Difference between revisions of "ספר הכללים במספר"

::<span style=color:green>'''common denominator:'''</span> say: 3 times 4 is 12.

::<math>\scriptstyle{\color{blue}{3\sdot4=12}}</math>

::The third is 4 dinar.

::The quarter is 3 dinar.

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

::<math>\scriptstyle{\color{blue}{\frac{1}{3}-\frac{1}{4}=\frac{\left(\frac{1}{3}\sdot12\right)-\left(\frac{1}{4}\sdot12\right)}{12}=\frac{4-3}{12}=\frac{1}{12}}}</math>

::<span style=color:green>'''common denominator:'''</span> do as we did above and say: 4 times 5 is 20.

::<math>\scriptstyle{\color{blue}{4\sdot5=20}}</math>

::The quarter is 5-twentieths.

::The fifth is 4-twentieths.

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

::<math>\scriptstyle{\color{blue}{\frac{1}{4}-\frac{1}{5}=\frac{\left(\frac{1}{4}\sdot20\right)-\left(\frac{1}{5}\sdot20\right)}{20}=\frac{5-4}{20}=\frac{1}{20}}}</math>

::<span style=color:green>'''common denominator:'''</span> say: 3 times 5 is 15.

::<math>\scriptstyle{\color{blue}{3\sdot5=15}}</math>

::The third is 5-fifteenths.

::The fifth is 3-fifteenths.

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

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

::<span style=color:green>'''common denominator:'''</span> you should divide the pašuṭ into the number in which you find all these fractions.

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

::<math>\scriptstyle{\color{blue}{\left[\left[\left(3\sdot4\right)\sdot5\right]\sdot6\right]\sdot7=\left[\left(12\sdot5\right)\sdot6\right]\sdot7=\left(60\sdot6\right)\sdot7=360\sdot7=2520}}</math>

::The third is 849 parts.

::<math>\scriptstyle{\color{blue}{\frac{1}{3}=\frac{\frac{1}{3}\sdot2520}{2520}=\frac{840}{2520}}}</math>

::The quarter is 630 parts.

::<math>\scriptstyle{\color{blue}{\frac{1}{4}=\frac{\frac{1}{4}\sdot2520}{2520}=\frac{630}{2520}}}</math>

::The fifth is 504 parts.

::<math>\scriptstyle{\color{blue}{\frac{1}{5}=\frac{\frac{1}{5}\sdot2520}{2520}=\frac{504}{2520}}}</math>

::The sixth is 420 parts.

::<math>\scriptstyle{\color{blue}{\frac{1}{6}=\frac{\frac{1}{6}\sdot2520}{2520}=\frac{420}{2520}}}</math>

::The seventh is 360 parts.

::<math>\scriptstyle{\color{blue}{\frac{1}{7}=\frac{\frac{1}{7}\sdot2520}{2520}=\frac{360}{2520}}}</math>

::The total is 2754.

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

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

::<math>\scriptstyle{\color{blue}{\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}=\frac{840+630+504+420+360}{2520}=\frac{2754}{2520}=1+\frac{234}{2520}=1+\frac{117}{1260}}}</math>

::<span style=color:green>'''False Position:'''</span> We look for a number that has a third and a quarter; it is 12.

|style="width:45%; text-align:right;"|נבקש מניין שיש לו שלישית ורביעית והוא י"ב

::The sum of its third and its quarter is 7.

::We subtract it from 12; 5 remains.

::<math>\scriptstyle{\color{blue}{12-\left[\left(\frac{1}{3}\sdot12\right)+\left(\frac{1}{4}\sdot12\right)\right]=12-7=5}}</math>

::<span style=color:green>'''Rule of Three:'''</span>

::We multiply the extremes; the result is 120.

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

::<math>\scriptstyle{\color{blue}{x=\frac{12\sdot10}{5}=\frac{120}{5}=24}}</math>

::<span style=color:green>'''Check:'''</span> Its third is 8; its quarter is 6; we subtract them from 24; 10 integers remain no more and no less.

::<math>\scriptstyle{\color{blue}{24-\left(\frac{1}{3}\sdot24\right)-\left(\frac{1}{4}\sdot24\right)=24-8-6=10}}</math>

::<span style=color:green>'''False Position - common denominator:'''</span> We look for the denominator; it is 315 and the parts are 143.

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

::<math>\scriptstyle{\color{blue}{\frac{1}{5}\sdot315=63}}</math>

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

::<math>\scriptstyle{\color{blue}{\frac{1}{7}\sdot315=45}}</math>

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

::<math>\scriptstyle{\color{blue}{\frac{1}{9}\sdot315=35}}</math>

::Sum them up; the result is 143.

::<math>\scriptstyle{\color{blue}{63+45+35=143}}</math>

::<span style=color:green>'''Rule of Three:'''</span> We multiply 315 by 10; the result is 3150.

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

::<math>\scriptstyle{\color{blue}{x=\frac{315\sdot10}{143}=\frac{3150}{143}=22+\frac{4}{143}}}</math>

::<span style=color:green>'''Check:'''</span> Take a fifth of 22; it is 4 and 2 remain that are 2 times 143, which is 286.  

::Add to it 4 parts; it is 290.

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

::<math>\scriptstyle{\color{blue}{\frac{1}{5}\sdot\left(22+\frac{4}{143}\right)=4+\frac{2}{5}+\frac{4}{5\sdot143}=4+\frac{\left(2\sdot143\right)+4}{5\sdot143}=4+\frac{286+4}{5\sdot143}=4+\frac{290}{5\sdot143}=4+\frac{58}{143}}}</math>

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

::<math>\scriptstyle{\color{blue}{\left[\frac{1}{5}\sdot\left(22+\frac{4}{143}\right)\right]+\left[\frac{1}{7}\sdot\left(22+\frac{4}{143}\right)\right]+\left[\frac{1}{9}\sdot\left(22+\frac{4}{143}\right)\right]=10}}</math>

::It has another method:

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

::<math>\scriptstyle{\color{blue}{315-143=172}}</math>

::<span style=color:green>'''Rule of Three:'''</span> say: if 143 equals 10, how much is 172 equal to?

::<math>\scriptstyle{\color{blue}{143:10=172:a}}</math>

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

::<math>\scriptstyle{\color{blue}{a=\frac{10\sdot172}{143}=\frac{1720}{143}=12+\frac{4}{143}}}</math>

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

::<math>\scriptstyle{\color{blue}{x=\left(12+\frac{4}{143}\right)+10=22+\frac{4}{143}}}</math>

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

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

::<math>\scriptstyle{\color{blue}{315-143=172}}</math>

::<span style=color:green>'''Rule of Three:'''</span> we do as follows: we multiply 10 by 315; the result is 3150. We divide it by 172; the result is 18 integers and 54 parts of 172.

::<math>\scriptstyle{\color{blue}{x=\frac{10\sdot315}{172}=\frac{3150}{172}=18+\frac{54}{172}}}</math>

::<span style=color:green>'''Check:'''</span> we take a fifth, a seventh, and a ninth of this number; 10 integers remain.

::<math>\scriptstyle{\color{blue}{\left(18+\frac{54}{172}\right)-\left[\left[\frac{1}{5}\sdot\left(18+\frac{54}{172}\right)\right]+\left[\frac{1}{7}\sdot\left(18+\frac{54}{172}\right)\right]+\left[\frac{1}{9}\sdot\left(18+\frac{54}{172}\right)\right]\right]=10}}</math>

::<span style=color:green>'''False Position:'''</span> we know that a half, a third, and a sixth are one integer. We consider [the amount of money] as one; so they are two plus a fifth.

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

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

::<math>\scriptstyle{\color{blue}{x=\frac{40}{2+\frac{1}{5}}=\frac{40}{\frac{\left(5\sdot2\right)+1}{5}}=\frac{40\sdot5}{11}=\frac{200}{11}=18+\frac{2}{11}}}</math>

::<span style=color:green>'''False Position - common denominator:'''</span> we look for the common denominator; it is 315.

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

::<math>\scriptstyle{\color{blue}{\left(\frac{1}{5}\sdot315\right)+\left(\frac{1}{7}\sdot315\right)+\left(\frac{1}{9}\sdot315\right)=143}}</math>

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

::<math>\scriptstyle{\color{blue}{\frac{1}{5}X+\frac{1}{7}X+\frac{1}{9}X=\frac{143}{315}=\frac{\frac{143}{35}}{9}=\frac{4}{9}+\frac{\frac{3}{35}}{9}=\frac{4}{9}+\left(\frac{3}{5}\sdot\frac{1}{7}\sdot\frac{1}{9}\right)}}</math>

::<span style=color:green>'''Rule of Three:'''</span> we say: 7 times 13 is 91. We divide it by 23; the result is 3 measures and 22 of 23 parts of one measure.

::<math>\scriptstyle{\color{blue}{x=\frac{7\sdot13}{23}=\frac{91}{23}=3+\frac{22}{23}}}</math>

::<span style=color:green>'''Rule of Three:'''</span> we say: 7 times 23 is 161. We divide it by 17; it is 12 and 5 parts of 13 of one pašuṭ.

::<math>\scriptstyle{\color{blue}{x=\frac{7\sdot23}{17}=\frac{161}{17}=12+\frac{5}{13}}}</math>

::<span style=color:green>'''Rule of Three:'''</span> <math>\scriptstyle{\color{blue}{x=\frac{5\sdot4}{3}=\frac{20}{3}=6+\frac{2}{3}}}</math> of Cortona

::<span style=color:green>'''Rule of Three:'''</span> we multiply the miles he walks by 10 days; it is 290. We divide it by the difference between the two velocities, which is 8; it is 36 days and a quarter of a day.

::<math>\scriptstyle{\color{blue}{x=\frac{29\sdot10}{37-29}=\frac{290}{8}=36+\frac{1}{4}}}</math>

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

::<math>\scriptstyle{\color{blue}{x=\frac{100}{19+17}=\frac{100}{36}=2+\frac{28}{36}=2+\frac{7}{9}}}</math>

::<span style=color:green>'''False Position:'''</span> we take one for their number and one "like us" - it is two; and "half of us" - it is two and a half. We add a quarter; it is [2 and] 3-quarters.

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

::<math>\scriptstyle{\color{blue}{1+1+\frac{1}{2}+\frac{1}{4}=2+\frac{1}{2}+\frac{1}{4}=2+\frac{3}{4}=\frac{\left(4\sdot2\right)+3}{4}=\frac{8+3}{4}=\frac{11}{4}}}</math>

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

::We convert them into quarters; they are 396.

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

::<math>\scriptstyle{\color{blue}{x=\frac{4\sdot\left(100-1\right)}{11}=\frac{4\sdot99}{11}=\frac{396}{11}=36}}</math>

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

::<math>\scriptstyle{\color{blue}{\frac{50}{1+\frac{1}{4}}=\frac{4\sdot50}{5}=\frac{200}{5}=40}}</math>

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

::<math>\scriptstyle{\color{blue}{\frac{50}{1-\frac{1}{4}}=\frac{4\sdot50}{3}=\frac{200}{3}=66+\frac{2}{3}}}</math>

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

::<math>\scriptstyle{\color{blue}{\left(\frac{50}{1+\frac{1}{4}}+\frac{50}{1-\frac{1}{4}}\right)-100=\left[40+\left(66+\frac{2}{3}\right)\right]-100=6+\frac{2}{3}}}</math>

::You have three known numbers:

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

::<math>\scriptstyle{\color{blue}{\frac{1}{3}\sdot10=3+\frac{1}{3}}}</math>

::The second is the 8 cooked measures.

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

::<span style=color:green>'''Rule of Three:'''</span> we multiply 6 by 3 and a third; it is 20. Divide it by 8; it is 2 and a half.

::<math>\scriptstyle{\color{blue}{x=\frac{\left(3+\frac{1}{3}\right)\sdot6}{8}=\frac{20}{8}=2+\frac{1}{2}}}</math>

::<span style=color:green>'''False Position:'''</span> we sum up all four amounts of their money; it is 56.

::<math>\scriptstyle{\color{blue}{11+13+15+17=56}}</math>

::<span style=color:green>'''Rule of Three:'''</span> as the ratio of each to 56, so each takes from 19.

::<math>\scriptstyle{\color{blue}{\frac{a_i}{56}=\frac{x_i}{19}}}</math>

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

::<math>\scriptstyle{\color{blue}{x_1=\frac{11\sdot19}{56}=\frac{209}{56}=3+\frac{41}{56}}}</math>

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

::<math>\scriptstyle{\color{blue}{x_2=\frac{13\sdot19}{56}=\frac{247}{56}=4+\frac{23}{56}}}</math>

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

::<math>\scriptstyle{\color{blue}{x_3=\frac{15\sdot19}{56}=\frac{285}{56}=5+\frac{5}{56}}}</math>

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

::<math>\scriptstyle{\color{blue}{x_4=\frac{17\sdot19}{56}=\frac{323}{56}=5+\frac{43}{56}}}</math>

::<span style=color:green>'''Check:'''</span> we sum up the integers and the fractions; the result is 19 integers, because the fractions are 56 parts.

::<math>\scriptstyle{\color{blue}{\left(3+\frac{41}{56}\right)+\left(4+\frac{23}{56}\right)+\left(5+\frac{5}{56}\right)+\left(5+\frac{43}{56}\right)=19}}</math>

::<span style=color:green>'''False Position:'''</span> look for a [number] that has a third, a quarter, and a sixth; it is 12.

::The parts are 9 and this is a dinar.

::<math>\scriptstyle{\color{blue}{\left(\frac{1}{3}\sdot12\right)+\left(\frac{1}{4}\sdot12\right)+\left(\frac{1}{6}\sdot12\right)=9}}</math>

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

::<math>\scriptstyle{\color{blue}{x=\frac{12}{9}}}</math>

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

::<math>\scriptstyle{\color{blue}{12\sdot\frac{12}{9}=12\sdot\left(1+\frac{1}{3}\right)=12+4=16}}</math>

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

::<math>\scriptstyle{\color{blue}{\left(7\sdot6\right)\sdot5=42\sdot5=210}}</math>

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

::<math>\scriptstyle{\color{blue}{\left(6\sdot5\right)\sdot4=30\sdot4=120}}</math>

::<span style=color:green>'''Rule of Three:'''</span> we multiply 11 times 120; it is 1320. Divide it by 210; the result is 6 integers and 60 remains, which is 2-sevenths of one pašuṭ.

::<math>\scriptstyle{\color{blue}{x=\frac{11\sdot120}{210}=\frac{1320}{210}=6+\frac{60}{210}=6+\frac{2}{7}}}</math>

::<span style=color:green>'''common denominator:'''</span> look for the common denominator: multiply 5 by 7; it is 35.

::<math>\scriptstyle{\color{blue}{5\sdot7=35}}</math>

::Its 3-fifths are 21.

::<math>\scriptstyle{\color{blue}{x+1=\frac{3}{5}\sdot35=21}}</math>

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

::<math>\scriptstyle{\color{blue}{x=\frac{4}{7}\sdot35=20}}</math>

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

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

::<math>\scriptstyle{\color{blue}{\frac{4}{7}>\frac{5}{9}}}</math>

::<span style=color:green>'''common denominator:'''</span> the common denominator is 63.

::Its 5-ninths are 35.

::<math>\scriptstyle{\color{blue}{x=\frac{5}{9}\sdot63=35}}</math>

::Its 4-sevenths are 36.

::<math>\scriptstyle{\color{blue}{x+1=\frac{4}{7}\sdot63=36}}</math>

::<span style=color:green>'''Check:'''</span> you can check this: as he buys 4-sevenths of a liṭra for one pašuṭ, and his money is 35, he has 20 liṭra.  

::Convert them into ninths; they are 180.

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

::<math>\scriptstyle{\color{blue}{\frac{\frac{4}{7}\sdot35}{\frac{5}{9}}=\frac{20}{\frac{5}{9}}=\frac{20\sdot9}{5}=\frac{180}{5}=36=35+1}}</math>

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

::<math>\scriptstyle{\color{blue}{x=2\sdot35}}</math>

::<math>\scriptstyle{\color{blue}{\frac{\frac{4}{7}\sdot\left(2\sdot35\right)}{\frac{5}{9}}=\frac{40}{\frac{5}{9}}=\left(2\sdot35\right)+2}}</math>

::He multiplies them by 35.

::<math>\scriptstyle{\color{blue}{x=3\sdot35}}</math>

::And so on for any number.

|style="text-align:right;"|וכן יבא בין פפריני' ופריביסיני' ולכל [חשבון] כזה

:20) <span style=color:green>'''Rule of Three:'''</span> with fractions:

|style="text-align:right;"|כ <big>אם</big> ישאל לך חשבון שיהיה בו מאלו הג' מינים ולפני הב' חלקי' יהיו שבורים<br>

אחרי כן נרבה אותו הדבר שנרצה לדעת כנגד אותו הדבר שאנינה ממינה עצמה<br>

ונחלק אותה על השלישית וכה"ל

:21) <span style=color:green>'''Rule of Three:'''</span> with fraction:

|style="text-align:right;"|כא <big>ואם</big> נשאל לך חשבון אחר שיהיה שבור מחלק אחד<br>

נרבה הב' חלקי' באותו המספר דהיינו עם אותו השבור<br>

אחרי כן נרבה אותו הדבר שנרצה לידע כנגד אותו הדבר שאנינה ממינה<br>

ונחלק אותה על השלישית

|Example for the two cases:

*{{#annot:two currencies|632|ppYb}}If 5¼ of Pisa are worth 7 of Cortona, how many of Cortona are 1000 liṭra of Pisa worth?<br>

|style="text-align:right;"|אם ה' פיסני' ורביע שוים ז' קורטוט' כמה ישוו אלף ליט' מפיסני&#x202B;'{{#annotend:ppYb}}

::converting to quarters

:::<math>\scriptstyle{\color{blue}{4\sdot\left(5+\frac{1}{4}\right)=21}}</math>

:::<math>\scriptstyle{\color{blue}{4\sdot7=28}}</math>

:::21 of Pisa = 28 of Cortona

::reducing by 7

:::<math>\scriptstyle{\color{blue}{\frac{1}{7}\sdot21=3}}</math>

:::<math>\scriptstyle{\color{blue}{\frac{1}{7}\sdot28=4}}</math>

:::3 of Pisa = 4 of Cortona

::<span style=color:green>'''Rule of Three:'''</span> <math>\scriptstyle{\color{blue}{3:4=1000:X}}</math>

|style="text-align:right;"|ואנחנו נרצה לידע כמה ישוו אלף ליט' מפיסני' הפיסני הוא הדבר שנרצה לידע והקורטו' הוא הדבר שאנינה ממינה עצמה

::<math>\scriptstyle{\color{blue}{X=\frac{1000\sdot4}{3}=\frac{4000}{3}=1333+\frac{6}{20}+\frac{\frac{8}{12}}{20}}}</math>

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

|style="text-align:right;"|לכן נרבה ד' פעמים אלף יבא ד' אלפי' מקורטו&#x202B;'<br>

ונחלק בג' שיבא אלף ושל"ג ליט' וו' דינרי' וח"פ וכן ישוו אלף ליט' מפיסני לקורטוט&#x202B;'

*{{#annot:two measures|867|p0EU}}If 7⅓ ounce of Pisa are worth 73 liṭra and 5 dinar of Cortona, how many of Cortona are 19 ounce of Pisa worth?

|style="text-align:right;"|ונשים המשל אל החשבון שהוא שבור מב' חלקיו אם ז' אונקיו' ושליש כסף שוים ע"ג ליט' וה' דינרי' מפיסני&#x202B;'<br>

::converting to parts of 12

|style="text-align:right;"|ואלו השבורים הם שליש ורביע אנה ימצאו בי"ב<br>

לכן יש לנו לרבות הב' חלקי' בי"ב

:::<math>\scriptstyle{\color{blue}{12\sdot\left(7+\frac{1}{3}\right)=88}}</math>

:::<math>\scriptstyle{\color{blue}{12\sdot\left(73+\frac{1}{4}\right)=879}}</math>

:::88 ounce = 879 liṭra

::<span style=color:green>'''Rule of Three:'''</span> <math>\scriptstyle{\color{blue}{88:879=19:X}}</math>

::<math>\scriptstyle{\color{blue}{X=\frac{19\sdot879}{88}=\frac{16701}{88}=\frac{16701}{8\sdot11}=189+\frac{15}{20}+\frac{\frac{8+\frac{2}{11}}{12}}{20}}}</math>

|style="text-align:right;"|נרבה י"ט פעמי' תתע"ט ליט' יהיו י"ו אלפי' ותש"א<br>

|}

{|

:{{#annot:for some liṭra|660|XXox}}22) One kikkar yields 36 liṭra and 18 dinar a year.

::<span style=color:green>'''Rule of Three:'''</span> multiply the 100 liṭra by the 12 months of the year, say: 12 times 100 are 1200.

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

::<math>\scriptstyle{\color{blue}{x=\frac{12\sdot100}{65}=\frac{1200}{65}=18+\frac{13}{30}+\frac{\frac{11}{13}}{30}}}</math>

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

:{{#annot:for some liṭra, some months|638|uzR3}}23) I lent one liṭra for 3 pešuṭim a month.

:How many [liṭra] will 60 liṭra make in 8 months?

::<math>\scriptstyle{\color{blue}{3\sdot8=24}}</math> pešuṭim = 2 dinar

::<math>\scriptstyle{\color{blue}{2\sdot60=120}}</math> dinar

::= <math>\scriptstyle{\color{blue}{20\sdot6}}</math> = <math>\scriptstyle{\color{blue}{X=6}}</math> liṭra

|style="text-align:right;"|אחר כן תרבה אלו הב' די' עם הס' ליט' ואמור ס' פעמי' ב' דינרי' הם ק"כ די' שהם ו' ליט&#x202B;'<br>

:{{#annot:liṭra for one day|666|Habg}}24) I lent one liṭra for so and so pešuṭim a month.

כמה ליט' ירויחו היום אח&#x202B;'{{#annotend:Habg}}

::30&divide;(the pešuṭim earned a month)

::<math>\scriptstyle{\color{blue}{X=\frac{30}{a}}}</math>

:*{{#annot:liṭra for one day|666|xRwx}}I lent one liṭra for 2½ pešuṭim a month.

::How many liṭra will yield one pašuṭ a day?

::<math>\scriptstyle\frac{2+\frac{1}{2}}{30\sdot1}=\frac{1}{1\sdot X}</math>

ותרצה לידע כמה ליט' ירויחו היום אח&#x202B;'{{#annotend:xRwx}}

:::<math>\scriptstyle{\color{blue}{X=\frac{30}{2+\frac{1}{2}}=12}}</math> liṭra

|style="text-align:right;"|נחלק ל' ליט' בב' וחצי שיבוא י"ב ליט&#x202B;'<br>

:{{#annot:days, for one liṭra|660|1jH8}}25) I lent one liṭra for so and so a month.

:In how many days will one liṭra yield one?

::<math>\scriptstyle X=\frac{30}{a}</math>

:*{{#annot:days, for one liṭra|660|xUoL}}I lent [one liṭra] for 3 [a month].

::In how many days will one liṭra yield one?

::<math>\scriptstyle\frac{3}{30\sdot1}=\frac{1}{X\sdot1}</math>

:::<math>\scriptstyle{\color{blue}{x=\frac{30}{3}=10}}</math> days

|style="text-align:right;"|חלק ל' יום בג' חלקי' שיבוא י' ימים הרי שבי' ימים ירויח הליט' א"פ

|'''Find the Fund'''

:{{#annot:liṭra for one day|666|pclr}}26) One kikkar yields so and so liṭra a year.

::150&divide;(the liṭra earned a year)

::<math>\scriptstyle{\color{blue}{X=\frac{150}{a}}}</math>

:*{{#annot:liṭra for one day|666|Vv31}}One kikkar yields 12 liṭra a year.

::How many liṭra will yield one pašuṭ a day?

::<math>\scriptstyle\frac{\left(20\sdot12\right)\sdot12}{360\sdot100}=\frac{1}{1\sdot X}</math>

ותרצה לידע כמה ליט' ירויחו היום אח&#x202B;'{{#annotend:Vv31}}

::<math>\scriptstyle{\color{blue}{X=\frac{150}{12}=12+\frac{1}{2}}}</math> liṭra

|style="text-align:right;"|חלק ק"נ ליט' בי"ב חלקי' שיבוא &#x202B;<ref>90v</ref>י"ב וחצי<br>

:{{#annot:days, for one liṭra|660|JrEQ}}27) One kikkar yields so and so liṭra a year.

|style="text-align:right;"|כז) אם ישאלך אדם הככר ירויח כל כך ליט' השנה<br>

בכמה ימים ירויח הליט' אח&#x202B;'{{#annotend:JrEQ}}

::150&divide;(the liṭra earned a year)

::<math>\scriptstyle{\color{blue}{X=\frac{150}{a}}}</math>

:*{{#annot:days, for one liṭra|660|U8hO}}One kikkar yields 9 liṭra a year.

::In how many days will one liṭra yield one pašuṭ?

::<math>\scriptstyle\frac{\left(20\sdot12\right)\sdot9}{360\sdot100}=\frac{1}{X\sdot1}</math>

|style="text-align:right;"|המשל בזה הככר ירויח ט' ליט' השנה ותרצה לידע בכמה ימים ירויח הליט' אח&#x202B;'{{#annotend:U8hO}}

::<math>\scriptstyle{\color{blue}{X=\frac{150}{9}=16+\frac{2}{3}}}</math> days

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

:{{#annot:months, for some liṭra|660|LLGv}}28) So and so liṭra yield so and so [liṭra] in so and so [months].

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

|style="text-align:right;"|כח) אם ישאלך אדם כל כך ליט' ירויחו כל כך בכל כך זמן כמה ליט' כמה כל כך ליט' בכל כך זמן בכמה זמן ירויחו כהם או בכל חדשים כמה ליט' ירויחו כל{{#annotend:LLGv}}

::<span style=color:green>'''Rule of Three:'''</span> <math>\scriptstyle X=\frac{b\sdot c}{d}</math>

:*{{#annot:months, for some liṭra|660|XRzd}}50 liṭra yield 3 liṭra in 4 months.

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

::<math>\scriptstyle\frac{3}{4\sdot50}=\frac{3}{X\sdot75}</math>

:<math>\scriptstyle{\color{blue}{x=\frac{50\sdot4}{75}=\frac{200}{75}=2+\frac{20}{30}}}</math> months

|style="text-align:right;"|הרבה סכום הליט' כנגד סכום החדשים ואמור נ' פעמי' ד' חדשים יבא ר' חדשים וחלקם בע"ה יבא ב' חדשים וכ' ימים נמצא שבב' חדשי' וב' ימים ירויחו הע"ה ליט' כל כך כמו הנ' ליט' ד' חדשים וכה

|'''Find the Fund'''

*{{#annot:liṭra for some months|666|gU7s}}29) So and so liṭra yield so and so liṭra in so and so [months].  

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

|style="text-align:right;"|כט) אם ישאלך אדם כל כך ליט' ירויחו בכל כך זמן כל כך ליט&#x202B;'<br>

::(months₂&times;liṭra)&divide;months₁

::<math>\scriptstyle{\color{blue}{X=\frac{c\sdot b}{d}}}</math>

:*{{#annot:liṭra for some months|666|EOv3}}25 liṭra yield 40 dinar in 6 months.

::How many liṭra will yield the same in 8 months?

::<math>\scriptstyle\frac{40}{6\sdot25}=\frac{40}{8\sdot X}</math>

::<math>\scriptstyle{\color{blue}{X=\frac{6\sdot25}{8}=\frac{150}{8}=18+\frac{15}{20}}}</math>

|'''Find the Time'''

:{{#annot:years, for some liṭra|660|ucM7}}30) I lent one liṭra for so and so [pešuṭim] a month.

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

|style="text-align:right;"|ל) אם ישאלך אדם הלותי הליט' בכך וכך החדש<br>

::20&divide;(the pešuṭim earned a month)

::<math>\scriptstyle{\color{blue}{X=\frac{20}{a}}}</math>

:*{{#annot:years, for some liṭra|660|uGZJ}}I lent one liṭra for 3 pešuṭim a month.

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

::<math>\scriptstyle\frac{3}{1\sdot1}=\frac{\left(20\sdot12\right)\sdot95}{12\sdot X\sdot95}</math>

|style="text-align:right;"|המשל בזה הלותי הליט' בגפ"ה הצ"ה ליט' בכמה זמן יכפלו{{#annotend:uGZJ}}

::<math>\scriptstyle{\color{blue}{X=\frac{20}{3}=6+\frac{8}{12}}}</math>

:{{#annot:years, for some liṭra|660|9FcN}}31) One kikkar yields so and so liṭra a year.

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

|style="text-align:right;"|לא) אם ישאלך אדם הככר ירויח השנה כל כך ליט&#x202B;'

::100&divide;(the liṭra earned a year)

::<math>\scriptstyle{\color{blue}{X=\frac{100}{a}}}</math>

:*{{#annot:years, for some liṭra|660|uvRl}}One kikkar yields 8 liṭra a year.

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

::<math>\scriptstyle\frac{8}{1\sdot100}=\frac{45}{X\sdot45}</math>

::<math>\scriptstyle{\color{blue}{X=\frac{100}{8}=12+\frac{6}{12}}}</math>

|'''Pricing Problem - Find the Price'''

:{{#annot:lint|629|Jc98}}32) One kantar [measurement of weight] of lint is worth 17 liṭra‏ and 5 dinar.

|style="text-align:right;"|לב) אם ישאלך אדם הקנטרו' מן המוך שוה י"ז ליט' וה' דינרי&#x202B;'<br>

::<math>\scriptstyle{\color{blue}{x=\frac{\frac{\frac{\left(20\sdot17\right)+5}{3}}{5}}{11}=\frac{\frac{\left(20\sdot5\right)+15}{5}}{11}=\frac{23}{11}=2+\frac{1}{11}}}</math> pešuṭim

|style="text-align:right;"|חלק אלו הי"ז ליט' וה' דינרי' בג' חלקי' שיבא לכל חלק ה' ליט' וט"ו די&#x202B;'<br>

אחרי כן חלק אלו הה' ליט' וט"ו די' בה' חלקי' שיבא לכל חלק כ"ג דינרי&#x202B;'<br>

:::<math>\scriptstyle{\color{blue}{11\sdot15=165}}</math>

|style="text-align:right;"|יסוד זה החשבון הוא כי הם ט"ו וי"א פעמי' ט"ו הם קס"ה א"כ בחלק הזמן

|'''Find the Earned Interest'''

:{{#annot:for one liṭra a month|638|nEKA}}33) So and so liṭra yield one pašuṭ a day.

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

|style="text-align:right;"|לג) אם ישאלך אדם כל כך ליט' ירויחו היום אח&#x202B;'<br>

::30&divide;(the liṭra that yield one pašuṭ a day)

::<math>\scriptstyle{\color{blue}{X=\frac{30}{a}}}</math>

|style="text-align:right;"|חלק ליט' בכל כך חלקי' כמו הליט' שירויחו היום אח&#x202B;'

:*{{#annot:for one liṭra a month|638|VBTh}}12 liṭra yield one pašuṭ a day.

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

::<math>\scriptstyle\frac{1}{1\sdot12}=\frac{X}{30\sdot1}</math>

::<math>\scriptstyle{\color{blue}{X=\frac{30}{12}=2+\frac{1}{2}}}</math> pešuṭim

נמצא שיצא החדש הליט' ב"פ וחצי וכה

:{{#annot:for some liṭra a year|638|RDKN}}34) So and so liṭra yield one pašuṭ a day.

|style="text-align:right;"|לד) אם ישאלך אדם כל כך ליט' ירויחו היום א"פ<br>

::<math>\scriptstyle x=\frac{150}{a}</math>

|style="text-align:right;"|חלק ק"נ בכל כך חלקי' כמו הליט' שירויחו היום א"פ

:*{{#annot:for some liṭra a year|638|Ins5}}15 liṭra yield one pašuṭ a day.

::How many liṭra will 100 liṭra yield a year?

::<math>\scriptstyle\frac{1}{1\sdot15}=\frac{\left(20\sdot12\right)\sdot X}{360\sdot100}</math>

:::<math>\scriptstyle{\color{blue}{x=\frac{150}{15}=10}}</math> liṭra for 100 liṭra a year

:{{#annot:liṭra for some months|666|nqXZ}}35) One kikkar yields 29 liṭra 13 dinar and 7 pešuṭim a year.

|style="text-align:right;"|לה) אם ישאלך אדם הככר ירויח השנה כ"ט ליט' וי"ג דינרי' וז"פ<br>

::<math>\scriptstyle{\color{blue}{X=\frac{12\sdot100}{7}=\frac{1200}{7}=171+\frac{8}{20}+\frac{6+\frac{6}{7}}{20\sdot12}}}</math>

::171 liṭra, 8 dinar, and <math>\scriptstyle6\frac{6}{7}</math> pešuṭim

וחלקם בכל כך חלקי' כמו החדשי' שנרצה לידע דהיינו הז' חדשים<br>