] days, and finally the owner of the house gave him 1 dinar.

:He worked for a known [number of] days, and finally the owner of the house gave him 1 dinar.

:We would like to know how many days did he work.

:We would like to know how many days did he work.

נרצה לדעת כמה ימים עבד{{#annotend:JLEa}}

נרצה לדעת כמה ימים עבד{{#annotend:JLEa}}

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::So, he worked 19 days and 5 parts of 14 and he rested from his work 10 days and 9 parts of 14.

::So, he worked 19 days and 5 parts of 14 and he rested from his work 10 days and 9 parts of 14.

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:{{#annot:donation|627|FPlr}}177) Question: A man goes to trade.

:{{#annot:donation|627|FPlr}}177) Question: A man goes to trade.

:He vows that if God doubles his money every day, he will give 3 dinars to charity and within 3 days all his money is gone.

:He vows that if God doubles his money every day, he will give 3 dinars to charity and within 3 days all his money is gone.

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:At the end of 4 days, all his money is gone.

:At the end of 4 days, all his money is gone.

:We ask how much money did he have on the first day?

:We ask how much money did he have on the first day?

נשאל כמה ממון היה לו ביום הראשון{{#annotend:0zsJ}}

נשאל כמה ממון היה לו ביום הראשון{{#annotend:0zsJ}}

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:They pay 12 pešuṭim and they are 12 also.

:They pay 12 pešuṭim and they are 12 also.

:How many are the men, the women and the children?

:How many are the men, the women and the children?

האנשים נותנים ב' פשוטי' והנשים חצי פשוט והטף רביעית ופרעו י"ב פשוטים ובין כלם י"ב ג"כ<br>

האנשים נותנים ב' פשוטי' והנשים חצי פשוט והטף רביעית ופרעו י"ב פשוטים ובין כלם י"ב ג"כ<br>

כמה הם האנשים והנשים והטף{{#annotend:VPgv}}

כמה הם האנשים והנשים והטף{{#annotend:VPgv}}

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::Answer: assume the men are as many as you wish, and so are the women and the children.

::Answer: assume the men are as many as you wish, and so are the women and the children.

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::Hence, we should find a number that will raise [the number of coins] by one: we turn [2] of the 4 [women] into [1] man, so there are [5 men] and we turn [one woman] into 2 children; then the numbers are the same.

::Hence, we should find a number that will raise [the number of coins] by one: we turn [2] of the 4 [women] into [1] man, so there are [5 men] and we turn [one woman] into 2 children; then the numbers are the same.

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:After spending, there are in his hand 4 hundreds minyanim.

:After spending, there are in his hand 4 hundreds minyanim.

:We would like to know how many dinar he had when he left Naples.

:We would like to know how many dinar he had when he left Naples.

ואח"כ הולך מרומא עד פלורינצא וכל עוד שיש לו ב' דינרין ישובו ג' ופזר לשם ו' דינרין<br>

ואח"כ הולך מרומא עד פלורינצא וכל עוד שיש לו ב' דינרין ישובו ג' ופזר לשם ו' דינרין<br>

אח"כ הולך מפלורינציאה עד וינישיאה וכל עוד שיש לו דינר א' ישוב ג' ופזר ו' בויניציאה י"ד מנינים<br>

אח"כ הולך מפלורינציאה עד וינישיאה וכל עוד שיש לו דינר א' ישוב ג' ופזר ו' בויניציאה י"ד מנינים<br>

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=== <span style=color:Green>Buy and Sell Problem - Silk</span> ===

=== <span style=color:Green>Buy and Sell Problem - Silk</span> ===

:He spends on the way to Naples 2-thirds of the Florentine money.

:He spends on the way to Naples 2-thirds of the Florentine money.

:In what way will he sell in Naples, so that he will earn 25 minyanim for every hundred?

:In what way will he sell in Naples, so that he will earn 25 minyanim for every hundred?

באיזה אופן ימכור לנאפולי באופן שירויח כ"ה מנינים בעד כל מאה ומאה{{#annotend:7zn3}}

באיזה אופן ימכור לנאפולי באופן שירויח כ"ה מנינים בעד כל מאה ומאה{{#annotend:7zn3}}

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::You find it is worth 4 minyanim and 4 parts of 15. So, as long as the Florentine cane is worth 4 minyanim and a sixth of Florence, the cane of Naples is worth 4 minyanim and 4 parts of 15 of a minyan of Naples.  

::You find it is worth 4 minyanim and 4 parts of 15. So, as long as the Florentine cane is worth 4 minyanim and a sixth of Florence, the cane of Naples is worth 4 minyanim and 4 parts of 15 of a minyan of Naples.  

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:{{#annot:one kind|617|AZ2k}}182) Question: a man has 17 ՚oqya of silver, each ՚oqya is worth 7 carlini.

:{{#annot:one kind|617|AZ2k}}182) Question: a man has 17 ՚oqya of silver, each ՚oqya is worth 7 carlini.

:How much lead should be added to these 17 ՚oqya, so that each ՚oqya will be worth only 5 carlini and a half?

:How much lead should be added to these 17 ՚oqya, so that each ՚oqya will be worth only 5 carlini and a half?

לכלול בו עופרת באופן לא ישוה כל אוק' כי אם ה' קרליני וחצי כמה עופרת יכלול באלו הי"ז אוק&#x202B;'{{#annotend:AZ2k}}

לכלול בו עופרת באופן לא ישוה כל אוק' כי אם ה' קרליני וחצי כמה עופרת יכלול באלו הי"ז אוק&#x202B;'{{#annotend:AZ2k}}

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:{{#annot:silver|629|RsiE}}183) Question: a man has 3 liter, plus 3 ՚oqya and 3-quarters of silver containing 222 pure esterlini [sterling ?] per liter that are worth 8 minyanim.

:{{#annot:silver|629|RsiE}}183) Question: a man has 3 liter, plus 3 ՚oqya and 3-quarters of silver containing 222 pure esterlini [sterling ?] per liter that are worth 8 minyanim.

:How much are 1 liter plus 1 ՚oqya and 7-eighths of silver containing only 111 esterlini worth?

:How much are 1 liter plus 1 ՚oqya and 7-eighths of silver containing only 111 esterlini worth?

א' ליט' וא' אוק' וז' שמיניות אוק' מכסף שאינו מכיל הליט' כי אם קי"א איצטרליני כמה שוה{{#annotend:RsiE}}

א' ליט' וא' אוק' וז' שמיניות אוק' מכסף שאינו מכיל הליט' כי אם קי"א איצטרליני כמה שוה{{#annotend:RsiE}}

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:The master wants to pay them, but he has only 100 minyanim.

:The master wants to pay them, but he has only 100 minyanim.

:We would like to know how much will each be paid.

:We would like to know how much will each be paid.

ויש מהם שעבדו ו' חדשים ויש ו' וחצי ויש ז' חדשים וכו&#x202B;'<br>

ויש מהם שעבדו ו' חדשים ויש ו' וחצי ויש ז' חדשים וכו&#x202B;'<br>

הנה תמצא שכל מה שהרויחו בין כלם לפי יחס הזמן שעבדו עולה ק"ט מנינים וז' פשוטים<br>

הנה תמצא שכל מה שהרויחו בין כלם לפי יחס הזמן שעבדו עולה ק"ט מנינים וז' פשוטים<br>

נרצה לדעת כמה יקח כל א' וא&#x202B;'{{#annotend:zgAa}}

נרצה לדעת כמה יקח כל א' וא&#x202B;'{{#annotend:zgAa}}

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::100 should be divided by 109 minyanim and 7 pešuṭim.

::100 should be divided by 109 minyanim and 7 pešuṭim.

::Divide 1000 by 1091; you receive 1000 parts of 1091.

::Divide 1000 by 1091; you receive 1000 parts of 1091.

|style="text-align:right;"|הנה ראוי לחלק ק' על ק"ט מנינים וז' פשוטים<br>

|style="text-align:right;"|הנה ראוי לחלק ק' על ק"ט מנינים וז' פשוטים<br>

חלק אלף על אלף וצ"א ויצא לך אלף חלקים מאלף וצ"א

חלק אלף על אלף וצ"א ויצא לך אלף חלקים מאלף וצ"א

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:He did the so, and gave him all the money.

:He did the so, and gave him all the money.

:We would like to know how much money is required to be given to him in such a way that for every hundred of the profit he receives twenty.

:We would like to know how much money is required to be given to him in such a way that for every hundred of the profit he receives twenty.

ובכניסתו לבית נותן לו חצי השכר מאותה השנה ובסוף השנה נותן לו מ' מנינים עשרים בעד תשלום פרעון אותה שנה ועשרים בעד תשלום השנה האחרת וכן עד כלות הג' שנים<br>

ובכניסתו לבית נותן לו חצי השכר מאותה השנה ובסוף השנה נותן לו מ' מנינים עשרים בעד תשלום פרעון אותה שנה ועשרים בעד תשלום השנה האחרת וכן עד כלות הג' שנים<br>

והנה הבעל הבית אמר לאותו ששוכר הבית בבקשה ממך תפרע לי עתה מכל הג' שנים ואתן לך עשרים מריוח על כל מאה לשנה תמימה<br>

והנה הבעל הבית אמר לאותו ששוכר הבית בבקשה ממך תפרע לי עתה מכל הג' שנים ואתן לך עשרים מריוח על כל מאה לשנה תמימה<br>

::The answer: say: if a hundred and twenty is equal to 100, how much is 20 equal to? and this is the twenty that he is obliged to pay him at the end of the year.

::The answer: say: if a hundred and twenty is equal to 100, how much is 20 equal to? and this is the twenty that he is obliged to pay him at the end of the year.

::<math>\scriptstyle{\color{blue}{120:100=20:X}}</math>

::<math>\scriptstyle{\color{blue}{120:100=20:X}}</math>

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::If you want to weigh five [liter], put that of 3 liter and that of 1 [liter] on one scale. Then, put that of 9 [liter] on the second scale. Put the 5 you want to weigh on the first scale, where the two stones weighing 4 [liter] are. So, there are 9 on both the first and the second scales.

::If you want to weigh five [liter], put that of 3 liter and that of 1 [liter] on one scale. Then, put that of 9 [liter] on the second scale. Put the 5 you want to weigh on the first scale, where the two stones weighing 4 [liter] are. So, there are 9 on both the first and the second scales.

אח"כ תשים בכף שניה אותה של ט&#x202B;'<br>

אח"כ תשים בכף שניה אותה של ט&#x202B;'<br>

אח"כ תשים בכף ראשנה ששם הב' אבנים ששוקלות ד' ושים ה' שתרצה לשקול ויהיו ט' בכף ראשנה גם בשניה

אח"כ תשים בכף ראשנה ששם הב' אבנים ששוקלות ד' ושים ה' שתרצה לשקול ויהיו ט' בכף ראשנה גם בשניה

:They agreed that he will pay the host an ՚oqya of silver every day.  

:They agreed that he will pay the host an ՚oqya of silver every day.  

:In what way will he pay him every night and night, so that he does not give away the goblets and he pays him a whole ՚oqya each day.

:In what way will he pay him every night and night, so that he does not give away the goblets and he pays him a whole ՚oqya each day.

באיזה אופן יפרענו בכל לילה ולילה שלא יתן הגביעים ויתן לו אוק' שלימה בכל יום{{#annotend:IY0T}}

באיזה אופן יפרענו בכל לילה ולילה שלא יתן הגביעים ויתן לו אוק' שלימה בכל יום{{#annotend:IY0T}}

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:When you wash the third one, the kikkar becomes 99 liter of clean wool.

:When you wash the third one, the kikkar becomes 99 liter of clean wool.

:We would like to know which of the three is better to buy.

:We would like to know which of the three is better to buy.

הראשנה מוכרה ככר שהם ק' ליט' עשרה מנינים<br>

הראשנה מוכרה ככר שהם ק' ליט' עשרה מנינים<br>

והב' מוכר הככר עשרה מנינים וחצי<br>

והב' מוכר הככר עשרה מנינים וחצי<br>

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::You receive one eighth and 4-sevenths of an eighth <span style=color:red>[this calculation is wrong - the result should be less than one eighth]</span>; so this is one is better than the first.

::You receive one eighth and 4-sevenths of an eighth <span style=color:red>[this calculation is wrong - the result should be less than one eighth]</span>; so this is one is better than the first.

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:Now, the lord, who pays, has only 10 ՚oqya and 5 tarini, and from this money he wants to pay the harvesters according to their work.

:Now, the lord, who pays, has only 10 ՚oqya and 5 tarini, and from this money he wants to pay the harvesters according to their work.

:How much should he pay each?

:How much should he pay each?

אחר שעבד שנה רצופה בצימצום ושואל אוק' וב' טריני<br>

אחר שעבד שנה רצופה בצימצום ושואל אוק' וב' טריני<br>

והב' עבד יותר מעט עד ששואל אוק' א' וג' טריני<br>

והב' עבד יותר מעט עד ששואל אוק' א' וג' טריני<br>

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:2 canna of the cloth are worth 3 barrels of wine.

:2 canna of the cloth are worth 3 barrels of wine.

:2 barrels of wine are worth 8 dinars.

:2 barrels of wine are worth 8 dinars.

:We wish to know how much 4 liter of silver are worth according to this ratio and how much 5 canna of the cloth are worth.

:We wish to know how much 4 liter of silver are worth according to this ratio and how much 5 canna of the cloth are worth.

וב' קנים מהבגד שוה ג' חביות של יין<br>

וב' קנים מהבגד שוה ג' חביות של יין<br>

וב' חביות מיין שוה ח' דינרין<br>

וב' חביות מיין שוה ח' דינרין<br>

::The answer: you know that 2 barrels are worth 8 dinar, so say: if 2 is equal to 8, how much is 3 equal to?

::The answer: you know that 2 barrels are worth 8 dinar, so say: if 2 is equal to 8, how much is 3 equal to?

::<math>\scriptstyle{\color{blue}{2:8=3:X}}</math>

::<math>\scriptstyle{\color{blue}{2:8=3:X}}</math>

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::You receive 12. Hence, three barrels of [wine] are worth 12 dinar.

::You receive 12. Hence, three barrels of [wine] are worth 12 dinar.

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:Then, he had 300 measures in his hand.

:Then, he had 300 measures in his hand.

:I ask: how many measurements he carried [for each equally]?

:I ask: how many measurements he carried [for each equally]?

בא שני ואמר לו אם תשא את שלי אתן לך השליש<br>

בא שני ואמר לו אם תשא את שלי אתן לך השליש<br>

בא אחר ואמר אם תשא את שלי אתן לך רביעית<br>

בא אחר ואמר אם תשא את שלי אתן לך רביעית<br>

ואח"כ נמצא בידו ש' מדות<br>

ואח"כ נמצא בידו ש' מדות<br>

אשאל כמה מדות נשא{{#annotend:8uum}}

אשאל כמה מדות נשא{{#annotend:8uum}}

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::The answer: look for a number that has a half, a third, a quarter, and a fifth; you find it is 60.

::The answer: look for a number that has a half, a third, a quarter, and a fifth; you find it is 60.

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:He wants to melt all of them and produce goblets from them, such that each liter contains 9 ՚oqya of pure silver.

:He wants to melt all of them and produce goblets from them, such that each liter contains 9 ՚oqya of pure silver.

:We wish to know how many liter will he take of the silver that contains 11 ՚oqya per a liter.

:We wish to know how many liter will he take of the silver that contains 11 ՚oqya per a liter.

עוד יש לו ה' ליט' מכסף שיש לו בכל ליט' ו' אוק' מכסף טוב<br>

עוד יש לו ה' ליט' מכסף שיש לו בכל ליט' ו' אוק' מכסף טוב<br>

ועוד יש לו ליט' בלתי נודעות שבכל ליט' יש כסף י"א אוק&#x202B;'<br>

ועוד יש לו ליט' בלתי נודעות שבכל ליט' יש כסף י"א אוק&#x202B;'<br>

::Answer: you should know how many ՚oqya are included in the first number, which is 6 liter that contain 7 ՚oqya per liter; they are 42 and this is the number of pure ՚oqya.

::Answer: you should know how many ՚oqya are included in the first number, which is 6 liter that contain 7 ՚oqya per liter; they are 42 and this is the number of pure ՚oqya.

::<math>\scriptstyle{\color{blue}{6\sdot7=42}}</math>

::<math>\scriptstyle{\color{blue}{6\sdot7=42}}</math>

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::<math>\scriptstyle{\color{blue}{\frac{\left(9\sdot11\right)-72}{11-9}=\frac{99-72}{2}=\frac{27}{2}=13+\frac{1}{2}}}</math>

::<math>\scriptstyle{\color{blue}{\frac{\left(9\sdot11\right)-72}{11-9}=\frac{99-72}{2}=\frac{27}{2}=13+\frac{1}{2}}}</math>

|style="text-align:right;"|וחלק אלו הכ"ז על היתרון שיש בין ט' לי"א ותמצא ב&#x202B;'<br>

|style="text-align:right;"|וחלק אלו הכ"ז על היתרון שיש בין ט' לי"א ותמצא ב&#x202B;'<br>

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:One asks for half the amount plus 2, the other asks for a third of the amount plus 3, and another asks for a quarter of the amount plus 4.

:One asks for half the amount plus 2, the other asks for a third of the amount plus 3, and another asks for a quarter of the amount plus 4.

:We ask: how much will each take?

:We ask: how much will each take?

הא' שואל חצי הממון עם תוספת ב' והאחר שואל שלישית הממון עם תוספת ג' והאחר שואל רביעית הממון עם תוספת ד&#x202B;'<br>

הא' שואל חצי הממון עם תוספת ב' והאחר שואל שלישית הממון עם תוספת ג' והאחר שואל רביעית הממון עם תוספת ד&#x202B;'<br>

נשאל כמה יקח כל א' וא&#x202B;'{{#annotend:jm00}}

נשאל כמה יקח כל א' וא&#x202B;'{{#annotend:jm00}}

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::The answer: all the additions they ask for, which are 2, 3, and 4, should be summed; it is 9.

::The answer: all the additions they ask for, which are 2, 3, and 4, should be summed; it is 9.

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::Divide it by [1]3; the result is 9 parts of 13.

::Divide it by [1]3; the result is 9 parts of 13.

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:One asks for half the amount minus 3, the other asks for a quarter plus 4, and another asks for a fifth plus 5.

:One asks for half the amount minus 3, the other asks for a quarter plus 4, and another asks for a fifth plus 5.

:We wish to know how much will each take.

:We wish to know how much will each take.

הא' שואל חצי הממון פחות ג' והאחר שואל הרביעית עם תוספת ד' והאחר שואל החומש עם תוספת ה&#x202B;'<br>

הא' שואל חצי הממון פחות ג' והאחר שואל הרביעית עם תוספת ד' והאחר שואל החומש עם תוספת ה&#x202B;'<br>

נרצה לדעת כמה יקח כל א' וא&#x202B;'{{#annotend:9mWf}}

נרצה לדעת כמה יקח כל א' וא&#x202B;'{{#annotend:9mWf}}

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::The answer: first, sum the additions that two of them ask for; it is 9.

::The answer: first, sum the additions that two of them ask for; it is 9.

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::Take a fifth of the number, multiply it by 10, then divide the product by 15 and a fifth; you receive 2 integers and 8 parts of 76; and since he said "plus 5", he takes 7 integers and 8 parts of 76.

::Take a fifth of the number, multiply it by 10, then divide the product by 15 and a fifth; you receive 2 integers and 8 parts of 76; and since he said "plus 5", he takes 7 integers and 8 parts of 76.

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:{{#annot:two men sharing money|645|N0LR}}194) Question: There are two here who want to divide twenty minyanim.

:{{#annot:two men sharing money|645|N0LR}}194) Question: There are two here who want to divide twenty minyanim.

:One asks for a third minus 4 and the other asks for a fifth minus 3.

:One asks for a third minus 4 and the other asks for a fifth minus 3.

הא' שואל השליש פחות ד' והאחר שואל החומש פחות ג&#x202B;'{{#annotend:N0LR}}

הא' שואל השליש פחות ד' והאחר שואל החומש פחות ג&#x202B;'{{#annotend:N0LR}}

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:We find each has 100 minyanim in his hand.

:We find each has 100 minyanim in his hand.

:How much did each has when they started playing?

:How much did each has when they started playing?

הב' משימים כל ממונם נגד הג' והג' המצחק הפסיד ופרע לכל א' סך כל ממונם נגדו והוא הפסיד ופרע לכל א' ממונו<br>

הב' משימים כל ממונם נגד הג' והג' המצחק הפסיד ופרע לכל א' סך כל ממונם נגדו והוא הפסיד ופרע לכל א' ממונו<br>

עתה מצחק הג' והב' הנשארים שמו כל ממונם והוא הפסיד ופרע לכל א' וא&#x202B;'<br>

עתה מצחק הג' והב' הנשארים שמו כל ממונם והוא הפסיד ופרע לכל א' וא&#x202B;'<br>

::Answer: take the number of the people, which is 3, add 1 to it, for the first duplication; it is 4.

::Answer: take the number of the people, which is 3, add 1 to it, for the first duplication; it is 4.

::<math>\scriptstyle{\color{blue}{3+1=4}}</math>

::<math>\scriptstyle{\color{blue}{3+1=4}}</math>

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:{{#annot:barrel|625|xzby}}196) Question: There is a barrel here that has 3 openings. From the first opening, all the wine comes out in 4 hours. From the second opening, without the first, all the wine comes out in 3 hours. From the third opening, without the others, all the wine comes out in 2 hours. If we open all three of them, how long will it take for all the wine to come out?

:{{#annot:barrel|625|xzby}}196) Question: There is a barrel here that has 3 openings. From the first opening, all the wine comes out in 4 hours. From the second opening, without the first, all the wine comes out in 3 hours. From the third opening, without the others, all the wine comes out in 2 hours. If we open all three of them, how long will it take for all the wine to come out?

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::The answer can be done in two ways:

::The answer can be done in two ways:

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::You find that 5 log and [7] parts of 13 come out.

::You find that 5 log and [7] parts of 13 come out.

::<math>\scriptstyle{\color{blue}{X=5+\frac{7}{13}}}</math>

::<math>\scriptstyle{\color{blue}{X=5+\frac{7}{13}}}</math>

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=== <span style=color:Green>Compound Barter Problem - Cloth and Wool</span> ===

=== <span style=color:Green>Compound Barter Problem - Cloth and Wool</span> ===

:The woolen cloth is worth [7] minyanim for a cloth.

:The woolen cloth is worth [7] minyanim for a cloth.

:We wish to know by how much will the price of wool increase, as the price of cloth increases?

:We wish to know by how much will the price of wool increase, as the price of cloth increases?

מי שמוכר הבגד מוכר אותה הקנה שוה ג' מנינים במעות מנויים ומצד שהוא רוצה להחליף העלה אותו מנין א' לקנה ושם אותו ה' מנינים<br>

מי שמוכר הבגד מוכר אותה הקנה שוה ג' מנינים במעות מנויים ומצד שהוא רוצה להחליף העלה אותו מנין א' לקנה ושם אותו ה' מנינים<br>

ומזה רוצה שינתן לו ב' חמישיות במעות מנויים והב' חמישיות רוצה שיתן לו האחר מצמר<br>

ומזה רוצה שינתן לו ב' חמישיות במעות מנויים והב' חמישיות רוצה שיתן לו האחר מצמר<br>

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::<span style=color:Green>'''Check:'''</span> if you want to check it: we suppose he bought 2 cane that were worth 80 minyanim in cash and now are worth 100 in their barter.

::<span style=color:Green>'''Check:'''</span> if you want to check it: we suppose he bought 2 cane that were worth 80 minyanim in cash and now are worth 100 in their barter.

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:Three of them, without the second, contributed 900 minyanim.

:Three of them, without the second, contributed 900 minyanim.

:We wish to know how much did each contribute?

:We wish to know how much did each contribute?

הג' בלתי הרביעי שמו ו' מאות מנינים<br>

הג' בלתי הרביעי שמו ו' מאות מנינים<br>

והג' האחרים בלתי הג' שמו ז' מאות<br>

והג' האחרים בלתי הג' שמו ז' מאות<br>

:Three of them, without the fourth, contributed 24 [minyanim].

:Three of them, without the fourth, contributed 24 [minyanim].

:We wish to know how much did each contribute?

:We wish to know how much did each contribute?

הג' בלתי הראשון שמו י"ח<br>

הג' בלתי הראשון שמו י"ח<br>

והג' מהם בלתי האחד שמו כ' מנינים<br>

והג' מהם בלתי האחד שמו כ' מנינים<br>

:{{#annot:five partners|661|oLL3}}197)[L] Question: there are five - four of them contributed 22, four of them contributed 24, four of them contributed 26, four of them contributed 28, and four of them contributed 30.

:{{#annot:five partners|661|oLL3}}197)[L] Question: there are five - four of them contributed 22, four of them contributed 24, four of them contributed 26, four of them contributed 28, and four of them contributed 30.

:How much did each have?

:How much did each have?

כמה לכל אחד ואחד{{#annotend:oLL3}}

כמה לכל אחד ואחד{{#annotend:oLL3}}

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::<math>\scriptstyle{\color{blue}{a_1=\left(32+\frac{1}{2}\right)-30=2+\frac{1}{2}}}</math>

::<math>\scriptstyle{\color{blue}{a_1=\left(32+\frac{1}{2}\right)-30=2+\frac{1}{2}}}</math>

|style="text-align:right;"|תסיר מהם ל' וישארו ב' וחצי הוא ממון הראשון

|style="text-align:right;"|תסיר מהם ל' וישארו ב' וחצי הוא ממון הראשון

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:A man comes and says to him: sell to me of the two kinds of wheat for a price of 14 minyanim and I will buy 20 measures.

:A man comes and says to him: sell to me of the two kinds of wheat for a price of 14 minyanim and I will buy 20 measures.

:How much will he sell to him from each [kind of] wheat, so that the wheat will be worth 14 minyanim?

:How much will he sell to him from each [kind of] wheat, so that the wheat will be worth 14 minyanim?

כמה יתן לו מכל חטה וחטה באופן שיבא י"ד מנינים החטה{{#annotend:LZP3}}

כמה יתן לו מכל חטה וחטה באופן שיבא י"ד מנינים החטה{{#annotend:LZP3}}

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::The answer: do as follows: see how much is the difference between 14 and 10; you find it 4. Place it above 14 that is its counter.

::The answer: do as follows: see how much is the difference between 14 and 10; you find it 4. Place it above 14 that is its counter.

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::<math>\scriptstyle{\color{blue}{14\sdot20=280}}</math>

::<math>\scriptstyle{\color{blue}{14\sdot20=280}}</math>

|style="text-align:right;"|ואם תרצה לבחון אותו ידוע הוא כי הוא שאל כ' מדות לערך י"ד מנינים במדה<br>

|style="text-align:right;"|ואם תרצה לבחון אותו ידוע הוא כי הוא שאל כ' מדות לערך י"ד מנינים במדה<br>

הנה א"כ הכ' מדות שוות ר"פ מנינים

הנה א"כ הכ' מדות שוות ר"פ מנינים

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::Then, go back to the wheat that is worth 10 minyanim. You say that [he] should buy 6 measures and 2-thirds of [this kind]. Multiply 6 and 2-thirds by ten; you receive [6]6 and 2-thirds.

::Then, go back to the wheat that is worth 10 minyanim. You say that [he] should buy 6 measures and 2-thirds of [this kind]. Multiply 6 and 2-thirds by ten; you receive [6]6 and 2-thirds.

::<math>\scriptstyle{\color{blue}{10\sdot\left(6+\frac{2}{3}\right)=66+\frac{2}{3}}}</math>

::<math>\scriptstyle{\color{blue}{10\sdot\left(6+\frac{2}{3}\right)=66+\frac{2}{3}}}</math>

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::<math>\scriptstyle{\color{blue}{\left(66+\frac{2}{3}\right)+\left(213+\frac{1}{3}\right)=280}}</math>

::<math>\scriptstyle{\color{blue}{\left(66+\frac{2}{3}\right)+\left(213+\frac{1}{3}\right)=280}}</math>

|style="text-align:right;"|חבר ס"ו וב' שלישיות אל רי"ג ושליש ויצא לך ר"פ שלמים והוא שוה אל הר"פ השמורים

|style="text-align:right;"|חבר ס"ו וב' שלישיות אל רי"ג ושליש ויצא לך ר"פ שלמים והוא שוה אל הר"פ השמורים

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:{{#annot:five coins|642|cQoZ}}200) Question: there is one here who has one magen and he wants to be given carlini, grani, tornesi, dinar, and cavalli for this magen equally, so that he will take of the one the same as of the others.

:{{#annot:five coins|642|cQoZ}}200) Question: there is one here who has one magen and he wants to be given carlini, grani, tornesi, dinar, and cavalli for this magen equally, so that he will take of the one the same as of the others.

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:{{#annot:herdsman and bulls|612|SD1R}}201) Question: there is someone here who has bulls and he told the herdsman that if he will graze his bulls 5 years, he will give him half of all the profit and also half the bulls after paying for the 5 years, but the herdsman died at the end of 31 and a half months and the bulls are 3 hundred and 50.

:{{#annot:herdsman and bulls|612|SD1R}}201) Question: there is someone here who has bulls and he told the herdsman that if he will graze his bulls 5 years, he will give him half of all the profit and also half the bulls after paying for the 5 years, but the herdsman died at the end of 31 and a half months and the bulls are 3 hundred and 50.

:We want to know how much the herdsman should have been paid, given that if he had not died by the end of the 5 years he would have taken half the bulls.

:We want to know how much the herdsman should have been paid, given that if he had not died by the end of the 5 years he would have taken half the bulls.

נרצה לדעת כמה יקח הרועה שאלו לא מת עד סוף ה' שנים לקח חצי השורים{{#annotend:SD1R}}

נרצה לדעת כמה יקח הרועה שאלו לא מת עד סוף ה' שנים לקח חצי השורים{{#annotend:SD1R}}

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:He tells them that everything they loot would be theirs and they agree that each horse rider will take 5 minyanim and the infantryman 3 minyanim and the loot is a thousand minyanim.

:He tells them that everything they loot would be theirs and they agree that each horse rider will take 5 minyanim and the infantryman 3 minyanim and the loot is a thousand minyanim.

:We wish to know how much will the horse riders take and how much will the infantrymen take?

:We wish to know how much will the horse riders take and how much will the infantrymen take?

אמר להם כל מה שיבוזו יהיה שלהם והתנו ביניהם כי רוכב הסוס יקח כל א' ה' מנינים והרגליים ג' מנינים והשלל הוא אלף מנינים<br>

אמר להם כל מה שיבוזו יהיה שלהם והתנו ביניהם כי רוכב הסוס יקח כל א' ה' מנינים והרגליים ג' מנינים והשלל הוא אלף מנינים<br>

נרצה לדעת כמה יקחו רוכבי על סוסים וכמה יקחו הרוגליים{{#annotend:0n0a}}

נרצה לדעת כמה יקחו רוכבי על סוסים וכמה יקחו הרוגליים{{#annotend:0n0a}}

::It should be done as follows: multiply the [number of] minyanim the infantrymen should take by 4 hundred, because they are 4 hundred. Multiply it by 3; you receive 12 hundred.

::It should be done as follows: multiply the [number of] minyanim the infantrymen should take by 4 hundred, because they are 4 hundred. Multiply it by 3; you receive 12 hundred.

::<math>\scriptstyle{\color{blue}{3\sdot400=1200}}</math>

::<math>\scriptstyle{\color{blue}{3\sdot400=1200}}</math>

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:One comes with 100 [measures]. He wants to grind them quickly and he wants that the measures will be divided among the three grinders, so that it will be done at once.

:One comes with 100 [measures]. He wants to grind them quickly and he wants that the measures will be divided among the three grinders, so that it will be done at once.

:How many measures should we assign to each?

:How many measures should we assign to each?

כמה מדות נשים בכל א' וא&#x202B;'{{#annotend:r6u8}}

כמה מדות נשים בכל א' וא&#x202B;'{{#annotend:r6u8}}

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:She asked that he would pay for them and said: when I counted them by two - 1 was left; by three - 1; by four - 1; by five - 1; by six - 1; and by 7 - [all gone].

:She asked that he would pay for them and said: when I counted them by two - 1 was left; by three - 1; by four - 1; by five - 1; by six - 1; and by 7 - [all gone].

:How many were they?

:How many were they?

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:He asked that he would pay for them and said: when I counted them by two - 1 was left; by three - 2; by five - 4; by six - 5; by seven - [all gone].

:He asked that he would pay for them and said: when I counted them by two - 1 was left; by three - 2; by five - 4; by six - 5; by seven - [all gone].

:How many were they?

:How many were they?

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:He wants to melt the two kinds of silver, and mixture them, so that each liter will contain 9 ՚oqya of pure silver.

:He wants to melt the two kinds of silver, and mixture them, so that each liter will contain 9 ՚oqya of pure silver.

:We ask: how much should he take of each?

:We ask: how much should he take of each?

נשאל כמה יקח מכל א' וא&#x202B;'{{#annotend:qdWs}}

נשאל כמה יקח מכל א' וא&#x202B;'{{#annotend:qdWs}}

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:{{#annot:three kinds|617|KpV9}}207) Question: a man has three kinds of silver, one is 5 ՚oqya of pure [silver] per liter, another is 7 [՚oqya per liter], and another is 11 [՚oqya per liter].

:{{#annot:three kinds|617|KpV9}}207) Question: a man has three kinds of silver, one is 5 ՚oqya of pure [silver] per liter, another is 7 [՚oqya per liter], and another is 11 [՚oqya per liter].

:He wants to mixture the three, so that each liter will contain 8 ՚oqya [of pure silver].

:He wants to mixture the three, so that each liter will contain 8 ՚oqya [of pure silver].

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:{{#annot:three kinds|617|SbRx}}208) Question: if you have 3 liter of silver that contain 7 ՚oqya of pure [silver] in each liter, and you wish to mixture them, so that each liter will contain 9 ՚oqya.  

:{{#annot:three kinds|617|SbRx}}208) Question: if you have 3 liter of silver that contain 7 ՚oqya of pure [silver] in each liter, and you wish to mixture them, so that each liter will contain 9 ՚oqya.  

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:{{#annot:one kind|617|wP40}}209) Question: a man has 6 liter of silver that contain 8 [՚oqya] of pure [silver in each liter], and we wish to mixture them, so that each liter will contain only 4 ՚oqya of pure [silver].

:{{#annot:one kind|617|wP40}}209) Question: a man has 6 liter of silver that contain 8 [՚oqya] of pure [silver in each liter], and we wish to mixture them, so that each liter will contain only 4 ՚oqya of pure [silver].

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:{{#annot:group of people|649|ab3M}}210) 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.

:{{#annot:group of people|649|ab3M}}210) 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.

:<math>\scriptstyle X+X+\frac{1}{2}X+\frac{1}{4}X+1=100</math>

:<math>\scriptstyle X+X+\frac{1}{2}X+\frac{1}{4}X+1=100</math>

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:We want to know: did he earn or lose?

:We want to know: did he earn or lose?

:<math>\scriptstyle\left(\frac{50}{1+\frac{1}{4}}+\frac{50}{\frac{3}{4}}\right)-100</math>

:<math>\scriptstyle\left(\frac{50}{1+\frac{1}{4}}+\frac{50}{\frac{3}{4}}\right)-100</math>

נשאל אם הרויח או הפסיד{{#annotend:nbpd}}

נשאל אם הרויח או הפסיד{{#annotend:nbpd}}

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:How much money did he have originally?

:How much money did he have originally?

:<math>\scriptstyle\frac{\frac{3}{5}X}{\frac{4}{7}}=X+1</math>

:<math>\scriptstyle\frac{\frac{3}{5}X}{\frac{4}{7}}=X+1</math>

כמה ממונו{{#annotend:eErQ}}

כמה ממונו{{#annotend:eErQ}}

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:{{#annot:WP|641|8u66}}213) Question: a man bought four sevenths of a liṭra for one pašuṭ, then he sold five ninths of a liṭra for one pašuṭ and he earned one pašuṭ.

:{{#annot:WP|641|8u66}}213) Question: a man bought four sevenths of a liṭra for one pašuṭ, then he sold five ninths of a liṭra for one pašuṭ and he earned one pašuṭ.

:<math>\scriptstyle\frac{\frac{4}{7}X}{\frac{5}{9}}=X+1</math>

:<math>\scriptstyle\frac{\frac{4}{7}X}{\frac{5}{9}}=X+1</math>

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: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.

: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.

:<math>\scriptstyle\frac{1}{3}X+\frac{1}{4}X+\frac{1}{6}X=1</math>

:<math>\scriptstyle\frac{1}{3}X+\frac{1}{4}X+\frac{1}{6}X=1</math>

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:How much should he be paid?  

:How much should he be paid?  

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:When will he catch up with him?

:When will he catch up with him?

:<math>\scriptstyle29X=37\sdot\left(X-10\right)</math>

:<math>\scriptstyle29X=37\sdot\left(X-10\right)</math>

מתי ישיגנו{{#annotend:GEab}}

מתי ישיגנו{{#annotend:GEab}}

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

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

:<math>\scriptstyle\frac{8}{\frac{1}{3}\sdot10}=\frac{8-2}{X}</math>

:<math>\scriptstyle\frac{8}{\frac{1}{3}\sdot10}=\frac{8-2}{X}</math>

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:How much should they be boiled?

:How much should they be boiled?

:<math>\scriptstyle\frac{8}{\frac{1}{3}\sdot10}=\frac{8-2}{X}</math>

:<math>\scriptstyle\frac{8}{\frac{1}{3}\sdot10}=\frac{8-2}{X}</math>

כמה ראוי לבשל{{#annotend:1PBL}}

כמה ראוי לבשל{{#annotend:1PBL}}

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::<span style=color:Green>'''Rule of Three:'''</span> Multiply 2 by 3; the result is 6. Divide it by 6; the result is 1 and this is the required.

::<span style=color:Green>'''Rule of Three:'''</span> Multiply 2 by 3; the result is 6. Divide it by 6; the result is 1 and this is the required.

::<math>\scriptstyle{\color{blue}{x=\frac{\left(\frac{1}{3}\sdot9\right)\sdot\left(6-4\right)}{6}=\frac{3\sdot2}{6}=\frac{6}{6}=1}}</math>

::<math>\scriptstyle{\color{blue}{x=\frac{\left(\frac{1}{3}\sdot9\right)\sdot\left(6-4\right)}{6}=\frac{3\sdot2}{6}=\frac{6}{6}=1}}</math>

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:How many cubits long is the whole tree?

:How many cubits long is the whole tree?

:<math>\scriptstyle\frac{1}{7}x+\frac{1}{9}x+8=x</math>

:<math>\scriptstyle\frac{1}{7}x+\frac{1}{9}x+8=x</math>

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:We lowered the top of the ladder by 2 cubits from the top.

:We lowered the top of the ladder by 2 cubits from the top.

:We would like to know: how far is the ladder from the base of the wall?

:We would like to know: how far is the ladder from the base of the wall?

הורדנו ראש הסולם מלמעלה ב' אמות<br>

הורדנו ראש הסולם מלמעלה ב' אמות<br>

נרצה לדעת כמה רחוק ראש הסולם מיסוד הקיר{{#annotend:GG8y}}

נרצה לדעת כמה רחוק ראש הסולם מיסוד הקיר{{#annotend:GG8y}}

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:How much water was spilled?

:How much water was spilled?

כמה חסרו המים

כמה חסרו המים

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:The worker worked [a] known [number of] days and the other likewise.

:The worker worked [a] known [number of] days and the other likewise.

:He paid him ten coins.

:He paid him ten coins.

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:223) Question: If I lent you 150 minyanim for 6 months, but then I need [them] and [someone] told me that he would lend me money for 2 months that would be equal to the 150 I lent you for 6 months.

:223) Question: If I lent you 150 minyanim for 6 months, but then I need [them] and [someone] told me that he would lend me money for 2 months that would be equal to the 150 I lent you for 6 months.

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:{{#annot:moneychanger|642|ljzv}}224) Question: the moneychanger has four [kinds of] coins. Of the first [kind] - 6 are worth one zahuv; of the second - 4; of another - 3; and of another - 2. [Someone] wants to be given from each [kind] for a quarter of a zahuv.

:{{#annot:moneychanger|642|ljzv}}224) Question: the moneychanger has four [kinds of] coins. Of the first [kind] - 6 are worth one zahuv; of the second - 4; of another - 3; and of another - 2. [Someone] wants to be given from each [kind] for a quarter of a zahuv.

:How much will he take from each?

:How much will he take from each?

כמה יקח מכל אחת{{#annotend:ljzv}}

כמה יקח מכל אחת{{#annotend:ljzv}}

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::The first [kind of] coin, which is 6, should be divided by 4; the result is 1 and a half and so he will take from it.

::The first [kind of] coin, which is 6, should be divided by 4; the result is 1 and a half and so he will take from it.

::<math>\scriptstyle{\color{blue}{\frac{6}{4}=1+\frac{1}{2}}}</math>

::<math>\scriptstyle{\color{blue}{\frac{6}{4}=1+\frac{1}{2}}}</math>

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=== <span style=color:Green>Multiple Quantities Problems</span> ===

=== <span style=color:Green>Multiple Quantities Problems</span> ===

:After this was done we find that everyone has an equal share.

:After this was done we find that everyone has an equal share.

:We would like to know how many sons he had and how much was the share of each.

:We would like to know how many sons he had and how much was the share of each.

ואמר לראשון אתה תקח מממוני ז' דינרין ועשור הנשאר<br>

ואמר לראשון אתה תקח מממוני ז' דינרין ועשור הנשאר<br>

ואמ' לאחר ואתה קח לך י"ד מהנשאר והעשור<br>

ואמ' לאחר ואתה קח לך י"ד מהנשאר והעשור<br>

:He divided this money into 3 equal parts, then we find that each one has 40 minyanim in his hand.

:He divided this money into 3 equal parts, then we find that each one has 40 minyanim in his hand.

:We would like to know how much each one took.

:We would like to know how much each one took.

ויהי ריב וה[.]לו ממונם וכל א' וא' לקח מה שיכול לקחת ויתקוטטו ביניהם<br>

ויהי ריב וה[.]לו ממונם וכל א' וא' לקח מה שיכול לקחת ויתקוטטו ביניהם<br>

בא אדם ואמ' אל תרגזו<br>

בא אדם ואמ' אל תרגזו<br>

:They ate together and he gave 4 to the owner of the three and 1 to the owner of the 2.

:They ate together and he gave 4 to the owner of the three and 1 to the owner of the 2.

:I ask: why is this?

:I ask: why is this?

בא א' ואמר אם תרצו שאוכל עמכם אפרע כלם שווים<br>

בא א' ואמר אם תרצו שאוכל עמכם אפרע כלם שווים<br>

ואכלו ביחד ונתן לבעל הג' ד' ובעל הב' א&#x202B;'<br>

ואכלו ביחד ונתן לבעל הג' ד' ובעל הב' א&#x202B;'<br>

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

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

::<math>\scriptstyle{\color{blue}{3-\left(1+\frac{2}{3}\right)=\frac{4}{3}}}</math>

::<math>\scriptstyle{\color{blue}{3-\left(1+\frac{2}{3}\right)=\frac{4}{3}}}</math>

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:He is left with 2.

:He is left with 2.

:How much did he pick?  

:How much did he pick?  

ראשון יש לו לתת חצים עם תוספת ג&#x202B;'<br>

ראשון יש לו לתת חצים עם תוספת ג&#x202B;'<br>

ובב' יש לתת השליש מהנשאר עם תוספת ב&#x202B;'<br>

ובב' יש לתת השליש מהנשאר עם תוספת ב&#x202B;'<br>

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=== <span style=color:Green>Find the Fund Problem, Find the Interest Problem</span> ===

=== <span style=color:Green>Find the Fund Problem, Find the Interest Problem</span> ===

:At the end of 4 months he gave him 4 hundred and at the end of 3 [months] he gave him 2 hundred.

:At the end of 4 months he gave him 4 hundred and at the end of 3 [months] he gave him 2 hundred.

:We would like to know how much the owner of the money should receive at the end of the year and how much the interest was.

:We would like to know how much the owner of the money should receive at the end of the year and how much the interest was.

ובסוף ו' בעל הממון נצטרך למעות ואמר למחוייב לו תן לי ק' מנינים<br>

ובסוף ו' בעל הממון נצטרך למעות ואמר למחוייב לו תן לי ק' מנינים<br>

וכן עשה ובסוף ד' חדשים נתן לו ד' מאות עוד לסוף ג' נתן לו ב' מאות<br>

וכן עשה ובסוף ד' חדשים נתן לו ד' מאות עוד לסוף ג' נתן לו ב' מאות<br>

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::Then, sum up [12] and a half with 33 and a third and with 12 and a half; subtract [the sum] from 250; 1[9]1 and 2-thirds remains and this is the interest he should get at the end of the year.

::Then, sum up [12] and a half with 33 and a third and with 12 and a half; subtract [the sum] from 250; 1[9]1 and 2-thirds remains and this is the interest he should get at the end of the year.

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:At the same price he bought the first time, he bought the second time.  

:At the same price he bought the first time, he bought the second time.  

:I would like to know for how much did he buy the oil and for how much did he buy the wine.

:I would like to know for how much did he buy the oil and for how much did he buy the wine.

אח"כ קנה ט' חביות שמן וב' חביות יין ק' מנינים<br>

אח"כ קנה ט' חביות שמן וב' חביות יין ק' מנינים<br>

ובאותו ערך שקנה בראשנה קנה בשניה<br>

ובאותו ערך שקנה בראשנה קנה בשניה<br>

::We turn to the second [purchase] and say that the 9 barrels [of oil] are worth 72 and the 2 [barrels of wine] are worth 12; the sum is 84.

::We turn to the second [purchase] and say that the 9 barrels [of oil] are worth 72 and the 2 [barrels of wine] are worth 12; the sum is 84.

::<math>\scriptstyle{\color{blue}{\left(9\sdot8\right)+\left(2\sdot6\right)=72+12=84}}</math>

::<math>\scriptstyle{\color{blue}{\left(9\sdot8\right)+\left(2\sdot6\right)=72+12=84}}</math>

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=== <span style=color:Green>Pursuit Problem Problem - Dog and Rabbit</span> ===

=== <span style=color:Green>Pursuit Problem Problem - Dog and Rabbit</span> ===

:Each walks the same as the other.

:Each walks the same as the other.

:I would like to know in how many steps they will catch up?

:I would like to know in how many steps they will catch up?

והארנבת רחוקה מהכלב ק' פסיעות<br>

והארנבת רחוקה מהכלב ק' פסיעות<br>

וכל פסיעה מהכלב שוה ה' מהארנבת<br>

וכל פסיעה מהכלב שוה ה' מהארנבת<br>

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=== <span style=color:Green>Divide a Number Problem - 30 into 4 parts</span> ===

=== <span style=color:Green>Divide a Number Problem - 30 into 4 parts</span> ===

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:{{#annot:30 into 4 parts|619|7fd8}}232) Question: make me four parts out of 30, such that the ratio of the fourth to the third is as the third to the second and as the second to the first.

:{{#annot:30 into 4 parts|619|7fd8}}232) Question: make me four parts out of 30, such that the ratio of the fourth to the third is as the third to the second and as the second to the first.

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::Multiply 1 by [30] and divide by 15; the result is 2 and this is the first [part].

::Multiply 1 by [30] and divide by 15; the result is 2 and this is the first [part].

::<math>\scriptstyle{\color{blue}{a_1=\frac{1\sdot30}{15}=2}}</math>

::<math>\scriptstyle{\color{blue}{a_1=\frac{1\sdot30}{15}=2}}</math>

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=== <span style=color:Green>How Many Problem - Hours</span> ===

=== <span style=color:Green>How Many Problem - Hours</span> ===

:He replied: 2-thirds of those that passed are equal to 3-quarters of those that will pass.

:He replied: 2-thirds of those that passed are equal to 3-quarters of those that will pass.

:We ask: how many hours have passed?

:We ask: how many hours have passed?

השיב הב' שלישיות מהמוקשות שוות לג' שרביעיות לאותם שראוי להקיש<br>

השיב הב' שלישיות מהמוקשות שוות לג' שרביעיות לאותם שראוי להקיש<br>

נשאל כמה שעות הקישו{{#annotend:wo4B}}

נשאל כמה שעות הקישו{{#annotend:wo4B}}

:I wish to know: how much money will each of them receive and how will they sell [the eggs]?

:I wish to know: how much money will each of them receive and how will they sell [the eggs]?

:<math>\scriptstyle a_1=2a_2=3a_3=4a_4=5a_5=6a_6=7a_7</math>

:<math>\scriptstyle a_1=2a_2=3a_3=4a_4=5a_5=6a_6=7a_7</math>

ארצה לדעת כמה ממון נתן לכל א' וא' ובאיזה אופן מכרו{{#annotend:5vBy}}

ארצה לדעת כמה ממון נתן לכל א' וא' ובאיזה אופן מכרו{{#annotend:5vBy}}

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::The money of one is the same as the other's, so you find that the money of each is 15.

::The money of one is the same as the other's, so you find that the money of each is 15.

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::Deduce from this.

::Deduce from this.

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:The same as one sells so does the other.

:The same as one sells so does the other.

:How much does each have and how will he sell it?

:How much does each have and how will he sell it?

ובאופן שמכר הא' ימכור האחר<br>

ובאופן שמכר הא' ימכור האחר<br>

כמה לכל א' ובאיזה אופן מכרו{{#annotend:0z25}}

כמה לכל א' ובאיזה אופן מכרו{{#annotend:0z25}}

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=== <span style=color:Green>Find the Volume - Suit</span> ===

=== <span style=color:Green>Find the Volume - Suit</span> ===

:{{#annot:suit|681|UCK9}}236) Question: a man wants to make a suit from a cloth that is 5 spans wide, which will contain 14 spans, and he wants to make another suit from a cloth that is 4 spans wide.

:{{#annot:suit|681|UCK9}}236) Question: a man wants to make a suit from a cloth that is 5 spans wide, which will contain 14 spans, and he wants to make another suit from a cloth that is 4 spans wide.

:How many spans will he include in it?

:How many spans will he include in it?

כמה זרתות ישים בו{{#annotend:UCK9}}

כמה זרתות ישים בו{{#annotend:UCK9}}

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::Hence, 17 spans and a half should be taken from the other cloth.

::Hence, 17 spans and a half should be taken from the other cloth.

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