10 arith operations
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Addition
Back to Algebra–Ready Review Content.
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AdditionTo “add” means to combine two quantities A and B.
The digit–sum table
(Wikipedia)
![Page 3: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/3.jpg)
AdditionTo “add” means to combine two quantities A and B.
The digit–sum table
(Wikipedia)
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
![Page 4: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/4.jpg)
AdditionTo “add” means to combine two quantities A and B.
The digit–sum table
(Wikipedia)
The combined result is called the sum or the total of A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).
![Page 5: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/5.jpg)
AdditionTo “add” means to combine two quantities A and B.
The digit–sum table
(Wikipedia)
To add two numbers,
Example A.
Add 8,978 + 657
The combined result is called the sum or the total of A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).
![Page 6: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/6.jpg)
AdditionTo “add” means to combine two quantities A and B.
The digit–sum table
(Wikipedia)
To add two numbers,
Example A.
Add 8,978 + 6578,978
657+
The combined result is called the sum or the total of A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).
1. line up the numbers vertically
to match the place values,
![Page 7: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/7.jpg)
AdditionTo “add” means to combine two quantities A and B.
The digit–sum table
(Wikipedia)
To add two numbers,
Example A.
Add 8,978 + 6578,978
657+
2. add the digits from right to left and
“carry” when necessary.
The combined result is called the sum or the total of A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).
1. line up the numbers vertically
to match the place values,
![Page 8: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/8.jpg)
AdditionTo “add” means to combine two quantities A and B.
The digit–sum table
(Wikipedia)
To add two numbers,
Example A.
Add 8,978 + 6578,978
657+
1
5
2. add the digits from right to left and
“carry” when necessary.
The combined result is called the sum or the total of A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).
1. line up the numbers vertically
to match the place values,
![Page 9: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/9.jpg)
AdditionTo “add” means to combine two quantities A and B.
The digit–sum table
(Wikipedia)
To add two numbers,
Example A.
Add 8,978 + 6578,978
657+
1
53
1
2. add the digits from right to left and
“carry” when necessary.
The combined result is called the sum or the total of A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).
1. line up the numbers vertically
to match the place values,
![Page 10: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/10.jpg)
AdditionTo “add” means to combine two quantities A and B.
The digit–sum table
(Wikipedia)
To add two numbers,
Example A.
Add 8,978 + 6578,978
657+
1
53
1
6
1
2. add the digits from right to left and
“carry” when necessary.
The combined result is called the sum or the total of A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).
1. line up the numbers vertically
to match the place values,
![Page 11: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/11.jpg)
AdditionTo “add” means to combine two quantities A and B.
The digit–sum table
(Wikipedia)
To add two numbers,
Example A.
Add 8,978 + 6578,978
657+
1
53
1
6
1
9,So the sum is 9,635.
2. add the digits from right to left and
“carry” when necessary.
The combined result is called the sum or the total of A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
A, B are called the addends and the sum is often denoted as S i.e. A + B = S (Sum).
1. line up the numbers vertically
to match the place values,
![Page 12: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/12.jpg)
Addition
+
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Addition
+
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Addition
+ +
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Addition
=+ +
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If we are to add two apples to a pile of three apples, the outcome
is the same as adding three apples to the pile of two apples.
Addition
+
=+
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In general, if A and B are two numbers, then A + B = B + A
and we say that “the addition operation is commutative.”
If we are to add two apples to a pile of three apples, the outcome
is the same as adding three apples to the pile of two apples.
Addition
+
=+
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In general, if A and B are two numbers, then A + B = B + A
and we say that “the addition operation is commutative.”
If we are to add two apples to a pile of three apples, the outcome
is the same as adding three apples to the pile of two apples.
Addition
=
The subtraction operation is not commutative, that is,
– –≠
In practical terms, this means that when doing addition,
we don’t care who is added to whom,
or A – B ≠ B – A
but when doing subtraction,
be sure “who” is taken away from “whom.”
+ +
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SubtractionTo subtract is to take away, or to undo an addition.
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SubtractionTo subtract is to take away, or to undo an addition.
We write “A – B” for taking the amount B away from A.
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SubtractionTo subtract is to take away, or to undo an addition.
We write “A – B” for taking the amount B away from A.
We call the outcome “the difference of A and B” and we write
A – B = D (for difference).
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SubtractionTo subtract is to take away, or to undo an addition.
The following phrases are also translated as “A – B”:
“A subtract B,” “A minus B,” “A less B,” “A is decreased or
reduced by B,” “B is subtracted, or is taken away from A.”
We write “A – B” for taking the amount B away from A.
We call the outcome “the difference of A and B” and we write
A – B = D (for difference).
![Page 23: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/23.jpg)
SubtractionTo subtract is to take away, or to undo an addition.
The following phrases are also translated as “A – B”:
“A subtract B,” “A minus B,” “A less B,” “A is decreased or
reduced by B,” “B is subtracted, or is taken away from A.”
We write “A – B” for taking the amount B away from A.
Hence the statements “five apples take away three apples,”
all mean 5 – 3
“three apples are taken away from five apples,”
“five apples minus three apples,”
= 2 .
We call the outcome “the difference of A and B” and we write
A – B = D (for difference).
![Page 24: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/24.jpg)
SubtractionTo subtract is to take away, or to undo an addition.
If “who is taken away from whom” is not specified, then it is
assumed that we are taking the smaller number away from the
bigger one. So “the difference between $10 and $50” is
50 –10 = $40. (In fact, we can’t do 10 – 50, yet.)
The following phrases are also translated as “A – B”:
“A subtract B,” “A minus B,” “A less B,” “A is decreased or
reduced by B,” “B is subtracted, or is taken away from A.”
We write “A – B” for taking the amount B away from A.
Hence the statements “five apples take away three apples,”
all mean 5 – 3
“three apples are taken away from five apples”
“five apples minus three apples,”
= 2 .
We call the outcome “the difference of A and B” and we write
A – B = D (for difference).
![Page 25: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/25.jpg)
SubtractionTo subtract,
1. lineup the numbers vertically,
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SubtractionTo subtract,
1. lineup the numbers vertically,
For example, 634 – 87 is: 6 3 48 7–
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Subtraction
For example, 634 – 87 is: 6 3 48 7–
To subtract,
1. lineup the numbers vertically,
2. subtract the digits from right to left and “borrow” when it
is necessary.
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Subtraction
For example, 634 – 87 is: 6 3 48 7–
To subtract,
1. lineup the numbers vertically,
2. subtract the digits from right to left and “borrow” when it
is necessary. need to borrow
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Subtraction
For example, 634 – 87 is: 6 3 48 7–
To subtract,
1. lineup the numbers vertically,
2. subtract the digits from right to left and “borrow” when it
is necessary. need to borrow
142
7
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Subtraction
For example, 634 – 87 is: 6 3 48 7–
To subtract,
1. lineup the numbers vertically,
2. subtract the digits from right to left and “borrow” when it
is necessary. need to borrow
142
7
12
5
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Subtraction
For example, 634 – 87 is: 6 3 48 7–
To subtract,
1. lineup the numbers vertically,
2. subtract the digits from right to left and “borrow” when it
is necessary. need to borrow
142
7
12
5
45
![Page 32: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/32.jpg)
Subtraction
For example, 634 – 87 is: 6 3 48 7–
To subtract,
1. lineup the numbers vertically,
2. subtract the digits from right to left and “borrow” when it
is necessary. need to borrow
142
7
12
5
45
When reading mathematical expressions or translating real life problems involving subtraction into mathematics, always ask the question “who subtracts whom?”, answer it clearly, then proceed.
![Page 33: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/33.jpg)
Subtraction
For example, 634 – 87 is: 6 3 48 7–
To subtract,
1. lineup the numbers vertically,
2. subtract the digits from right to left and “borrow” when it
is necessary. need to borrow
142
7
12
5
45
Example A.
The store price of a Thingamajig is $500. How much money do we save if we buy one for $400 online?
When reading mathematical expressions or translating real life problems involving subtraction into mathematics, always ask the question “who subtracts whom?”, answer it clearly, then proceed.
![Page 34: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/34.jpg)
Subtraction
For example, 634 – 87 is: 6 3 48 7–
To subtract,
1. lineup the numbers vertically,
2. subtract the digits from right to left and “borrow” when it
is necessary. need to borrow
142
7
12
5
45
Example A.
The store price of a Thingamajig is $500. How much money do we save if we buy one for $400 online?
The amount saved is: the expensive price – the cheaper price, so we saved 500 – 400 = $100.
When reading mathematical expressions or translating real life problems involving subtraction into mathematics, always ask the question “who subtracts whom?”, answer it clearly, then proceed.
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SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd
floor. After two hours, we were at the 67th floor.
108th floortop
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SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd
floor. After two hours, we were at the 67th floor.
108th floortop
1st hr 42th floor
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SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd
floor. After two hours, we were at the 67th floor.
108th floortop
1st hr 42th floor
2nd hr 67th floor
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SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd
floor. After two hours, we were at the 67th floor. a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?
108th floortop
1st hr 42th floor
2nd hr 67th floor
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SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd
floor. After two hours, we were at the 67th floor.
After the 1st hour, we still have 108 – 42 = 66 floors to the top.
a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?
108th floortop
1st hr 42th floor
2nd hr 67th floor
![Page 40: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/40.jpg)
SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd
floor. After two hours, we were at the 67th floor.
After the 1st hour, we still have 108 – 42 = 66 floors to the top.
a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?
108th floortop
1st hr 42th floor
2nd hr 67th floor
During the 2nd hour we climbed from the 42nd floor to the 67th
floor hence we climbed 67 – 42 = 25 floors during the 2nd hour.
![Page 41: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/41.jpg)
SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd
floor. After two hours, we were at the 67th floor.
After the 1st hour, we still have 108 – 42 = 66 floors to the top.
a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?
108th floortop
1st hr 42th floor
2nd hr 67th floor
During the 2nd hour we climbed from the 42nd floor to the 67th
floor hence we climbed 67 – 42 = 25 floors during the 2nd hour.
b. We are on the Nth floor, how many floors are wefrom the 108th floor? Write the answer as a subtraction.
![Page 42: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/42.jpg)
SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd
floor. After two hours, we were at the 67th floor.
After the 1st hour, we still have 108 – 42 = 66 floors to the top.
a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?
108th floortop
1st hr 42th floor
2nd hr 67th floor
During the 2nd hour we climbed from the 42nd floor to the 67th
floor hence we climbed 67 – 42 = 25 floors during the 2nd hour.
b. We are on the Nth floor, how many floors are wefrom the 108th floor? Write the answer as a subtraction.
Nth fl.
108th fl.
![Page 43: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/43.jpg)
SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd
floor. After two hours, we were at the 67th floor.
After the 1st hour, we still have 108 – 42 = 66 floors to the top.
a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?
108th floortop
1st hr 42th floor
2nd hr 67th floor
During the 2nd hour we climbed from the 42nd floor to the 67th
floor hence we climbed 67 – 42 = 25 floors during the 2nd hour.
b. We are on the Nth floor, how many floors are wefrom the 108th floor? Write the answer as a subtraction.
Nth fl.
108th fl.
?
![Page 44: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/44.jpg)
SubtractionExample B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd
floor. After two hours, we were at the 67th floor.
After the 1st hour, we still have 108 – 42 = 66 floors to the top.
a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?
108th floortop
1st hr 42th floor
2nd hr 67th floor
During the 2nd hour we climbed from the 42nd floor to the 67th
floor hence we climbed 67 – 42 = 25 floors during the 2nd hour.
b. We are on the Nth floor, how many floors are wefrom the 108th floor? Write the answer as a subtraction.
We are on the Nth floor out of total 108 floors, so the number of remaining floors to the topis “108 – N” as shown. (Not “N – 108”!)
Nth fl.
108th fl.
108 – N
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We simplify the notation for adding the same quantity
repeatedly.
Multiplication
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We simplify the notation for adding the same quantity
repeatedly.
2 + 2 + 2 = 6
3 copies
as 3 x 2 or 3*2 or 3(2) = 6.
For example, we shall write
Multiplication
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We simplify the notation for adding the same quantity
repeatedly.
We call this operation multiplication and we say that
“3 times 2 is 6” or “3 multiplied with 2 is 6”.
2 + 2 + 2 = 6
3 copies
as 3 x 2 or 3*2 or 3(2) = 6.
For example, we shall write
Multiplication
![Page 48: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/48.jpg)
We simplify the notation for adding the same quantity
repeatedly.
We call this operation multiplication and we say that
“3 times 2 is 6” or “3 multiplied with 2 is 6”.
Note that 3 copies = 2 copies
so that 3 x 2 = 2 x 3.
2 + 2 + 2 = 6
3 copies
as 3 x 2 or 3*2 or 3(2) = 6.
For example, we shall write
Multiplication
![Page 49: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/49.jpg)
We simplify the notation for adding the same quantity
repeatedly.
We call this operation multiplication and we say that
“3 times 2 is 6” or “3 multiplied with 2 is 6”.
Note that 3 copies = 2 copies
so that 3 x 2 = 2 x 3.
2 + 2 + 2 = 6
3 copies
as 3 x 2 or 3*2 or 3(2) = 6.
For example, we shall write
In general, just as addition,
Multiplication s commutative, i.e. A x B = B x A.
Multiplication
![Page 50: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/50.jpg)
We simplify the notation for adding the same quantity
repeatedly.
We call this operation multiplication and we say that
“3 times 2 is 6” or “3 multiplied with 2 is 6”.
Note that 3 copies = 2 copies
so that 3 x 2 = 2 x 3.
2 + 2 + 2 = 6
3 copies
as 3 x 2 or 3*2 or 3(2) = 6.
For example, we shall write
In the expression: 3 x 2 = 2 x 3 = 6
In general, just as addition,
Multiplication s commutative, i.e. A x B = B x A.
Multiplication
![Page 51: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/51.jpg)
We simplify the notation for adding the same quantity
repeatedly.
We call this operation multiplication and we say that
“3 times 2 is 6” or “3 multiplied with 2 is 6”.
the multiplicands 2 and 3are called factors (of 6).
Note that 3 copies = 2 copies
so that 3 x 2 = 2 x 3.
2 + 2 + 2 = 6
3 copies
as 3 x 2 or 3*2 or 3(2) = 6.
For example, we shall write
In the expression: 3 x 2 = 2 x 3 = 6
In general, just as addition,
Multiplication s commutative, i.e. A x B = B x A.
Multiplication
![Page 52: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/52.jpg)
We simplify the notation for adding the same quantity
repeatedly.
We call this operation multiplication and we say that
“3 times 2 is 6” or “3 multiplied with 2 is 6”.
the multiplicands 2 and 3are called factors (of 6).
the result 6 is
called the product
(of 2 and 3).
Note that 3 copies = 2 copies
so that 3 x 2 = 2 x 3.
2 + 2 + 2 = 6
3 copies
as 3 x 2 or 3*2 or 3(2) = 6.
For example, we shall write
In the expression: 3 x 2 = 2 x 3 = 6
In general, just as addition,
Multiplication s commutative, i.e. A x B = B x A.
Multiplication
![Page 53: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/53.jpg)
We simplify the notation for adding the same quantity
repeatedly.
We call this operation multiplication and we say that
“3 times 2 is 6” or “3 multiplied with 2 is 6”.
the multiplicands 2 and 3are called factors (of 6).
the result 6 is
called the product
(of 2 and 3).
Note that 3 copies = 2 copies
so that 3 x 2 = 2 x 3.
2 + 2 + 2 = 6
3 copies
as 3 x 2 or 3*2 or 3(2) = 6.
For example, we shall write
In the expression: 3 x 2 = 2 x 3 = 6
(Note: 1 and 6 are also factors of 6 because 1 x 6 = 6 x 1 = 6.)
In general, just as addition,
Multiplication s commutative, i.e. A x B = B x A.
Multiplication
![Page 54: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/54.jpg)
The multiplication table shown
here is to be memorized and
below are some features and
tricks that might help.
Multiplication
![Page 55: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/55.jpg)
* (0 * x = 0 * x = 0)
The product of zero with any
number is 0.
The multiplication table shown
here is to be memorized and
below are some features and
tricks that might help.
Multiplication
![Page 56: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/56.jpg)
* (0 * x = 0 * x = 0)
The product of zero with any
number is 0.
* (1 * x = x * 1 = x)
The product of 1 with any
number x is x.
The multiplication table shown
here is to be memorized and
below are some features and
tricks that might help.
Multiplication
![Page 57: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/57.jpg)
* For the products with 9 as a factor, the sum of their digits is 9.
* (0 * x = 0 * x = 0)
The product of zero with any
number is 0.
* (1 * x = x * 1 = x)
The product of 1 with any
number x is x.
The multiplication table shown
here is to be memorized and
below are some features and
tricks that might help.
Multiplication
![Page 58: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/58.jpg)
* For the products with 9 as a factor, the sum of their digits is 9.
6 x 9 = 54 7 x 9 = 63 8 x 9 = 72 9 x 9 = 81
For example,
* (0 * x = 0 * x = 0)
The product of zero with any
number is 0.
* (1 * x = x * 1 = x)
The product of 1 with any
number x is x.
The multiplication table shown
here is to be memorized and
below are some features and
tricks that might help.
all have digit sum equal to 9,
Multiplication
![Page 59: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/59.jpg)
* For the products with 9 as a factor, the sum of their digits is 9.
6 x 9 = 54 7 x 9 = 63 8 x 9 = 72 9 x 9 = 81
For example,
i.e. 5 + 4 = 9,
* (0 * x = 0 * x = 0)
The product of zero with any
number is 0.
* (1 * x = x * 1 = x)
The product of 1 with any
number x is x.
The multiplication table shown
here is to be memorized and
below are some features and
tricks that might help.
all have digit sum equal to 9,
Multiplication
![Page 60: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/60.jpg)
* For the products with 9 as a factor, the sum of their digits is 9.
6 x 9 = 54 7 x 9 = 63 8 x 9 = 72 9 x 9 = 81
For example,
i.e. 5 + 4 = 9, 6 + 3 = 9
* (0 * x = 0 * x = 0)
The product of zero with any
number is 0.
* (1 * x = x * 1 = x)
The product of 1 with any
number x is x.
The multiplication table shown
here is to be memorized and
below are some features and
tricks that might help.
7 + 2 = 9, 8 + 1 = 9
all have digit sum equal to 9,
Multiplication
![Page 61: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/61.jpg)
* The following 4-digit numbers
represent the products of the
higher digits 6 thru 9, the more
difficult part of the table:
Multiplication
![Page 62: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/62.jpg)
6636 6742 6848 6954
7749 7856 7963
8864 8972
9981
* The following 4-digit numbers
represent the products of the
higher digits 6 thru 9, the more
difficult part of the table:
Multiplication
![Page 63: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/63.jpg)
6636 6742 6848 6954
7749 7856 7963
8864 8972
9981
* The following 4-digit numbers
represent the products of the
higher digits 6 thru 9, the more
difficult part of the table:
6 x 7 = 42 (= 7 x 6)
For example,
7 x 8 = 56 (= 8 x 7).
Multiplication
![Page 64: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/64.jpg)
6636 6742 6848 6954
7749 7856 7963
8864 8972
9981
* The following 4-digit numbers
represent the products of the
higher digits 6 thru 9, the more
difficult part of the table:
6 x 7 = 42 (= 7 x 6)
For example,
The numbers with 2 as a factor: 0, 2, 4, 6, 8,…etcare called even numbers.
7 x 8 = 56 (= 8 x 7).
Multiplication
![Page 65: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/65.jpg)
6636 6742 6848 6954
7749 7856 7963
8864 8972
9981
* The following 4-digit numbers
represent the products of the
higher digits 6 thru 9, the more
difficult part of the table:
6 x 7 = 42 (= 7 x 6)
For example,
The numbers with 2 as a factor: 0, 2, 4, 6, 8,…etcare called even numbers.
The numbers 0(= 0*0), 1(= 1*1), 4(= 2*2), 9(= 3*3), 16(= 4*4),.., of the form x*x, down the diagonal, are called square numbers.
7 x 8 = 56 (= 8 x 7).
Multiplication
![Page 66: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/66.jpg)
The Vertical Format
We use a vertical format to multiply larger numbers.
The following demonstrates how this is done.
Multiplication
![Page 67: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/67.jpg)
The Vertical Format
We use a vertical format to multiply larger numbers.
The following demonstrates how this is done.
We start with a two-digit number times a single digit number.
Multiplication
![Page 68: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/68.jpg)
The Vertical Format
47
7x
For example,
We use a vertical format to multiply larger numbers.
The following demonstrates how this is done.
We start with a two-digit number times a single digit number.
Multiplication
![Page 69: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/69.jpg)
The Vertical Format
47
i. Starting from the right,
multiply the two unit-digits,
7x
For example,
We use a vertical format to multiply larger numbers.
The following demonstrates how this is done.
We start with a two-digit number times a single digit number.
record the unit-digit of the product,
and carry the 10’s digit of the product.
Multiplication
![Page 70: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/70.jpg)
The Vertical Format
47
i. Starting from the right,
multiply the two unit-digits,
7x
For example,
We use a vertical format to multiply larger numbers.
The following demonstrates how this is done.
We start with a two-digit number times a single digit number.
i. 4x7=28
record the unit-digit of the product,
and carry the 10’s digit of the product.
Multiplication
![Page 71: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/71.jpg)
The Vertical Format
47
i. Starting from the right,
multiply the two unit-digits,
7x8
record
the 8,
carry
the 2
For example,
We use a vertical format to multiply larger numbers.
The following demonstrates how this is done.
We start with a two-digit number times a single digit number.
i. 4x7=28
record the unit-digit of the product,
and carry the 10’s digit of the product.
Multiplication
![Page 72: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/72.jpg)
The Vertical Format
47
i. Starting from the right,
multiply the two unit-digits,
ii. Multiply the next digit of the double
digit number to the single digit,
7x8
record
the 8,
carry
the 2
For example,
We use a vertical format to multiply larger numbers.
The following demonstrates how this is done.
We start with a two-digit number times a single digit number.
i. 4x7=28
record the unit-digit of the product,
and carry the 10’s digit of the product.
Multiplication
![Page 73: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/73.jpg)
The Vertical Format
47
i. Starting from the right,
multiply the two unit-digits,
ii. Multiply the next digit of the double
digit number to the single digit,
7x8
record
the 8,
carry
the 2
For example,
We use a vertical format to multiply larger numbers.
The following demonstrates how this is done.
We start with a two-digit number times a single digit number.
i. 4x7=28 ii. 7x7=49,record the unit-digit of the product,
and carry the 10’s digit of the product.
Multiplication
![Page 74: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/74.jpg)
The Vertical Format
47
i. Starting from the right,
multiply the two unit-digits,
ii. Multiply the next digit of the double
digit number to the single digit,
7x8
record
the 8,
carry
the 2
For example,
We use a vertical format to multiply larger numbers.
The following demonstrates how this is done.
We start with a two-digit number times a single digit number.
i. 4x7=28 ii. 7x7=49,
49+2=51
add the previous carry to the product,
record the unit-digit of the product,
and carry the 10’s digit of the product.
Multiplication
![Page 75: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/75.jpg)
The Vertical Format
47
i. Starting from the right,
multiply the two unit-digits,
ii. Multiply the next digit of the double
digit number to the single digit,
7x8
record
the 8,
carry
the 2
For example,
We use a vertical format to multiply larger numbers.
The following demonstrates how this is done.
We start with a two-digit number times a single digit number.
i. 4x7=28 ii. 7x7=49,
1
record
the 1,
5
carry
the 5
49+2=51
add the previous carry to the product,
record the unit-digit of this sum and
carry the 10’s digit of this sum.
record the unit-digit of the product,
and carry the 10’s digit of the product.
Multiplication
![Page 76: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/76.jpg)
The Vertical Format
47
i. Starting from the right,
multiply the two unit-digits,
ii. Multiply the next digit of the double
digit number to the single digit,
7x8
record
the 8,
carry
the 2
For example,
We use a vertical format to multiply larger numbers.
The following demonstrates how this is done.
We start with a two-digit number times a single digit number.
i. 4x7=28 ii. 7x7=49,
1
record
the 1,
5
carry
the 5
49+2=51
add the previous carry to the product,
record the unit-digit of this sum and
carry the 10’s digit of this sum.
record the unit-digit of the product,
and carry the 10’s digit of the product.
To multiply a longer number against a
single digit number, repeat step ii until
all the digits are multiplied.
Multiplication
![Page 77: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/77.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
47
7x
9
Multiplication
6
![Page 78: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/78.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
47
7x
9For example,
Multiplication
6
![Page 79: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/79.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
47
7x
8
record
the 8
carry
the 2
4x7=28
9For example,
Multiplication
6
![Page 80: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/80.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
47
7x
8
record
the 8
carry
the 2
4x7=28 7x7=49,
49+2=51
9For example,
Multiplication
6
![Page 81: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/81.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
47
7x
8
record
the 8
carry
the 2
4x7=28 7x7=49,
1
record
the 1
49+2=51
9For example,
Multiplication carry
the 5
6
![Page 82: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/82.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
47
7x
8
record
the 8
carry
the 2
4x7=28 7x7=49,
1
record
the 1
carry
the 5
49+2=51
9
9x7=63,
63+5= 68
For example,
Multiplication
6
![Page 83: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/83.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
47
7x
8
record
the 8
carry
the 2
4x7=28 7x7=49,
1
record
the 1
carry
the 5
49+2=51
9
9x7=63,
63+5= 68
8
record
the 8
carry
the 6
6
For example,
Multiplication
6
![Page 84: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/84.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
47
7x
8
record
the 8
carry
the 2
4x7=28 7x7=49,
1
record
the 1
carry
the 5
49+2=51
9
9x7=63,
63+5= 68
8
record
the 8
carry
the 6
6When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example,
Multiplication
6
![Page 85: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/85.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
47
7x
8
record
the 8
carry
the 2
4x7=28 7x7=49,
1
record
the 1
carry
the 5
49+2=51
9
9x7=63,
63+5= 68
8
record
the 8
carry
the 6
6When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example,
Because we are in a
place value system, the
result of the multiplication
must be placed in the correct slots,
so it is shift one place to the left.
Multiplication
6
![Page 86: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/86.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example,
Because we are in a
place value system, the
result of the multiplication
must be placed in the correct slots,
so it is shift one place to the left.
47
7
8
record
the 8
1
record
the 1
9
8
record
the 8
carry
the 6
6
6
Multiplication
x
![Page 87: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/87.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example,
Because we are in a
place value system, the
result of the multiplication
must be placed in the correct slots,
so it is shift one place to the left.
47
7
8
record
the 8
4x6=24
1
record
the 1
9
8
record
the 8
carry
the 6
6
6
Multiplication
x
![Page 88: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/88.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example,
Because we are in a
place value system, the
result of the multiplication
must be placed in the correct slots,
so it is shift one place to the left.
47
7
8
record
the 8
4x6=24
1
record
the 1
←record
9
8
record
the 8
carry
the 6
6
6
carry
the 2
4
Multiplication
x
![Page 89: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/89.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example,
Because we are in a
place value system, the
result of the multiplication
must be placed in the correct slots,
so it is shift one place to the left.
47
7
8
record
the 8
4x6=24 7x6=42,
1
record
the 1
←record
42+2=44
9
8
record
the 8
carry
the 6
6
6
carry
the 2
4
Multiplication
x
![Page 90: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/90.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example,
Because we are in a
place value system, the
result of the multiplication
must be placed in the correct slots,
so it is shift one place to the left.
47
7
8
record
the 8
carry
the 4
4x6=24 7x6=42,
1
record
the 1
←record
42+2=44
9
8
record
the 8
carry
the 6
6
6
carry
the 2
44
Multiplication
x
![Page 91: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/91.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example,
Because we are in a
place value system, the
result of the multiplication
must be placed in the correct slots,
so it is shift one place to the left.
47
7
8
record
the 8
carry
the 4
4x6=24 7x6=42,
1
record
the 1
←record
42+2=44
9
9x6=54
54+4= 58
8
record
the 8
carry
the 6
6
6
carry
the 2
44
Multiplication
x
![Page 92: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/92.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example,
Because we are in a
place value system, the
result of the multiplication
must be placed in the correct slots,
so it is shift one place to the left.
47
7
8
record
the 8
carry
the 4
4x6=24 7x6=42,
1
record
the 1
←record
42+2=44
9
9x6=54
54+4= 58
8
record
the 8
carry
the 6
6
6
carry
the 2
4485
Multiplication
x
![Page 93: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/93.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example,
Because we are in a
place value system, the
result of the multiplication
must be placed in the correct slots,
so it is shift one place to the left.
47
7
8
record
the 8
carry
the 4
4x6=24 7x6=42,
1
record
the 1
←record
42+2=44
9
9x6=54
54+4= 58
8
record
the 8
carry
the 6
6
6
carry
the 2
Finally, we obtain the answer
by adding the two rows.
4485
Multiplication
x
![Page 94: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/94.jpg)
We treat the multiplication of two
multiple digit numbers as separate
problems of multiplying with a
single digit number.
we start the multiplication as
before by multiplying the top
with the bottom unit-digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example,
Because we are in a
place value system, the
result of the multiplication
must be placed in the correct slots,
so it is shift one place to the left.
47
7
8
carry
the 4
4x6=24 7x6=42,
1
←record
42+2=44
9
9x6=54
54+4= 58
86
6
carry
the 2
Finally, we obtain the answer
by adding the two rows.
4485
8526 5
Multiplication
+
x
![Page 95: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/95.jpg)
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
Division
![Page 96: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/96.jpg)
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
Division
![Page 97: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/97.jpg)
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
Division
![Page 98: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/98.jpg)
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
Division
![Page 99: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/99.jpg)
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.
Division
![Page 100: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/100.jpg)
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.
In general, the expression
T ÷ D = Q
Division
![Page 101: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/101.jpg)
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.
In general, the expression
T ÷ D = Q
The total T is the dividend,
Division
![Page 102: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/102.jpg)
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.
In general, the expression
T ÷ D = Q
The total T is the dividend,
The number of parts D is the divisor.
Division
![Page 103: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/103.jpg)
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.
In general, the expression
T ÷ D = Q
The total T is the dividend,
The number of parts D is the divisor.
Q is the quotient.
Division
![Page 104: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/104.jpg)
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.
In general, the expression
T ÷ D = Q
says that “if T is divided into D equal parts, then each part is Q.”
The total T is the dividend,
The number of parts D is the divisor.
Q is the quotient.
Division
![Page 105: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/105.jpg)
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.
In general, the expression
T ÷ D = Q
says that “if T is divided into D equal parts, then each part is Q.”
The total T is the dividend,
The number of parts D is the divisor.
Q is the quotient.
If T ÷ D = Q then T = D x Q or that D and Q are factors of T,
Division
![Page 106: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/106.jpg)
Division is the operation of dividing a given amount into a
prescribed number of equal parts.
For example, if three people share a dozen apples, then each
person gets four apples and there is no leftovers.
In this case, we say that “12 divides evenly by 3”.
We write this as “12 ÷ 3 = 4” which translates into
“if 12 is divided into 3 equal parts, then each part is 4”.
In general, the expression
T ÷ D = Q
says that “if T is divided into D equal parts, then each part is Q.”
The total T is the dividend,
The number of parts D is the divisor.
Q is the quotient.
If T ÷ D = Q then T = D x Q or that D and Q are factors of T,
e.g. 12 ÷ 3 = 4 so 12 = 3(4), so both 3 and 4 are factors of 12.
Division
![Page 107: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/107.jpg)
The Vertical Format Division
![Page 108: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/108.jpg)
We demonstrate the vertical long-division format below.
The Vertical Format Division
![Page 109: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/109.jpg)
We demonstrate the vertical long-division format below.
The Vertical Format
Steps. i. (Front-in Back-out)
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front-
number (the dividend) inside the
scaffold.
Division
![Page 110: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/110.jpg)
We demonstrate the vertical long-division format below.
The Vertical Format
Example C. a. Write 6 ÷ 2 as Steps. i. (Front-in Back-out)
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front-
number (the dividend) inside the
scaffold.“back-one”
outside)2 6
“front-one”
inside
Division
![Page 111: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/111.jpg)
We demonstrate the vertical long-division format below.
The Vertical Format
Example C. a. Write 6 ÷ 2 as
ii. Enter the quotient on top,
Steps. i. (Front-in Back-out)
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front-
number (the dividend) inside the
scaffold.“back-one”
outside)2 6
“front-one”
inside
Division
![Page 112: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/112.jpg)
We demonstrate the vertical long-division format below.
The Vertical Format
Example C. a. Write 6 ÷ 2 as
ii. Enter the quotient on top,
Steps. i. (Front-in Back-out)
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front-
number (the dividend) inside the
scaffold.“back-one”
outside)2 6
“front-one”
inside
Enter the quotient on top
3
Division
![Page 113: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/113.jpg)
We demonstrate the vertical long-division format below.
The Vertical Format
Example C. a. Write 6 ÷ 2 as
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend.)
Steps. i. (Front-in Back-out)
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front-
number (the dividend) inside the
scaffold.“back-one”
outside)2 6
“front-one”
inside
Enter the quotient on top
3
Division
![Page 114: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/114.jpg)
We demonstrate the vertical long-division format below.
The Vertical Format
Example C. a. Write 6 ÷ 2 as
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend.)
Steps. i. (Front-in Back-out)
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front-
number (the dividend) inside the
scaffold.“back-one”
outside)2 6
“front-one”
inside
Enter the quotient on top
3
multiply the quotient
back into the scaffold.
63 x 2
Division
![Page 115: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/115.jpg)
We demonstrate the vertical long-division format below.
The Vertical Format
Example C. a. Write 6 ÷ 2 as
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend.)
Steps. i. (Front-in Back-out)
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front-
number (the dividend) inside the
scaffold.“back-one”
outside)2 6
“front-one”
inside
Enter the quotient on top
3
multiply the quotient
back into the scaffold.
63 x 2 0
The new dividend is 0,
Division
![Page 116: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/116.jpg)
We demonstrate the vertical long-division format below.
The Vertical Format
Example C. a. Write 6 ÷ 2 as
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend.)
Steps. i. (Front-in Back-out)
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front-
number (the dividend) inside the
scaffold.“back-one”
outside)2 6
“front-one”
inside
Enter the quotient on top
3
iii. If the new dividend is not
enough to be divided by the divisor,
stop. This is the remainder R.
Otherwise, repeat steps i and ii.
multiply the quotient
back into the scaffold.
63 x 2 0
The new dividend is 0,
Division
![Page 117: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/117.jpg)
We demonstrate the vertical long-division format below.
The Vertical Format
Example C. a. Write 6 ÷ 2 as
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend.)
Steps. i. (Front-in Back-out)
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front-
number (the dividend) inside the
scaffold.“back-one”
outside)2 6
“front-one”
inside
Enter the quotient on top
3
iii. If the new dividend is not
enough to be divided by the divisor,
stop. This is the remainder R.
Otherwise, repeat steps i and ii.
multiply the quotient
back into the scaffold.
63 x 2 0
The new dividend is 0, not
enough to be divided again,
stop. This is the remainder R.
Division
![Page 118: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/118.jpg)
We demonstrate the vertical long-division format below.
The Vertical Format
Example C. a. Write 6 ÷ 2 as
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend.)
Steps. i. (Front-in Back-out)
Put the problem in the long division
format with the back-number (the
divisor) outside, and the front-
number (the dividend) inside the
scaffold.“back-one”
outside)2 6
“front-one”
inside
Enter the quotient on top
3
iii. If the new dividend is not
enough to be divided by the divisor,
stop. This is the remainder R.
Otherwise, repeat steps i and ii.
multiply the quotient
back into the scaffold.
63 x 2 0
The new dividend is 0, not
enough to be divided again,
stop. This is the remainder R.
So the remainder R is 0 and
we have that 6 ÷ 2 = 3 evenly.
Division
![Page 119: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/119.jpg)
b. Carry out the long division 7 ÷ 3.Division
![Page 120: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/120.jpg)
b. Carry out the long division 7 ÷ 3.
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the back-
number (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.
Division
![Page 121: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/121.jpg)
b. Carry out the long division 7 ÷ 3.
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the back-
number (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.
“back-one”
outside )3 7
“front-one”
inside
Division
![Page 122: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/122.jpg)
b. Carry out the long division 7 ÷ 3.
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the back-
number (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.
“back-one”
outside )3 7
“front-one”
inside
Enter the quotient on top
2
Division
![Page 123: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/123.jpg)
b. Carry out the long division 7 ÷ 3.
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend).
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the back-
number (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.
“back-one”
outside )3 7
“front-one”
inside
Division
Enter the quotient on top
2
![Page 124: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/124.jpg)
b. Carry out the long division 7 ÷ 3.
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend).
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the back-
number (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.
“back-one”
outside )3 7
“front-one”
inside
Enter the quotient on top
2
multiply the quotient
back into the scaffold.
62 x 3
1
Division
![Page 125: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/125.jpg)
b. Carry out the long division 7 ÷ 3.
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend).
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the back-
number (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.
“back-one”
outside )3 7
“front-one”
inside
Enter the quotient on top
2
iii. If the new dividend is not
enough to be divided by the
divisor, stop. This is the remainder.
Otherwise, repeat steps i and ii.
multiply the quotient
back into the scaffold.
62 x 3
1
Division
![Page 126: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/126.jpg)
b. Carry out the long division 7 ÷ 3.
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend).
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the back-
number (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.
“back-one”
outside )3 7
“front-one”
inside
Enter the quotient on top
2
iii. If the new dividend is not
enough to be divided by the
divisor, stop. This is the remainder.
Otherwise, repeat steps i and ii.
multiply the quotient
back into the scaffold.
62 x 3
1
The new dividend is 1, not
enough to be divided again, so
stop. This is the remainder.
Division
![Page 127: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/127.jpg)
b. Carry out the long division 7 ÷ 3.
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend).
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the back-
number (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.
“back-one”
outside )3 7
“front-one”
inside
Enter the quotient on top
2
iii. If the new dividend is not
enough to be divided by the
divisor, stop. This is the remainder.
Otherwise, repeat steps i and ii.
multiply the quotient
back into the scaffold.
62 x 3
1
The new dividend is 1, not
enough to be divided again, so
stop. This is the remainder.
So the remainder is 1 and
we have that 7 ÷ 3 = 2 with R = 1.
Division
![Page 128: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/128.jpg)
b. Carry out the long division 7 ÷ 3.
ii. Enter the quotient on top,
Multiply the quotient back into the
problem and subtract the results
from the dividend (and bring down
the rest of the digits, if any. This is
the new dividend).
Steps. i. (Front-in Back-out)
Put the problem in the long
division format with the back-
number (the divisor) outside, and
the front-number (the dividend)
inside the scaffold.
“back-one”
outside )3 7
“front-one”
inside
Enter the quotient on top
2
iii. If the new dividend is not
enough to be divided by the
divisor, stop. This is the remainder.
Otherwise, repeat steps i and ii.
multiply the quotient
back into the scaffold.
62 x 3
1
The new dividend is 1, not
enough to be divided again, so
stop. This is the remainder.
So the remainder is 1 and
we have that 7 ÷ 3 = 2 with R = 1.
Put the result in the multiplicative form, we have that
7 = 2 x 3 + 1.
Division
![Page 129: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/129.jpg)
Division
)3 7 7 4 3 1 7
c. Divide 74317 ÷ 37.
Find the Q and R.
![Page 130: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/130.jpg)
Division
)3 7 7 4 3 1 7
i. Starting from the left,
37 goes into 74 twice. 2
c. Divide 74317 ÷ 37.
Find the Q and R.
![Page 131: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/131.jpg)
Division
)3 7 7 4 3 1 7
i. Starting from the left,
37 goes into 74 twice.
ii. Subtract 2x37.
2
c. Divide 74317 ÷ 37.
Find the Q and R.
7 4
![Page 132: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/132.jpg)
Division
)3 7 7 4 3 1 7
i. Starting from the left,
37 goes into 74 twice.
ii. Subtract 2x37.
3 1 7iii. Bring down the rest of
the digits, this is the new
dividend.
2
c. Divide 74317 ÷ 37.
Find the Q and R.
7 4
![Page 133: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/133.jpg)
Division
)3 7 7 4 3 1 7
i. Starting from the left,
37 goes into 74 twice.
ii. Subtract 2x37.
3 1 7iii. Bring down the rest of
the digits, this is the new
dividend.
2
c. Divide 74317 ÷ 37.
Find the Q and R.
iv. We need the entire 317
to be divided by 37.
7 4
![Page 134: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/134.jpg)
Division
)3 7 7 4 3 1 7
i. Starting from the left,
37 goes into 74 twice.
ii. Subtract 2x37.
3 1 7iii. Bring down the rest of
the digits, this is the new
dividend.
2
c. Divide 74317 ÷ 37.
Find the Q and R.
iv. We need the entire 317
to be divided by 37.
v. The two skipped-spaces
must be filled by two “0’s”.
7 4
0 0
![Page 135: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/135.jpg)
Division
)3 7 7 4 3 1 7
i. Starting from the left,
37 goes into 74 twice.
ii. Subtract 2x37.
3 1 7iii. Bring down the rest of
the digits, this is the new
dividend.
2
c. Divide 74317 ÷ 37.
Find the Q and R.
iv. We need the entire 317
to be divided by 37.
v. The two skipped-spaces
must be filled by two “0’s”.
7 4
80 0
One checks that
the quotient is 8.
![Page 136: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/136.jpg)
Division
)3 7 7 4 3 1 7
i. Starting from the left,
37 goes into 74 twice.
ii. Subtract 2x37.
3 1 7iii. Bring down the rest of
the digits, this is the new
dividend.
vi. Continue, subtract 8x37=296
2
c. Divide 74317 ÷ 37.
Find the Q and R.
iv. We need the entire 317
to be divided by 37.
v. The two skipped-spaces
must be filled by two “0’s”.
7 4
80 0
2 9 6
One checks that
the quotient is 8.
![Page 137: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/137.jpg)
Division
)3 7 7 4 3 1 7
i. Starting from the left,
37 goes into 74 twice.
ii. Subtract 2x37.
3 1 7iii. Bring down the rest of
the digits, this is the new
dividend.
vi. Continue, subtract
8x37=296 so R=21,
which is not enough to be divided by 37, so stop.
2
c. Divide 74317 ÷ 37.
Find the Q and R.
iv. We need the entire 317
to be divided by 37.
v. The two skipped-spaces
must be filled by two “0’s”.
7 4
80 0
2 9 62 1
One checks that
the quotient is 8.
![Page 138: 10 arith operations](https://reader034.fdocuments.in/reader034/viewer/2022051314/55a26e101a28ab880a8b460c/html5/thumbnails/138.jpg)
Division
)3 7 7 4 3 1 7
i. Starting from the left,
37 goes into 74 twice.
ii. Subtract 2x37.
3 1 7iii. Bring down the rest of
the digits, this is the new
dividend.
vi. Continue, subtract
8x37=296 so R=21,
which is not enough to be divided by 37, so stop.
2
Hence 74317 ÷ 37 = 2008 with R = 21,
or that 74317 = 2008(37) + 21.
c. Divide 74317 ÷ 37.
Find the Q and R.
iv. We need the entire 317
to be divided by 37.
v. The two skipped-spaces
must be filled by two “0’s”.
7 4
80 0
2 9 62 1
One checks that
the quotient is 8.