Difference between revisions of "ספר חשבון"

The Nature of One

Example: 10 - its one side is 11 and its other side is 9; together they are 20 and 10 is half [the sum of] both sides.

It is also half [the sum of the numbers on] the sides of its sides, which are 12 and 8.

It is also half [the sum of the numbers on] the sides of the sides of its sides, which are 13 and 7.

Since one is not a number, you do not find two sides of it.

Since it is a number, it is half [the number on] its one side, i.e. half the number 2.

• The property of the integer is that when you multiply it by itself it increases.

As 2 times 2, which is 4.

3 times 3, which is 9.

And so on.

• Whereas [the property of] the fraction is that when you multiply it by itself it decreases.

As half times a half, which is a quarter.

A third times a third, which is a ninth.

A quarter times a quarter, which is a part of 16.

And so on.

Since one does not increase when you multiply it by itself, as the integers, and does not decrease, as the fractions, this indicates that it is not a number.

Shortcuts - Products of Ranks by Ranks

• As 6 times 40.

We take their analogous from the corresponding units: since 40 is 4 tens multiplied 6 times, and since 4 times 6 is 24, then 6 times 40 is 24 tens, which is 240.

• The same for 3 times 30.

Their analogous is 3 times 3, which is 9. So, it is 9 tens, which is 90.

• As 30 times 30.

We take their analogous from the corresponding units: since 30 is 3 tens, their analogous is 3 times 3 and as 3 times 3 is 9, so 30 times 30 is 9 hundred.

• Also 40 times 40.

Their analogous is 4 times [4], which is 16 hundred.

• Also 30 times 40.

Their analogous is 3 times 4, which is 12. So, 30 times 40 is 12 hundred.

• As 20 times 2 hundreds.

Their analogous is 2 times 2, which is 4. So, it is 4 thousand.

• Also 50 times 500.

Their analogous is 5 times 5, which is 25. So, it is 25 thousand.

• As 200 times 200.

Their analogous is 2 times 2, which is 4. So, it is 4 tens of thousands.

• Also 500 times 500.

Their analogous is 5 times 5, which is 25. So, it is 25 tens of thousands.

The Decimal Ranks‫^[66]

• The units are in the first rank.

• The tens are in the second rank.

• The hundreds are in the third rank.

• The thousands are in the fourth rank.

• The tens of thousands are in the fifth rank.

• The hundreds of Thousands are in the sixth rank.

• If we want to multiply 600 by 4 thousand.

Their analogous is 6 times 4, which is 24.

If you want to know what [is the rank of] this 24, take the [positional value of] the rank of hundreds, which is three, and count 3 ranks from the thousands. The rank you will reach is the rank of 24 that you have kept and it is the rank of hundreds of thousands.

Likewise, if you count 4 ranks, which is the rank of thousands, from the rank of hundreds, you reach also the rank of hundreds of thousands.

Know then that the units of the number you have kept, which is 24, is in the rank of hundreds of thousands, and its tens are in the rank of thousands of thousands. So, the 24 you have kept are 2 thousands of thousands and 400 thousand, because its units are four and its tens are 2.

As if you take [the positional value of] the rank of hundreds in this example, which is 3, and [the positional value of] the rank of thousands, which is 4; they are 7. Subtract 1 from it; 6 remains and this number is [the positional value of] the rank of hundreds of thousands.

Three Types of Numbers‫^[76]

The Ten Numerals‫^[78]

The Method of Finding the Square Numbers‫^[80]

• As if you sum one, which is the first square, with 3, which is the first of the odd numbers; you find 4, which is the second square, whose root is 2.

• If you sum 4, which is the second square, with 5, which is the second odd number, they are 9, which is the third square, whose root is 3.

• If you add the third odd number, which is 7, to 9, which is the third square, you find 16, which is the fourth square, whose root is 4.

Cubic Numbers‫^[84]

• 1 is the first cubic number. It is a place that is 1 in length, 1 in width, 1 in height and 1 in depth.

• 8 is the second cubic number derived from a place whose length, width, and depth are 2.

• 27 is the third cubic number derived from a place whose length, width, and depth are 3.

• 64 is the fourth cubic number derived from a place whose length, width, and depth are 4.

• 125 is the fifth cubic number derived from a place whose length, width, and depth are 5.

• 216 is the sixth cubic number derived from a place whose length, width, and depth are 6.

• 343 is the seventh cubic number derived from a place whose length, width, and depth are 7.

• 512 is the eighth cubic number derived from a place whose length, width, and depth are 8.

• 729 is the ninth cubic number derived from a place whose length, width, and depth are 9.

• One thousand is the tenth cubic number derived from a place whose length, width, and depth are 10.

• Because the cube of 1 is 1.

• The cube of 2, which is double 1, is 8.

• The cube of 4, which is double 2, is 64, which is 8 times the cube of 2.

• Also the cube of 6 is 8 times the cube of 3.

• Therefore, if someone says: dig me a hole 2 in length, 2 in breadth, 2 in depth, the payment should be 40 liṭra, which is 8 times 5 liṭra.

Primary Definitions‫^[85]

• The unity is the issue by which every one of the beings in the world is called one.

• The number is the multitude summed from units.

• The number counted by another number is the product of the counted number multiplied a number of times that equals to the number of the units contained in the other number by which it is counted.

• The square number is the product of a number by the number of the units contained in that number itself.

• The the original number is the root of the square number.

As the number nine, which is called a square, because it is a product of three times three.

The number three is called the root of the square.

• The cubic number is an equilateral body, whose length, breadth and depth are equal and this number is the product of a square number by its root.

As the number 9, which is a square; if you multiply it by 3, which is its root, the product is 27 and this number is called a cubic number.

• Relating a number to another number, or a shape to another shape - ?

Multiplication

Units and tens by units and tens

בראשונה נרבה האחדים עם האחדים
ואחרי כן^[88] נרבה האחדים עם ב העשרות
ואחרי כן^[89] נרבה העשרות עם העשרות

• We wish to multiply 22 times 22.

2 2  4   2 2  4   4 4   4 4     4 8 4

4 8 4   2 2   2 2

First, we multiply the units by each other and say: 2 times 2 is 4. We write 4 above the units on the right.

Then, we multiply the units by the tens and say: 2 times [2] and 2 times [2] is 8. We write 8.

Then, we multiply the tens by each other and say: 2 times 2 is 4. We write 4.

We receive that 22 times 22 is 484.

		4		84		484
22		22		22		22
22		22		22		22

• Also if we wish to multiply 44 times 44.

1 9 3 6     4 4     4 4

First, we multiply the units by each other and say: 4 times 4 is 16. We write 6 and keep the ten, which is 1.

Then, we multiply the units by the tens and say: 4 times 4 is 16 and 4 times 4 is 16; it is 32. Add to it the 1 you have kept; it is 33. We write 3 and keep 3.

Then, we multiply the tens by the tens and say: 4 times 4 is 16; with the 3 we have kept, it is 19. We write 19.

The result is that 44 times 44 is 1936.

		6		36		1936
24		44		44		44
24		44		44		44

• Also if we wish to multiply 20 times 20.

2 0   2 0   0 0 4 0   4 0 0

4 0 0   2 0   2 0

We say: zero times zero is zero. We write it.

Then, we say: zero times 2 and 2 times zero is zero. We write it.

Then, we sa: 2 times 2 is 4. We write it.

We receive that 20 times 20 is four hundred.

		0		00		400
20		20		20		20
20		20		20		20

Units by units and tens

• For example, when we wish to know [how much is] 7 times 45, as you see in the diagram:

0 7   4 5 3 1 5

3 1 5   0 7   4 5

We multiply them [by adding] a zero, so that they will be two digits by two digits and this will be done easily.

We say: 5 times 7 is 35. We write the units, which is 5, and keep the tens that are 3.

Then, we say: zero times 5 is zero and 4 times 7 is 28. Add the 3 that we have kept; it is 31. We write 31, for we have nothing left to multiply.

Because, zero times 4 we have left to multiply is zero; so the result is nothing.

We receive that 7 times 45 is 315.

		5		15		315
07		07		07		07
45		45		45		45

Check

• We do this way with the calculation of 22 times 22:

We sum up the upper digits and say: 2 and 2 is 4. We write it next to the two bottom digits.

Then, we multiply the two digits we wrote one above the other and say: 4 times [4] is 16.

We cast out the 9 from it; 7 remains. We write 7 between the two mentioned digits.

As the 7 we have left, so we should have 7 left from 484, which is the result of the calculation, after we cast out the 9 from it.

If we sum up 4, 8, and 4, which are the digits indicating 484; it is 16.

If we cast out 9 from it, 7 remains. So, the calculation is correct.

• Also in the second calculation we did, which is 44 times 44.

We sum up the upper digits and say: 4 and 4 is 8. We write it.

We multiply them by each other and say: 8 times 8 is 64.

Cast out the nines from it; 1 remains.

The same should remain from 1936, whose digits are 1, 9, 3, 6.

Sum them up; it is 19. Cast out the nines from it; the remainder is 1.

Units, tens, and hundreds by units, tens, and hundreds

• As when we wish to know [the product of] 222 times 222.

We multiply them five times [= i.e. five phases of multiplication, five ranks in the final result of multiplication] and this is the method of multiplication:

We write them 5 times, so that the first, second, third, fourth, and fifth multiplications will be clear to the eye.

If we want to multiply 222 times 222.

We start to multiply the units and say: 2 times 2 is 4. We write it.

Then, we multiply the units by the tens and say: 2 times 2 is 4 and 2 times 2 is 4; so, it is 8. We write it.

		4		84
222		222		222
222		222		222

Then, we multiply all three together and say: 2 times 2 is 4; 2 times 2 is 4; so it is 8; and 2 times 2 is 4; so it is 12. We write 2 and keep 1.

	284
	222
	222

We multiply the tens by the hundreds and say: 2 times 2 is 4 and 2 times 2 is 4; so it is 8; with the 1 we have kept, it is 9. We write it.

We multiply the hundreds by the hundreds and say: 2 times 2 is 4. We write it.

We receive that 222 times 222 is 49284.

	9284		49284
	222		222
	222		222

Check

We sum up the upper 222, whose digits are 2, 2, 2; the result is 6. We write it next to them.

We multiply 6 by 6; the result is 36.

We cast out the nines; we are left with nothing. We write a zero between them.

So, a zero should also remain from the sum of the calculation result, which is 49284, whose digits are 4, 9, 2, 8, 4. The sum is 27.

If you cast out the nines, a zero remains.

We find that the calculation is correct.

Units and tens by units, tens, and hundreds

• For example, when we wish to [find the product of] 222 times 22, as you see in the diagram:

We match them by [adding] a zero, so they are three digits by three digits and then the procedure is easier.

Units by units, tens, and hundreds

• As when we wish to know [the product of] 9 times 222.

We write them this way, as you see in the diagram:

Units, tens, hundreds, and thousands by units, tens, hundreds, and thousands

• As when we wish to multiply 2222 times 2222.

We multiply them seven times and write them seven times, in order that it will be clear to the quick and slow observer.

If we wish to multiply 2222 by 2222.

We do as we did in the previous calculations.

The result of the calculation is 4937284.

Units, tens, hundreds, and thousands by units, tens, and hundreds

• As when we want to multiply 2222 by 222.

We match them by [adding] a zero, as you see in the diagram:

The result is 493284.

Units, tens, hundreds, and thousands by units and tens

• As when we wish to multiply 2222 by 22.

We match them by [adding] zeros, as you see in the diagram:

The result is 48884.

Units, tens, hundreds, and thousands by units

• As 2 by 2222.

We match them by [adding] zeros, as you see in the diagram:

The result is 4444.

Units, tens, hundreds, thousands, and tens of thousands by units, tens, hundreds, thousands, and tens of thousands

• As when we wish to know [the product of] 22222 by 22222.

We multiply them nine times and this is the method of multiplication:

We write them nine times, in order that it will be clear to the quick and slow observer.

Units, tens, hundreds, thousands, tens of thousands, and hundreds of thousands by units, tens, hundreds, thousands, tens of thousands, and hundreds of thousands

• As when we wish to multiply 222222 by 222222.

We multiply them 11 times, in order that it will be clear to the quick and slow observer.

The Method of Summing Up Separate Numbers Called Congregare or Riunire

• Example: we wish to sum up 12345 with 67891, 23456, 78912, and 34567.

We start summing the units and say: 5, 1, 6, 2, 7 are 21. We write 1 and keep 2.

Then, we sum the tens and say: 4, 9, 5, 1, 6 are 25; with the 2 we have kept, they are 27. We write 7 and keep 2.

Then, we sum the hundreds and say: 3, 8, 4, 9, 5 are 29; with the 2 we have kept, they are 31. We write 1 and keep 3.

Then, we sum the thousands and say: 2, 7, 3, 8, 4 are 24; with the 3 we have kept, they are 27. We write 7 and keep 2.

Then, we sum the tens of thousands and say: 1, 6, 2, 7, 3 are 19; with the 2 we have kept, they are 21. We write everything, because we do not have another number to sum.

The total result is 217171.

• Example: we wish to sum up 12345 with 6789, 123, 45600, 80912, and 3406.

We start summing the units and say: 5, 9, 3, 2, 6 are 25. We write [5] and keep 2.

Then, we sum the tens and say: 4, 8, 2, 1 are 15; with the 2 we have kept, they are 17. We write 7 and keep 1.

Then, we sum the hundreds and say: 3, 7, 1, 6, 9, 4 are 30; with the 1 we have kept, they are 31. We write 1 and keep 3.

Then, we sum the thousands and say: 2, 6, 5, 3 are 16; with the 3 we have kept, they are 19. We write 9 and keep 1.

Then, we sum the tens of thousands and say: 1, 4, 8 are 13; with the 1 we have kept, they are 14. We write 14, because we have summed everything.

Their sum is 149175.

The Method of Subtracting a Number from a Number Called Sottrarre

• Example: we wish to subtract 645232 from 987654.

We do this way:

We start from the units and say: subtract 2 from 4; 2 remains. We write 2 beneath them.

Subtract 3 from 5; 2 remains. Write 2 beneath them.

Subtract 2 from 6; 4 remains. Write 4 beneath them.

Subtract 5 from 7; 2 remains. Write 2 beneath them.

Subtract 4 from 8; 4 remains. Write 4 beneath them.

Subtract 6 from 9; 3 remains. Write 3 beneath them.

So, if we subtract 645232 from 987654, 342422 remains and this procedure is easy.

Example: We sum 2 with 2; it is 4. We sum 3 with 2; it is 5. We sum 2 with 4; it is 6. We sum 5 with 2; it is 7. We sum 4 with 4; it is 8. We sum 6 with 3; it is 9. They are the same as the greater digits from which you subtracted the smaller digits. This indicates that the calculation is correct.

• Example: we wish to subtract 27654 from 93432.

We start from the units and say: subtract 4 from 2; we cannot.

We subtract 1 from the 3 next to it and add it to the 2; they are 12.

We say: subtract 4 from 12; 8 remains. We write it.

We return to the 3, from which we have subtracted 1; 2 remains.

We say: subtract 5 from 2; we cannot.

We subtract 1 from the 4 next to it and say: subtract 5 from 12; 7 remains. We write it.

93432		93432		93432
27654		27654		27654
		8		78

We return to the 4, from which we have subtracted 1; 3 remains.

We say: subtract 6 from 3; we cannot.

We subtract 1 from the 3 next to it and say: subtract 6 from 13; 7 remains. We write it.

We return to the 3, from which we have subtracted 1; 2 remains.

We say: subtract 7 from 2; we cannot.

We subtract 1 from the 9 next to it and say: subtract 7 from 12; 5 remains. We write it.

We return to the 9, from which we have subtracted 1; 8 remains.

We say: subtract 2 from 8; 6 remains. We write it.

Hence, we subtract 27654 from 93432 and 65778 remains

	93432		93432		93432
	27654		27654		27654
	778		5778		65778

If we sum 4 with 8; they are 12. We write 2 and keep 1.

If we sum 5 with 7; they are 12. With the 1 we have left, they are 13. We write 3 and keep 1.

If we sum 6 with 7; they are 13. With the 1 we have left, they are 14. We write 4 and keep 1.

If we sum 7 with 5; they are 12. With the 1 we have left, they are 13. We write 3 and keep 1.

If we sum 2 with 6; they are 8. With the 1 we have left, they are 9. We write it.

So, the final number is the same as the original [number].

The Method of Dividing a Number by a Number Called Dividere or Partire

• Example: we wish to divide 98756 by 4.

We do this way:

We start from the large number and say: we divide 9 by 4; it is 2. We write it and we are left with 1.

We add this 1 to the 8 next to it; they are 18.

We say: we divide 18 by 4; it is 4. We write it and we are left with 2.

98756		98756		98756
		2		24
4		4		4

We add this 2 to the 7 that follows it; they are 27.

We divide it by 4; it is 6. We write it and we are left with 3.

We add this 3 to the [5] that follows it; they are 35.

We divide it by 4; it is 8. We write it and we are left with 3.

	98756		98756
	246		2468
	4		4

We add this 3 to the 6 that follows it; they are 36.

We divide it by 4; it is 9. We write it.

Hence, we divide 98756 by 4 and the result is 24689.

	98756
	24689
	4

• If we want to divide the number 8976 by 5.

We do this way:

We say: divide 8 by 5; it is 1. We write it. 1 remains.

We add this 3 to the 9 that follows it; they are 39.

We divide it by 5; it is 7. We write it. 4 remains.

8976		8976		8976
		1		17
5		5		5

We add this 4 to the 7 that follows it; they are 47.

We divide it by 5; it is 9. We write it and we are left with 2.

We add this 2 to the 6 that follows it; they are 26.

We divide it by 5; it is 5. We write it and we are left with 1. We write it above the 5, by which we divided the number.

The result of the division mentioned is 1795 and a fifth.

	8976		8976
	179		1795
	5		5

• Example: we wish to divide 12345 by 9.

We say: divide 1 by 9; we cannot.

We add 1 to the 2 that follows it; they are 12.

Divide it by 9; it is 1. We write it and we are left with 3.

We add it to the 3 that follows it; they are 33.

Divide it by 9; it is 3. We write it and we are left with 6.

12345		12345		12345
		1		13
9		9		9

We add this 6 to the 4 that follows it; they are 64.

Divide it by 9; it is 7. We write it and we are left with 1.

We add the 1 to the 5 that follows it; they are 15.

We say: divide 15 by 9; it is 1 and we are left with 6. We write it above the 9, by which we divided the whole number.

The result is that if we divide 12345 by 9, the result of division is 2371 and 6-ninths.

	12345		12345
	137		1371
	9		9

• We will illustrate this in the calculation we wrote above:

First, we sum up the digits of the dividend and say: 6, 5, 7, 8, 9; the total is 35.

We cast out the nines; 8 remains. We write it next to the digits of the dividend.

Then, we sum up [the digits of] the result of division and say: 9, 8, 6, 4, 2; the total is 29.

We cast out the nines; 2 remains. We write it next to the result of division.

Then, we multiply it by the divisor and say: 2 times 4 is 8. Therefore, we receive the same as we received when we summed the digits of the dividend, so the calculation is correct.

• We will illustrate this in the second calculation we wrote above:

First, we sum up the digits of the dividend and say: 6, 7, 8, 9; the total is 30.

We cast out the nines; we are left with 3. We write it next to the digits of the dividend.

Then, we sum up [the digits of] the result of division and say: 5, 9, 7, 1; the total is 22.

We cast out the nines; we are left with 4. We write it next to the result of division.

Then, we multiply it by the divisor and say: 4 times 5 is 20.

We cast out the nines; we are left with 2. We add it to the 1 we received from the division, which is written above the 5; they are 3.

Therefore, we receive the same as we received when we summed the digits [of the dividend], so the calculation is correct.

• Example: we wish to divide 89765 by 12.

If we divide it by 12 once the result is 7480 and 5 parts of 12 and this is the order of the procedure:

We say: divide 8 by 12; we cannot.

We add the 8 to the 9 that follows it; they are 89.

Divide it by 12; it is 7. We write it and we are left with 5.

We add it to the 7 that follows it; they are 57.

Divide it by 12; it is 4. We write it and we are left with 9.

89765		89765		89765
		7		74
12		12		12

We add it to the 6 that follows it; they are 96.

Divide it by 12; it is 8. We write it.

We say: divide 5 by 12; we cannot.

Write a zero beneath the 5 and write the 5 above 12; it is 7780 and 5 parts of 12.

	89765		89765
	748		7480
	12		12

If we want to divide this same number in another way, we divide it by 3 once, then we divide the result by 4, so it is divided into 12 parts also.

We write this number again and divide it first by 3, then we divide it by 4.

We say: divide 8 by 3; it is 2. We write it and 2 remains.

We add it to the 9 that follows it; they are 29.

Divide it by 3; it is 9. We write it and 2 remains.

89765		89765		89765
		2		29
3		3		3

We add it to the 7 that follows it; they are 27.

We divide it by 3; it is 9. We write it.

Then, we say: divide 6 by 3; it is 2. We write it.

Then, we say: divide 5 by 3; it is 1. We write it and 2 remains. We write it above the 3, by which we divided this number.

	89765		89765		89765
	299		2992		29921
	3		3		3

Then, we divide by 4 the number we received from the division and say: divide 2 by 4; we cannot.

Add 2 to the 9 that follows it; they are 29.

Divide it by 4; it is 7. We write it and we are left with 1.

We add 1 to the 9 that follows it; they are 19.

Divide it by 4; it is 4. We write it and we are left with 3.

We add it to the 2 that follows it; they are 32.

We divide it by 4; it is 8. We write it.

	89765		89765		89765
	29921		29921		29921
	7		74		748
	4		4		4

We divide 1 by 4; we cannot. We write a zero beneath the 1 and we write the 1, which we cannot divide by 4, above the 4, by which we divide the number.

	89765
	29921
	7480
	4

So, you have divided the number by 3 and 4 and it is as if you have divided it by 12.

Multiply the 1 above the 4 by the 3 and say: 1 time 3 is 3; with the 2 above the 3, it is 5 and they are 5 parts of 12.

It follows from this that if we first divide by 3, then we divide the result by 4 as we did, the result of division is 7480 and 5 parts of 12, as we received when we divided it once by 12 and this is clear.

The Method of Knowing the Numbers Summed One after the Other Successively

• If a person says: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 - how much is the total?

Say: half 10 is 5. Multiply it by 11: 5 times 11 is 54 and so is [the sum of] the numbers mentioned.

• If he continues his question until 12.

Say: 6 times 13 is 78 and so is [their sum].

• If he continues his question until 20.

Say: 10 times 21 and so is [their sum].

• If he continues his question until one hundred.

Say: 50 times 101 and so is [their sum].

• As the one who says: I continue my calculation until 7. How much is the total?

Say: the greater half of 7 is 4. Multiply it by 7; it is 28 and so is [their sum].

• If he continues his question until 9.

Say: the greater half of 9 is 5. Multiply it by 9; it is 45 and so is [their sum].

• If he continues until 17.

Multiply 9 by 17 and so is [their sum].

• If he continues until 19.

Multiply 10 by 19 and so is [their sum].

Sum of Evens

• As 2, 4, 6, 8, 10, 12 - how much is the total?

Say: half 12 is 6 and the number that follows 6 is 7. 6 times 7 is 42 and so is [their sum].

• If he says: I continued until twenty.

Say: half twenty is 10 and the number that follows 10 is 11. 10 times [11] is 110 and so is [their sum].

Sum of Odds

• As 1, 3, 5, 7, 9 - how much is the total?

Say: the greater half of 9 is 5. 5 times 5 is 25 and so is [their sum].

• If he continues his question until 11.

Say: 6 times 6 and so is [their sum].

• If he continues until 13.

Say: 7 times 7 and so is [their sum].

Sum of Powers of Two

• As 1, 2, 4, 8, 16, 32.

• Example: one says: I summed the doubles from 1 to 32. How much is their sum?

Double 32, which is the last term; it is 64. Subtract the first term, which is 1; 63 remains and so is [their sum].

• If he says: I started from 1 and doubled until 64. How much is [their sum]?

Double 64; it is 128. Subtract one; it is 127.

• If he says: I started from one and doubled until 128. How much is [their sum]?

Double 128; it is 256. Subtract 1; it is 255.

• If he says: I started from 4 and doubled until 32. How much is their sum?

Double 32; it is 64. Subtract 4; it is 60.

Sum of Powers of Three

• As 1, 3, 9, 27.

He subtracts the first term, which is 1, from the last term, which is 27; 26 remains.

He says: 26 plus its half, which is 13, is 39. With 1, which is the first term, it is 40 and this is the [sum of] 1, 3, 9, 27.

Sum of Powers of Four

• As 1, 4, 16.

Example: 1, 4, 16 are 21.

The last term is 16. Subtract the first term, which is 1; 15 remains.

Add to 15 its third, which is 5; it is 20. Add to it [the first] term; it is 21.

Motion Problems - Pursuit

• If it is said: A man is walking ten miles a day.

Another man is walking one mile on the first day, two miles on the second day, three miles on the third day, four miles on the fourth day, and so on he goes on walking in each day [one mile more].

In how many days will they walk the same [total distance]?

ואדם אחר הולך ביום אחד א' מיל וביום ב' ב' מילי' וביום ג' ג' מילים וביום ד' ד' מילי' וכן מוסיף והולך בכל יום ויום
כמה ימים יעמדו בשוה

Double the 10 miles that the one who walks 10 miles every day walks; it is 20.

Subtract 1 from it; 19 remains.

We find that in 19 [days] they walk the same [total distance].

Check:

The one who walks 10 miles a day walks in 19 days 190 miles, as the product of 10 by 19, which is 190.

The one who adds another mile each day walks in 19 days 190 miles also: take the greater half of 19; it is 10. Multiply it by 19; it is 190.

כיצד תפוש הרוב מי"ט והם י' ותכפול אותו על י"ט ויהיה ק"צ

• If it is said: A man is walking ten miles a day.

Another man is walking two miles on the first day, four [miles] on the second day, six [miles] on the third day, eight [miles] on the fourth day, and so on he adds two [miles more each day].

In how many days will they walk the same [total distance]?

ואדם‫^[130] אחר הולך ביום אחד ב' מילין וביום שני ד' וביום שלישי ו' וביום רביעי ח' וכן לעולם הוא מוסיף ב' והולך על הסדר
בכמה ימים יעמדו בשוה

Subtract 1 from the miles that the one who walks the same distance every day walks; 9 remains.

We find that in 9 days they walk the same [total distance].

Check:

The one who walks 10 miles a day walks in 9 days 90 miles.

For the one who walks an increasing even number of miles [each day], i.e. 2, 4, 6, 8, 10, 12, 14, 16, 18:

Say: the half of 18 is 9

Multiply it by 10, which is the number that follows 9; it is 90.

We find that in 9 days they walk the same [total distance].

• If it is said: A man is walking ten miles a day.

Another man is walking one mile on the first day, three miles on the second day, five miles on the third day, seven miles on the fourth day, and so on he goes on walking [in each day] an odd number of miles.

ואדם אחר הולך בכל ביום אחד א' מיל וביום ב' ג' מילי' וביום ג' ה' מילי' וביום ד' ז' מילי' וכן מוסיף והולך לעולם החשבונות נפרדים

I.e. in 10 days.

Check:

Because, the one who walks 10 miles a day walks in 10 days 100 miles.

The one who walks 1 mile on the first day, 3 miles on the second day, and so on this way, walks 19 miles on the tenth day.

The greater half of 19 is 10.

If you multiply 10 by itself, it is 100.

Word Problems - The Rule of Three

Pricing Problems

• A man says: 3 eggs are worth 4 pešiṭim.

How many pešiṭim are 5 eggs worth at this price?

ה' ביצות לזה החשבון כמה פשיטים עולי‫'

Rule of Three: Do this way: return the calculation back: he asks if 3 is equal to 4, how much is 5 equal to?

Say: 5 times 4; divide it by 3; it is 6 and 2-thirds.

• If he asks: if 5 is equal to 7, how much is 9 equal to?

Rule of Three: Say: 9 times 7 is 63. Divide it by 5 and so it is.

חלקם בה' וכן יהיו

• If he says: 11 are worth 17

How much are 31 worth at this price?

ל"א לזה החשבון כמה שוים

Rule of Three: Say: 31 times 17; divide it by 11.

• If he says: 10 korim are worth 6 dinar.

How much are 4 korim worth at this price?

ד' כורי' לחשבון זה כמה שוי‫'

Rule of Three: Say: 4 times 6 is 24. Divide it by 10; the result is 2 and 4-tenths that are 2 dinar and 2-fifths of a dinar.

חלקם בי' יבואו ב' וד' עשיריות שהם ב' דינרי' וב' חומשי דינר

• If it is said: 3 sheaves are worth 4 dinar.

How much are 10 sheaves worth at this price?

י' עומרי' לחשבון זה כמה הם שוי‫'

Rule of Three: Say: 10 times 4 is 40. Divide it by 3; it is 13 dinar and a third of a dinar.

חלקם בג' יהיו י"ג דינרי' ושליש דינר

• Say: if it is said: I bought 10 korim of wheat for 6 dinar.

How many korim could I buy for 4 dinar?

כמה כורי' אוכל ליקח בד' דינרי‫'

Rule of Three: Say: if we find 10 korim for 6 dinar, how many korim will we find for 4 dinar?

Say: 4 times 10 is 40. Divide it by 6; it is 6 and 4-sixths, which is 6 and 2-thirds.

חלקם בו' יהיו ו' וד' ששיות שהם ו' וב' שלישיות

• If it is said: 6 korim of wheat are worth 4 dinar.

How many korim could I buy for 7 dinar?

כמה כורי אוכל לקנות בז' דינרי

Rule of Three: Say: if 4 dinar are worth 6 korim, how many korim are worth 7 dinar?

Say: 7 times 6 is 42. Divide it by 4; it is 10 korim and 2-quarters that are 10 korim and half a kor.

חלקם בד' יהיו ד' כורי י' כורי' וב' רביעיות שהם ו' כורי י' כורי' וחצי כור

Employment Problems - Payment Problems

• Example: the one who says: I hired a worker for 30 days for 10 zuzim but he worked with me 8 days.

How much should his payment be?

כמה שכרו

Rule of Three: Say: if for 30 days, his payment is 10 zuz, how much is his payment for 8 days?

Say: 8 times 10 is 80. Divide it by 30; the result is 2 zuzim and 20 parts of 30, which are 2 and 2-thirds, which are 3 dinar minus a third of a dinar.

dinar

חלקם בל' יבואו ב' זוזי' וכ' חלקי' מל' שהם ב' וב' שלישיות שהם ג' דינרי' פחות שליש דינר

Calculation of Fractions

• If you divide one into two [equal] parts, each part will be a half and the two halves are one whole.

• If you divide one into three [equal] parts, each part is a third and three thirds are one whole.

• If you divide one into four [equal] parts, each part is a quarter and four quarters are one whole.

Addition of Fractions

• If it is said: whoever has a half and a third of one thing, how much does he have of that thing?

Common denominator: Say: a half and a third are found in 2 times 3, which is 6.

Its half is 3 and its third is 2. If you sum them, they are 5. So, the one who has a half and a third of the thing, has 5 parts of 6 of that thing.

• Whoever has a third and a quarter of the thing, how much does he have of that thing?

Common denominator: Say: a third and a quarter are found in 3 times 4, which is 12.

Its third is 4 and its quarter is 3. If you sum them, they are 7. So, the one who has a third and a quarter of the thing, has 7 parts of 12 of that thing.

• Whoever has a third and a fifth of the thing.

Common denominator: Say: a third and a fifth are found in 3 times 5, which is 15.

Its third is 5 and its fifth is 3. Together they are 8. So, the one who has a third and a fifth of the thing, has 8 parts of 15 of that thing.

• Whoever has a quarter and a fifth of the thing.

Common denominator: Say: a quarter and a fifth are found in 4 times 5, which is 20.

Its quarter is 5 and its fifth is 4. Together they are 9. So, he has 9 parts of 20 of that thing.

• Whoever has a third and a seventh of the thing.

Common denominator: Say: a third and a seventh are found in 3 times 7, which is 21.

Its third is 7 and its seventh is 3. Together they are 10. So, he has 10 parts of 21.

• Whoever has a third, a quarter, a fifth, and a sixth of the thing.

Common denominator: Say: a third, a quarter, a fifth and a sixth are found in 60.

Its third is 20, its quarter is 15, its fifth is 12 and its sixth is 10. If you sum them, they are 57. So, he has 57 parts of 60 of that thing.

Subtraction of Fractions

• If a man says: by how much is the third greater than the quarter?

Common denominator: Do this way, say: a third and a quarter are found in 3 times 4, which is 12.

Its third is 4 and its quarter is 3. By how much is 4, which is the third, greater than 3, which is the quarter? By 1. So, the third is greater than the quarter by 1 part of 12.

• If a man says: by how much is the quarter greater than the fifth?

Common denominator: Say: a quarter and a fifth are found in 4 times 5, which is 20.

Its quarter is 5 and its fifth is 4. By how much is 5 greater than 4? By 1. So, the quarter is greater than the fifth by 1 part of twenty.

• If it is said: by how much is the third greater than the seventh?

Common denominator: Say: a third and a seventh are found in 3 times 7, which is 21.

Its third is 7 and its seventh is 3. By how much is 7 greater than 3? By 4. So, the third is greater than the seventh by 4 parts of 21.

• If it is said: by how much are the third and the quarter greater than the fifth and the eighth?

Common denominator: Say: a third, a quarter, a fifth and a sixth are found in 60.

Its third is 20 and its quarter is 15. If you sum them, they are 35. The fifth of 60 is 12 and its sixth is 10. If you sum them, they are 22. By how much is 35, which is the third and the quarter, greater than 22, which is the fifth and the sixth? By 13. So, the third and the quarter are greater than the fifth and the sixth by 13 parts of 60.

Fractions of Fractions

Common denominator: Say: a third and a quarter are found in 12.

Common denominator: Say: a third and a fifth are found in 3 times 5, which is 15.

Common denominator: Say: a fifth and a ninth are found in 5 times 9, which is 45.

Conversion of fractions

• How many tenths there are in three eighths

Common denominator:

Operations with fractions - numerator greater than 1

Common denominator: Say: an eighth and a tenth are found in 8 times 10, which is 80.

Common denominator: Say: a third and a seventh are found in 3 times 7, which is 21.

Common denominator: Say: a fifth and an eighth are found in 5 times 8, which is 40.

Common denominator: Say: an eighth and a fifth are found in 5 times 8, which is 40.

Common denominator: Say: an eighth and a fifth are found in 5 times 8, which is 40.

Multiplication of fractions

Proportions with fractions

Common denominator:

Rule of Three:

converting to thirds:

נחלק אח"כ ג' עשיריות בי' חלקי' ישאר ג' חלקי' ממאה‫^[147]
הרי שעולה החשבון ו' וו' עשיריות וג' חלקי' ממאה‫^[148]
וכדי שנוכל לקבץ הכל יחד ו' עשיריות חלקים ממאה ויהיו ס' חלקים מק' וג' חלקים מק' שהיו בידינו הרי ס"ג
תמצא מדברינו שאם ג' ושליש שוי' ד' ורביע ה' וחומש שוי' ו' וס"ג חלקים ממאה

Word Problems

Exchange Problems

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

ק' ליט' פיסאני כמה ליט' קוטורניסי שוים

Do it this way:

It is known that if 7 pisani are worth 9 cortonisi, then 7 dinar of Pisa are worth 9 dinar of Cortona; 7 liṭra of Pisa are worth 9 liṭra of Cortona; and 700 liṭra of Pisa are worth 900 liṭra of Cortona; and the same for any currency you want.

Rule of Three: So, this question is as if it is asked: If 700 liṭra of Pisa are worth 900 liṭra of Cortona, how much are 100 liṭra of Pisa worth?

Rule of Three: To make the calculation easier we say: if 7 is equal to 9, how much is 100 equal to?

Say: 100 times 9 is 900.

Divide it by 7; it is 128 liṭra, 11 dinar and [3-sevenths]; and this number of Cortona is equal to 100 of Pisa.

• If a man says: if 5 and a quarter of Pisa are worth 7 of Cortona, how many liṭra of Cortona are a thousand liṭra of Pisa worth?

אלף ליט' פיסאני כמה ליט מקוטורניסי שוים

You see that this calculation has a fraction in one part and this fraction is a quarter.

Therefore, we multiply both parts, which are 5 and a quarter and 7, by 4, i.e. we convert all into quarters.

Say: 4 times 5 pisani and a quarter; the result is 21 pisani.

4 times 7 cortonisi; the result is 28 cortonisi.

So, 21 pisani are worth 28 cortonosi.

To make the case easier, we reduce them by seven, because both parts have a seventh.

The seventh of 21 is 3.

The seventh of 28 is 4.

We find that 3 pisani are worth 4 cortonisi.

We want to know how much are a thousand liṭra of Pisa worth?

Rule of Three: Say: if 3 pisani are worth 4 cortonisi, how much are a thousand liṭra worth?

Say: a thousand times 4 are 4 thousand.

Divide them by 3; the result is 1333 [liṭra], 6 dinar and 8 pešuṭim and so are a thousand liṭra of Pisa worth in Cortona.

• If a man says: if 7 silver ounces and a third are worth 73 liṭra and a quarter, how much are 19 silver ounces worth?

י"ט אונקיו' מכסף כמה ישוו לפי חשבון זה

Do this way: the fractions in this calculation are a third and a quarter; a third and a quarter are found in 12.

Therefore, we have to multiply both parts, which are 7 and a third and 73 and a quarter, by 12, i.e. to convert all of them into parts of 12.

Say: 12 times 7 and a third; the result is 88 silver ounces.

12 times 73 liṭra and 5 dinar are 879 liṭra.

So, 88 silver ounces are worth 879 liṭra.

We want to know how much the 19 ounces are worth.

Rule of Three: Say: if 88 silver ounces are worth 879 liṭra, how much are 19 ounces worth?

Say: 19 times 879 is 16701.

Divide it by 88, i.e. by 8 and by 11; the result is 189 liṭra, 15 dinar, 8 pešiṭim and 2 parts of 11 of one pašuṭ and so the 19 silver ounces are worth.

Divide a Quantity Problems

• If you want to divide 5 pešiṭim into a third and a quarter without a remainder

False Position - denominator: Say: a third and a quarter are found in 3 times 4, which is 12.

The third is 4; the quarter is 3; their sum is 7 and this is the denominator.

Rule of Three: If you want to know how much is the portion of the one who has a third, say: 4 times 5, since they are 5; the result is 20. Divide them by 7; it is 2 and 6 parts of 7.

אמור ד' פעמי' ה' בעבור שהם ה' יבואו כ‫'
חלק אותם בז' יהיו ב' וו' חלקים מז‫'

Rule of Three: If you want to know the portion of the one who has a quarter, say: 3 times 5 is 15. Divide them by 7; it is 2 and one part of 7.

אמור ג' פעמי' ה' ט"ו
חלקם בז' יהיו ב' וא' חלק מז‫'

Check: If you sum 2 and 6 parts of 7 with 2 and one part of 7; they are 5 pešiṭim, as the amount of money you divided.

• If you want to divide 12 pešiṭim into a half, a third, and a quarter

False Position - denominator: Say: a half, a third and a quarter are found in 12.

The half is 6; the third is 4; the quarter is 3; their sum is 13 and this is the denominator.

If you want to know the share of the one who has a half:

Rule of Three: Say: a half of 12 is 6. Multiply it by 12, which is the amount of money; it is 72. Divide it by 13; it is 5 and 7 parts of 13.

חלקם בי"ג יהיו ה' וז' חלקים מי"ג

If you want to know the share of the one who has a third:

Rule of Three: Say: a third of 12 is 4. Multiply it by the amount of money, which is 12; it is 48. Divide it by 13; it is 3 and 9 parts of 13.

וחלקם בי"ג יהיו ג' וט' חלקים מי"ג

If you want to know the share of the one who has a quarter:

Rule of Three: Say: a quarter of 12 is 3. Multiply it by the amount of money, which is 12; it is 36. Divide it by 13; it is 2 and 10 parts of 13.

חלקם בי"ג יהיו ב' וי' חלקי' מי"ג

Check: If you sum up these three portions, they are 12, as the amount of money you divided.

• If you want to divide 12 pešiṭim into a third and a quarter

False Position - denominator: Say: a third and a quarter are found in 12.

The third is 4; the quarter is 3; their sum is 7 and this is the denominator.

If you want to know how much is [the share of] the one who has a third:

Rule of Three: Say: a third of 12 is 4. Multiply it by the amount of money, which is 12; it is 48. Divide it by 7; it is 6 and 6 parts of 7.

חלקם בז' יהיו ו' וו' חלקי' מז‫'

To know [how much is the share of] the one who has a quarter:

Rule of Three: Say: a quarter of 12 is 3. Multiply it by 12; the quarter is 36. Divide it by 7; it is 5 and 1 part of 7.

חלקם בז' יהיו ה' וא' חלק מז‫'

Check: The [sum of the] two is 12.

• If you have 30 pešiṭim and you want to divide them to a third, a quarter, a fifth, and a sixth

False Position - denominator: Say: these fractions are found in 60.

The third is 20; the quarter is 15; the fifth is 12; the sixth is 10; their sum is 57 and this is the denominator.

Rule of Three: For the one who has a third, say: a third of 60 is 20. Multiply it by 30, which is the amount of money; it is 600. Divide it by 57; the result is 10 and 30 parts of 57.

חלקי' בנ"ז יבואו י' ול' חלקי' מנ"ז

Rule of Three: For the one who has a quarter, which is 15, say: 15 times 30, is 450. Divide it by 57; it is 7 and 51 parts of 57.

חלקם בנ"ז יהיו ז' ונ"א חלקי' מנ"ז

Rule of Three: For the one who has a fifth of [60], which is 12, say: 12 times 30, is 360. Divide it by 57; the result is 6 and 18 parts of 57.

חלקם בנ"ז יבאו ו' וי"ח חלקי' מנ"ז

Rule of Three: For the one who has a sixth, which is 10, say: 10 times 30, is 300. Divide it by 57; it is 5 and 15 parts of 57.

חלקם בנ"ז יהיו ה' וט"ו חלקי' מנ"ז

Check: The total sum of all the mentioned portions is 30 pešiṭim.

• If you want to divide 60 liṭra into a half, a third, and a quarter

False Position - denominator: Say: a half, a third and a quarter are found in 12.

Their sum is 13 and this is the denominator.

If you want to know the share of the one who has a half:

Rule of Three: Say: a half of 12 is 6. Multiply it by 60, which is the number of liṭra; the result is 360. Divide them by 13; it is 27 and 9 parts of 13 of one liṭra.

חלקם בי"ג יהיו כ"ז וט' חלקים מי"ג מהליטר‫'

If you want to know the share of the one who has a third:

Rule of Three: Say: a third of 12 is 4. Multiply it by 60; it is 240. Divide them by 13; it is 18 and 6 parts of 13 of one liṭra.

חלקם בי"ג יהיו יהיו י"ח וו' חלקי' מי"ג מן הליטר‫'

If you want to know the share of the one who has a quarter:

Rule of Three: Say: a quarter of 12 is 3. Multiply it by 60; it is 180. Divide them by 13; it is 13 and 11 parts of 13 of one liṭra.

חלקם בי"ג הוא י"ג וי"א חלקי' מי"ג מן הליט‫'

1 part of 13 of one liṭra is 18 pešiṭim and [6] parts of 13.

So, the share of the one who has a half is 27 liṭra, 13 dinar, 10 pešiṭim and 2 parts of 13 of one pašuṭ.

The share of the one who has a third is 18 liṭra, 9 dinar, 2 pešiṭim and 10 parts of 13 of one pašuṭ.

The share of the one who has a quarter is 13 liṭra, 16 dinar, 11 pešiṭim and 1 part of 13 of one pašuṭ.

Check: If you sum up all the mentioned portions, they are 60 liṭra.

Partnership - for the same time

• Three people formed a partnership and contribute together 40 ounces of gold.

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

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

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

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

לאחד היה זהב שוה ג' ליטרי' האונקי‫'
ולשני זהב שוה ה' ליטרי' האונקי‫'
ולשלישי זהב שוה ח' לי' האונקי‫'
כמה זהב ישי' כל אחד שיהיו בין כלם מ' אונקיו' ויהיה חלק כלם שוה

False Position - denominator: Say: a third, a fifth, and an eighth are found is 120.