Basic addition tricks , digit sum equations

1. Names of Numbers

2. Names of Numbers
For conveniences, we name some of the larger units in our
decimal (base 10) system as shown.
1,000,000’s
100,000’s
10,000’s
1,000’s
million
hundred–
thousand
ten–
thousand
thousand
100’s
10’s
1’s
hundred
tens
ones

3. Names of Numbers
For conveniences, we name some of the larger units in our
decimal (base 10) system as shown.
1,000,000’s
100,000’s
10,000’s
1,000’s
million
hundred–
thousand
ten–
thousand
thousand
1,000 = one thousand
1,000,000 = one million
1,000,000,000 = one billion
1,000,000,000,000 = one trillion
100’s
10’s
1’s
hundred
tens
ones
(three 0’s)
(six 0’s)
(nine 0’s)
(twelve 0’s)

4. Names of Numbers
For conveniences, we name some of the larger units in our
decimal (base 10) system as shown.
1,000,000’s
100,000’s
10,000’s
1,000’s
million
hundred–
thousand
ten–
thousand
thousand
1,000 = one thousand
1,000,000 = one million
1,000,000,000 = one billion
1,000,000,000,000 = one trillion
Hence 3,054,208 is
100’s
10’s
1’s
hundred
tens
ones
(three 0’s)
(six 0’s)
(nine 0’s)
(twelve 0’s)

5. Names of Numbers
For conveniences, we name some of the larger units in our
decimal (base 10) system as shown.
1,000,000’s
100,000’s
10,000’s
1,000’s
million
hundred–
thousand
ten–
thousand
thousand
100’s
10’s
1’s
hundred
tens
ones
(three 0’s)
1,000 = one thousand
(six 0’s)
1,000,000 = one million
(nine 0’s)
1,000,000,000 = one billion
1,000,000,000,000 = one trillion (twelve 0’s)
Hence 3,054,208 is “three million

6. Names of Numbers
For conveniences, we name some of the larger units in our
decimal (base 10) system as shown.
1,000,000’s
100,000’s
10,000’s
1,000’s
million
hundred–
thousand
ten–
thousand
thousand
100’s
10’s
1’s
hundred
tens
ones
(three 0’s)
1,000 = one thousand
(six 0’s)
1,000,000 = one million
(nine 0’s)
1,000,000,000 = one billion
1,000,000,000,000 = one trillion (twelve 0’s)
Hence 3,054,208 is “three million fifty four thousand

7. Names of Numbers
For conveniences, we name some of the larger units in our
decimal (base 10) system as shown.
1,000,000’s
100,000’s
10,000’s
1,000’s
million
hundred–
thousand
ten–
thousand
thousand
100’s
10’s
1’s
hundred
tens
ones
(three 0’s)
1,000 = one thousand
(six 0’s)
1,000,000 = one million
(nine 0’s)
1,000,000,000 = one billion
1,000,000,000,000 = one trillion (twelve 0’s)
Hence 3,054,208 is “three million fifty four thousand two
hundred and eight.”

8. Names of Numbers
For conveniences, we name some of the larger units in our
decimal (base 10) system as shown.
1,000,000’s
100,000’s
10,000’s
1,000’s
million
hundred–
thousand
ten–
thousand
thousand
100’s
10’s
1’s
hundred
tens
ones
(three 0’s)
1,000 = one thousand
(six 0’s)
1,000,000 = one million
(nine 0’s)
1,000,000,000 = one billion
1,000,000,000,000 = one trillion (twelve 0’s)
Hence 3,054,208 is “three million fifty four thousand two
hundred and eight.”
The number 40 is 10 times as much as 4 since the 0 shifted
the 4 to a higher value slot.

9. Names of Numbers
For conveniences, we name some of the larger units in our
decimal (base 10) system as shown.
1,000,000’s
100,000’s
10,000’s
1,000’s
million
hundred–
thousand
ten–
thousand
thousand
100’s
10’s
1’s
hundred
tens
ones
(three 0’s)
1,000 = one thousand
(six 0’s)
1,000,000 = one million
(nine 0’s)
1,000,000,000 = one billion
1,000,000,000,000 = one trillion (twelve 0’s)
Hence 3,054,208 is “three million fifty four thousand two
hundred and eight.”
The number 40 is 10 times as much as 4 since the 0 shifted
the 4 to a higher value slot.
But the number 04 is the same as 4 since the 0 to the left
indicating an empty slot so it’s valueless.

10. Addition
To “add” means to combine two quantities A and B.
The digit–sum table
(Wikipedia)

11. Addition
To “add” means to combine two quantities A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
The digit–sum table
(Wikipedia)

12. Addition
To “add” means to combine two quantities A and B.
The digit–sum table
(Wikipedia)

13. Addition
To “add” means to combine two quantities A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
The digit–sum table
(Wikipedia)

14. Addition
To “add” means to combine two quantities A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
The combined result is called the sum or the total of A and B.
The digit–sum table
(Wikipedia)

15. Addition
To “add” means to combine two quantities A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
The combined result is called the sum or the total of A and B.
A, B are called the addends and the sum is often denoted as S
i.e. A + B = S (Sum).
The digit–sum table
(Wikipedia)

16. Addition
To “add” means to combine two quantities A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
The combined result is called the sum or the total of A and B.
A, B are called the addends and the sum is often denoted as S
i.e. A + B = S (Sum).
To add two numbers,
Example A.
Add 8,978 + 657
The digit–sum table
(Wikipedia)

17. Addition
To “add” means to combine two quantities A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
The combined result is called the sum or the total of A and B.
A, B are called the addends and the sum is often denoted as S
i.e. A + B = S (Sum).
To add two numbers,
1. line up the numbers vertically
to match the place values,
Example A.
Add 8,978 + 657
8,978
+ 657
The digit–sum table
(Wikipedia)

18. Addition
To “add” means to combine two quantities A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
The combined result is called the sum or the total of A and B.
A, B are called the addends and the sum is often denoted as S
i.e. A + B = S (Sum).
To add two numbers,
1. line up the numbers vertically
to match the place values,
2. add the digits from right to left and
“carry” when necessary.
Example A.
Add 8,978 + 657
8,978
+ 657
The digit–sum table
(Wikipedia)

19. Addition
To “add” means to combine two quantities A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
The combined result is called the sum or the total of A and B.
A, B are called the addends and the sum is often denoted as S
i.e. A + B = S (Sum).
To add two numbers,
1. line up the numbers vertically
to match the place values,
2. add the digits from right to left and
“carry” when necessary.
Example A.
Add 8,978 + 657
1
8,978
+ 657
5
The digit–sum table
(Wikipedia)

20. Addition
To “add” means to combine two quantities A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
The combined result is called the sum or the total of A and B.
A, B are called the addends and the sum is often denoted as S
i.e. A + B = S (Sum).
To add two numbers,
1. line up the numbers vertically
to match the place values,
2. add the digits from right to left and
“carry” when necessary.
Example A.
Add 8,978 + 657
1 1
8,978
+ 657
35
The digit–sum table
(Wikipedia)

21. Addition
To “add” means to combine two quantities A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
The combined result is called the sum or the total of A and B.
A, B are called the addends and the sum is often denoted as S
i.e. A + B = S (Sum).
To add two numbers,
1. line up the numbers vertically
to match the place values,
2. add the digits from right to left and
“carry” when necessary.
Example A.
Add 8,978 + 657
1 1 1
8,978
+ 657
63 5
The digit–sum table
(Wikipedia)

22. Addition
To “add” means to combine two quantities A and B.
All the following words mean to “add”: total, sum, combine,
increase by, count up, aggregate, augmented by, tally, etc..
The combined result is called the sum or the total of A and B.
A, B are called the addends and the sum is often denoted as S
i.e. A + B = S (Sum).
To add two numbers,
1. line up the numbers vertically
to match the place values,
2. add the digits from right to left and
“carry” when necessary.
Example A.
Add 8,978 + 657
So the sum is 9,635.
1 1 1
8,978
+ 657
9,63 5
The digit–sum table
(Wikipedia)

23. Addition
Visually, the Mayan numerals reveal
many basic addition relations that
Arabic numerals do not.

24. Addition
Visually, the Mayan numerals reveal
many basic addition relations that
Arabic numerals do not.
vs

25. Addition
Visually, the Mayan numerals reveal
many basic addition relations that
Arabic numerals do not.
The Mayan symbols visually show us that
6=5+1, 7=5+2, 8=5+3 and 9=5+4
vs

26. Addition
Visually, the Mayan numerals reveal
many basic addition relations that
vs
Arabic numerals do not.
The Mayan symbols visually show us that
6=5+1, 7=5+2, 8=5+3 and 9=5+4 so it’s easier to “see” and
memorize adding digits with Mayan symbols:

27. Addition
Visually, the Mayan numerals reveal
many basic addition relations that
vs
Arabic numerals do not.
The Mayan symbols visually show us that
6=5+1, 7=5+2, 8=5+3 and 9=5+4 so it’s easier to “see” and
memorize adding digits with Mayan symbols:
+
5+5
= 10
=

28. Addition
Visually, the Mayan numerals reveal
many basic addition relations that
vs
Arabic numerals do not.
The Mayan symbols visually show us that
6=5+1, 7=5+2, 8=5+3 and 9=5+4 so it’s easier to “see” and
memorize adding digits with Mayan symbols:
+
5+5
= 10
=
+
=
5+6
=5+5+1
= 11

29. Addition
Visually, the Mayan numerals reveal
many basic addition relations that
vs
Arabic numerals do not.
The Mayan symbols visually show us that
6=5+1, 7=5+2, 8=5+3 and 9=5+4 so it’s easier to “see” and
memorize adding digits with Mayan symbols:
+
5+5
= 10
=
+
+
=
=
5+6
5+7
=5+5+1 =5+5+2
= 11
= 12

30. Addition
Visually, the Mayan numerals reveal
many basic addition relations that
vs
Arabic numerals do not.
The Mayan symbols visually show us that
6=5+1, 7=5+2, 8=5+3 and 9=5+4 so it’s easier to “see” and
memorize adding digits with Mayan symbols:
+
5+5
= 10
=
+
+
=
=
5+6
5+7
=5+5+1 =5+5+2
= 11
= 12
+ =
+
=
5+9
5+8
=5+5+3 =5+5+4
= 14
= 13

31. Addition
Visually, the Mayan numerals reveal
many basic addition relations that
vs
Arabic numerals do not.
The Mayan symbols visually show us that
6=5+1, 7=5+2, 8=5+3 and 9=5+4 so it’s easier to “see” and
memorize adding digits with Mayan symbols:
+
5+5
= 10
=
+
+
=
=
5+6
5+7
=5+5+1 =5+5+2
= 11
= 12
+ =
+
=
5+9
5+8
=5+5+3 =5+5+4
= 14
= 13
+ =
+
+
+
=
=
=
6+9
6+6
6+8
6+7
=5+5+2 =5+5+3 =5+5+4 =5+5+5
= 15
= 12
= 14
= 13

32. Addition
Visually, the Mayan numerals reveal
many basic addition relations that
vs
Arabic numerals do not.
The Mayan symbols visually show us that
6=5+1, 7=5+2, 8=5+3 and 9=5+4 so it’s easier to “see” and
memorize adding digits with Mayan symbols:
+
5+5
= 10
=
+
+
=
=
5+6
5+7
=5+5+1 =5+5+2
= 11
= 12
+ =
+
=
5+9
5+8
=5+5+3 =5+5+4
= 14
= 13
+ =
+
+
+
=
=
=
6+9
6+6
6+8
6+7
=5+5+2 =5+5+3 =5+5+4 =5+5+5
= 15
= 12
= 14
= 13
Tables for the addition of 7, 8 and 9 are on the next slide.

33. Addition
+
+
=
7+7
=5+5+2+2
= 14
+
=
8+8
=5+5+3+3
= 16
+
=
7+8
=5+5+2+3
= 15
+
=
7+9
=5+5+2+4
= 16
=
8+9
=5+5+3+4
= 17
(The Mayan addition table of 7,8 and 9)
+
=
9+9
=5+5+4+4
= 18

34. Addition
+
+
=
7+7
=5+5+2+2
= 14
+
=
7+8
=5+5+2+3
= 15
+
8+8
=5+5+3+3
= 16
+
=
=
7+9
=5+5+2+4
= 16
=
8+9
=5+5+3+4
= 17
+
=
9+9
=5+5+4+4
= 18
(The Mayan addition table of 7,8 and 9)
Note the pattern of adding 9 with another nonzero digit.
To get the answer, we reduce the digit by 1 and carry.
9+1 9+2
= 10 = 11
9+3 9+4 9+5 9+6 9+7 9+8
= 12 = 13 = 14 = 15 = 16 = 17
9+9
= 18

35. Addition
+

36. Addition
+

37. Addition
+
+

38. Addition
+
=
+

39. Addition
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.
+
=
+

40. Addition
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.
+
=
+
In general, if A and B are two numbers, then A + B = B + A.
and we say that “addition is commutative.”

41. Addition
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.
+
=
+
In general, if A and B are two numbers, then A + B = B + A.
and we say that “addition is commutative.”
+
+

42. Addition
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.
+
=
+
In general, if A and B are two numbers, then A + B = B + A.
and we say that “addition is commutative.”
+
+

43. Addition
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.
+
+
=
In general, if A and B are two numbers, then A + B = B + A.
and we say that “addition is commutative.”
+
+
+
+

44. Addition
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.
+
+
=
In general, if A and B are two numbers, then A + B = B + A.
and we say that “addition is commutative.”
+
+
+
=
+

45. Addition
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.
+
+
=
In general, if A and B are two numbers, then A + B = B + A.
and we say that “addition is commutative.”
If we are gathering three piles of apples, it does not matter which
two piles we group together first, i.e. (A + B) + C = A + (B + C)
where the “( )” means “do first.”
+
+
+
=
+

46. Addition
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.
+
+
=
In general, if A and B are two numbers, then A + B = B + A.
and we say that “addition is commutative.”
If we are gathering three piles of apples, it does not matter which
two piles we group together first, i.e. (A + B) + C = A + (B + C)
where the “( )” means “do first.”
+
+
+
=
We say that “addition is associative.”
+

47. Addition
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.
+
+
=
In general, if A and B are two numbers, then A + B = B + A.
and we say that “addition is commutative.”
If we are gathering three piles of apples, it does not matter which
two piles we group together first, i.e. (A + B) + C = A + (B + C)
where the “( )” means “do first.”
+
+
+
+
=
We say that “addition is associative.” The addition operation
being commutative and associative allows us to add multiple
numbers in any order and we can take advantage of that.

48. Addition
To add multiple numbers, we may first collect those digits that
sum to 5, 10 or 15 as shown here:
then sum the rest of digits.

49. Addition
To add multiple numbers, we may first collect those digits that
sum to 5, 10 or 15 as shown here:
then sum the rest of digits.
Example A. a. Calculate.
2
3
4
7
+ 8
b. Calculate.
+
11
23
38
17
5
32
12 6

50. Addition
To add multiple numbers, we may first collect those digits that
sum to 5, 10 or 15 as shown here:
1+4=2+3=5
then sum the rest of digits.
Example A. a. Calculate.
2
3
4
7
+ 8
b. Calculate.
+
11
23
38
17
5
32
12 6

51. Addition
To add multiple numbers, we may first collect those digits that
sum to 5, 10 or 15 as shown here:
1+4=2+3=5
1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10
then sum the rest of digits.
Example A. a. Calculate.
b. Calculate.
2
3
11
4
23
7
38
+ 8
17
5
32
+
12 6

52. Addition
To add multiple numbers, we may first collect those digits that
sum to 5, 10 or 15 as shown here:
1+4=2+3=5
6 + 9 = 7 + 8 = 15
1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10
then sum the rest of digits.
Example A. a. Calculate.
b. Calculate.
2
3
11
4
23
7
38
+ 8
17
5
32
+

53. Addition
To add multiple numbers, we may first collect those digits that
sum to 5, 10 or 15 as shown here:
1+4=2+3=5
6 + 9 = 7 + 8 = 15
1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10
then sum the rest of digits.
Example A. a. Calculate.
b. Calculate.
2
3
11
10
4
10
23
7
38
+ 8
17
5
32
+

54. Addition
To add multiple numbers, we may first collect those digits that
sum to 5, 10 or 15 as shown here:
1+4=2+3=5
6 + 9 = 7 + 8 = 15
1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10
then sum the rest of digits.
Example A. a. Calculate.
b. Calculate.
2
3
11
10
4
10
23
7
38
+ 8
17
24
5
32
+

55. Addition
To add multiple numbers, we may first collect those digits that
sum to 5, 10 or 15 as shown here:
1+4=2+3=5
6 + 9 = 7 + 8 = 15
1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10
then sum the rest of digits.
Example A. a. Calculate.
b. Calculate.
2
3
11
10
4
10
23
7
38
+ 8
17
or
24
5
32
2+3+4+7+8 =
+

56. Addition
To add multiple numbers, we may first collect those digits that
sum to 5, 10 or 15 as shown here:
1+4=2+3=5
6 + 9 = 7 + 8 = 15
1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10
then sum the rest of digits.
Example A. a. Calculate.
b. Calculate.
2
3
11
10
4
10
23
7
38
+ 8
17
or
24
5
32
2+3+4+7+8 =
+
5
15

57. Addition
To add multiple numbers, we may first collect those digits that
sum to 5, 10 or 15 as shown here:
1+4=2+3=5
6 + 9 = 7 + 8 = 15
1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10
then sum the rest of digits.
Example A. a. Calculate.
b. Calculate.
2
3
11
10
4
10
23
7
38
+ 8
17
or
24
5
32
2 + 3 + 4 + 7 + 8 = 24
+
5
15
20

58. Addition
To add multiple numbers, we may first collect those digits that
sum to 5, 10 or 15 as shown here:
1+4=2+3=5
6 + 9 = 7 + 8 = 15
1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10
then sum the rest of digits.
Example A. a. Calculate.
b. Calculate.
2
3
11
10
4
10
23
7
38
10
+ 8
17
or
10
24
5
32
2 + 3 + 4 + 7 + 8 = 24
+
5
15
20

59. Addition
To add multiple numbers, we may first collect those digits that
sum to 5, 10 or 15 as shown here:
1+4=2+3=5
6 + 9 = 7 + 8 = 15
1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10
then sum the rest of digits.
Example A. a. Calculate.
b. Calculate.
2
2
3
11
10
4
10
23
7
38
10
+ 8
17
or
10
24
5
32
2 + 3 + 4 + 7 + 8 = 24
+
total 26,
6
carry the 2
5
15
20

60. Addition
To add multiple numbers, we may first collect those digits that
sum to 5, 10 or 15 as shown here:
1+4=2+3=5
6 + 9 = 7 + 8 = 15
1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10
then sum the rest of digits.
Example A. a. Calculate.
b. Calculate.
2
2
3
11
10
4
10
5
23
7
38
10
+ 8
17
5
or
10
24
5
32
2 + 3 + 4 + 7 + 8 = 24
+
total 26,
6
carry the 2
5
15
20

61. Addition
To add multiple numbers, we may first collect those digits that
sum to 5, 10 or 15 as shown here:
1+4=2+3=5
6 + 9 = 7 + 8 = 15
1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10
then sum the rest of digits.
Example A. a. Calculate.
b. Calculate.
2
2
3
11
10
4
10
5
23
7
38
10
+ 8
17
5
or
10
24
5
32
2 + 3 + 4 + 7 + 8 = 24
+
total 26,
12 6
carry the 2
5
15
20

62. Addition
Given a number, the sum of all its digits is called the
“digit sum” of that number.

63. Addition
Given a number, the sum of all its digits is called the
“digit sum” of that number. For example, the digit sum of
35,328,792 is
3+5+3+2+8+7+9+2

64. Addition
Given a number, the sum of all its digits is called the
“digit sum” of that number. For example, the digit sum of
35,328,792 is
3 + 5 + 3 + 2 + 8 + 7 + 9 + 2 = 39.
10
10
10

65. Addition
Given a number, the sum of all its digits is called the
“digit sum” of that number. For example, the digit sum of
35,328,792 is
3 + 5 + 3 + 2 + 8 + 7 + 9 + 2 = 39.
10
10
10
Example B. a. The digit sum of the number 3,X52,00X is 16
where X is an unknown digit, what is the number?

66. Addition
Given a number, the sum of all its digits is called the
“digit sum” of that number. For example, the digit sum of
35,328,792 is
3 + 5 + 3 + 2 + 8 + 7 + 9 + 2 = 39.
10
10
10
Example B. a. The digit sum of the number 3,X52,00X is 16
where X is an unknown digit, what is the number?
We a are given that 3 + X + 5 + 2 + 0 + 0 + X = 16,

67. Addition
Given a number, the sum of all its digits is called the
“digit sum” of that number. For example, the digit sum of
35,328,792 is
3 + 5 + 3 + 2 + 8 + 7 + 9 + 2 = 39.
10
10
10
Example B. a. The digit sum of the number 3,X52,00X is 16
where X is an unknown digit, what is the number?
We a are given that 3 + X + 5 + 2 + 0 + 0 + X = 16,
10

68. Addition
Given a number, the sum of all its digits is called the
“digit sum” of that number. For example, the digit sum of
35,328,792 is
3 + 5 + 3 + 2 + 8 + 7 + 9 + 2 = 39.
10
10
10
Example B. a. The digit sum of the number 3,X52,00X is 16
where X is an unknown digit, what is the number?
We a are given that 3 + X + 5 + 2 + 0 + 0 + X = 16,
10
Hence we must have 10 + X + X = 16.

69. Addition
Given a number, the sum of all its digits is called the
“digit sum” of that number. For example, the digit sum of
35,328,792 is
3 + 5 + 3 + 2 + 8 + 7 + 9 + 2 = 39.
10
10
10
Example B. a. The digit sum of the number 3,X52,00X is 16
where X is an unknown digit, what is the number?
We a are given that 3 + X + 5 + 2 + 0 + 0 + X = 16,
10
Hence we must have 10 + X + X = 16.
Therefore X + X = 6 or X = 3
and the number must be 3,352,003

70. Addition
Given a number, the sum of all its digits is called the
“digit sum” of that number. For example, the digit sum of
35,328,792 is
3 + 5 + 3 + 2 + 8 + 7 + 9 + 2 = 39.
10
10
10
Example B. a. The digit sum of the number 3,X52,00X is 16
where X is an unknown digit, what is the number?
We a are given that 3 + X + 5 + 2 + 0 + 0 + X = 16,
10
Hence we must have 10 + X + X = 16.
Therefore X + X = 6 or X = 3
and the number must be 3,352,003
Qn: Is it possible for the digit sum of the number 3,X52,00X
to be 15? Explain.

71. Addition
b. The digit sum of the number 2X,58Y is 16 where X and Y
are two possibly different digits, what could the number be?

72. Addition
b. The digit sum of the number 2X,58Y is 16 where X and Y
are two possibly different digits, what could the number be?
We have that 2 + X + 5 + 8 + Y = 16.

73. Addition
b. The digit sum of the number 2X,58Y is 16 where X and Y
are two possibly different digits, what could the number be?
We have that 2 + X + 5 + 8 + Y = 16.
15

74. Addition
b. The digit sum of the number 2X,58Y is 16 where X and Y
are two possibly different digits, what could the number be?
We have that 2 + X + 5 + 8 + Y = 16.
15
Hence we must have X + Y = 1 and
X and Y must consist of a “0” and a “1”

75. Addition
b. The digit sum of the number 2X,58Y is 16 where X and Y
are two possibly different digits, what could the number be?
We have that 2 + X + 5 + 8 + Y = 16.
15
Hence we must have X + Y = 1 and
X and Y must consist of a “0” and a “1”
So the number must be 20,581 or 21,580.

76. Addition
b. The digit sum of the number 2X,58Y is 16 where X and Y
are two possibly different digits, what could the number be?
We have that 2 + X + 5 + 8 + Y = 16.
15
Hence we must have X + Y = 1 and
X and Y must consist of a “0” and a “1”
So the number must be 20,581 or 21,580.
Digit sums is one of the basic procedures used in a computer
program designed for checking to see if the transmitted data is
corrupted, i.e. transmitted incorrectly. We will use this sum
later on when we address the division operation.

77. Addition
b. The digit sum of the number 2X,58Y is 16 where X and Y
are two possibly different digits, what could the number be?
We have that 2 + X + 5 + 8 + Y = 16.
15
Hence we must have X + Y = 1 and
X and Y must consist of a “0” and a “1”
So the number must be 20,581 or 21,580.
Digit sums is one of the basic procedures used in a computer
program designed for checking to see if the transmitted data is
corrupted, i.e. transmitted incorrectly. We will use this sum
later on when we address the division operation.
We call “0” the additive identity because x + 0 = 0 + x = x.
i.e. when 0 is added with another value x, we get back x.

78. Addition
b. The digit sum of the number 2X,58Y is 16 where X and Y
are two possibly different digits, what could the number be?
We have that 2 + X + 5 + 8 + Y = 16.
15
Hence we must have X + Y = 1 and
X and Y must consist of a “0” and a “1”
So the number must be 20,581 or 21,580.
Digit sums is one of the basic procedures used in a computer
program designed for checking to see if the transmitted data is
corrupted, i.e. transmitted incorrectly. We will use this sum
later on when we address the division operation.
We call “0” the additive identity because x + 0 = 0 + x = x.
i.e. when 0 is added with another value x, we get back x.
We end this section with some step by step exercises for
mental mathematics.

79. Addition
Ideally, one should be comfortable with mental addition of two
two-digit numbers such as totaling $28 and $45. Below are a
series of mental exercises that will help one to do that.

80. Addition
Ideally, one should be comfortable with mental addition of two
two-digit numbers such as totaling $28 and $45. Below are a
series of mental exercises that will help one to do that.
Step 1. Memorize the sums of two digits.

81. Addition
Ideally, one should be comfortable with mental addition of two
two-digit numbers such as totaling $28 and $45. Below are a
series of mental exercises that will help one to do that.
Step 1. Memorize the sums of two digits.
Step 2. Practice the sum of a two-digit number with a digit;
for example, do mentally,
33 + 4 = 37
57 + 8 = 65
9 + 84 = 93

82. Addition
Ideally, one should be comfortable with mental addition of two
two-digit numbers such as totaling $28 and $45. Below are a
series of mental exercises that will help one to do that.
Step 1. Memorize the sums of two digits.
Step 2. Practice the sum of a two-digit number with a digit;
for example, do mentally,
33 + 4 = 37
57 + 8 = 65
9 + 84 = 93
Step 3. Practice the sum of a two-digit number with a
multiple of 10;

83. Addition
Ideally, one should be comfortable with mental addition of two
two-digit numbers such as totaling $28 and $45. Below are a
series of mental exercises that will help one to do that.
Step 1. Memorize the sums of two digits.
Step 2. Practice the sum of a two-digit number with a digit;
for example, do mentally,
33 + 4 = 37
57 + 8 = 65
9 + 84 = 93
Step 3. Practice the sum of a two-digit number with a
multiple of 10;
for example, do mentally,
13 + 40 = 53
29 + 90 = 119
57 + 40 = 97

84. Addition
Ideally, one should be comfortable with mental addition of two
two-digit numbers such as totaling $28 and $45. Below are a
series of mental exercises that will help one to do that.
Step 1. Memorize the sums of two digits.
Step 2. Practice the sum of a two-digit number with a digit;
for example, do mentally,
33 + 4 = 37
57 + 8 = 65
9 + 84 = 93
Step 3. Practice the sum of a two-digit number with a
multiple of 10;
for example, do mentally,
13 + 40 = 53
29 + 90 = 119
57 + 40 = 97
Step 4. Do the sum of a two-digit number in two steps by adding
the 10’s first, then add the unit-digits;

85. Addition
Ideally, one should be comfortable with mental addition of two
two-digit numbers such as totaling $28 and $45. Below are a
series of mental exercises that will help one to do that.
Step 1. Memorize the sums of two digits.
Step 2. Practice the sum of a two-digit number with a digit;
for example, do mentally,
33 + 4 = 37
57 + 8 = 65
9 + 84 = 93
Step 3. Practice the sum of a two-digit number with a
multiple of 10;
for example, do mentally,
13 + 40 = 53
29 + 90 = 119
57 + 40 = 97
Step 4. Do the sum of a two-digit number in two steps by adding
the 10’s first, then add the unit-digits; for example, do mentally,
53 + 28 =

86. Addition
Ideally, one should be comfortable with mental addition of two
two-digit numbers such as totaling $28 and $45. Below are a
series of mental exercises that will help one to do that.
Step 1. Memorize the sums of two digits.
Step 2. Practice the sum of a two-digit number with a digit;
for example, do mentally,
33 + 4 = 37
57 + 8 = 65
9 + 84 = 93
Step 3. Practice the sum of a two-digit number with a
multiple of 10;
for example, do mentally,
13 + 40 = 53
29 + 90 = 119
57 + 40 = 97
Step 4. Do the sum of a two-digit number in two steps by adding
the 10’s first, then add the unit-digits; for example, do mentally,
53 + 28 = 53 + 20 + 8

87. Addition
Ideally, one should be comfortable with mental addition of two
two-digit numbers such as totaling $28 and $45. Below are a
series of mental exercises that will help one to do that.
Step 1. Memorize the sums of two digits.
Step 2. Practice the sum of a two-digit number with a digit;
for example, do mentally,
33 + 4 = 37
57 + 8 = 65
9 + 84 = 93
Step 3. Practice the sum of a two-digit number with a
multiple of 10;
for example, do mentally,
13 + 40 = 53
29 + 90 = 119
57 + 40 = 97
Step 4. Do the sum of a two-digit number in two steps by adding
the 10’s first, then add the unit-digits; for example, do mentally,
53 + 28 = 53 + 20 + 8
= 73 + 8

88. Addition
Ideally, one should be comfortable with mental addition of two
two-digit numbers such as totaling $28 and $45. Below are a
series of mental exercises that will help one to do that.
Step 1. Memorize the sums of two digits.
Step 2. Practice the sum of a two-digit number with a digit;
for example, do mentally,
33 + 4 = 37
57 + 8 = 65
9 + 84 = 93
Step 3. Practice the sum of a two-digit number with a
multiple of 10;
for example, do mentally,
13 + 40 = 53
29 + 90 = 119
57 + 40 = 97
Step 4. Do the sum of a two-digit number in two steps by adding
the 10’s first, then add the unit-digits; for example, do mentally,
53 + 28 = 53 + 20 + 8
= 73 + 8 = 81

89. Addition
Ideally, one should be comfortable with mental addition of two
two-digit numbers such as totaling $28 and $45. Below are a
series of mental exercises that will help one to do that.
Step 1. Memorize the sums of two digits.
Step 2. Practice the sum of a two-digit number with a digit;
for example, do mentally,
33 + 4 = 37
57 + 8 = 65
9 + 84 = 93
Step 3. Practice the sum of a two-digit number with a
multiple of 10;
for example, do mentally,
13 + 40 = 53
29 + 90 = 119
57 + 40 = 97
Step 4. Do the sum of a two-digit number in two steps by adding
the 10’s first, then add the unit-digits; for example, do mentally,
97 + 55 =
53 + 28 = 53 + 20 + 8
= 73 + 8 = 81

90. Addition
Ideally, one should be comfortable with mental addition of two
two-digit numbers such as totaling $28 and $45. Below are a
series of mental exercises that will help one to do that.
Step 1. Memorize the sums of two digits.
Step 2. Practice the sum of a two-digit number with a digit;
for example, do mentally,
33 + 4 = 37
57 + 8 = 65
9 + 84 = 93
Step 3. Practice the sum of a two-digit number with a
multiple of 10;
for example, do mentally,
13 + 40 = 53
29 + 90 = 119
57 + 40 = 97
Step 4. Do the sum of a two-digit number in two steps by adding
the 10’s first, then add the unit-digits; for example, do mentally,
97 + 55 = 97 + 50 + 5
53 + 28 = 53 + 20 + 8
= 73 + 8 = 81

91. Addition
Ideally, one should be comfortable with mental addition of two
two-digit numbers such as totaling $28 and $45. Below are a
series of mental exercises that will help one to do that.
Step 1. Memorize the sums of two digits.
Step 2. Practice the sum of a two-digit number with a digit;
for example, do mentally,
33 + 4 = 37
57 + 8 = 65
9 + 84 = 93
Step 3. Practice the sum of a two-digit number with a
multiple of 10;
for example, do mentally,
13 + 40 = 53
29 + 90 = 119
57 + 40 = 97
Step 4. Do the sum of a two-digit number in two steps by adding
the 10’s first, then add the unit-digits; for example, do mentally,
97 + 55 = 97 + 50 + 5
53 + 28 = 53 + 20 + 8
= 147 + 5 = 152
= 73 + 8 = 81

92. Addition
Your Turn: Do the following mentally in two steps.
26 + 27 = 26 + 20 + 7 =
44 + 39 = 44 + 30 + 9 =
87 + 48 = 87 + 40 + 8 =

93. Addition
HW A. Fill in the values.
Number
of nickels
2
4
6
8
10
12
14
16
18
20
1
3
5
7
9
11
13
15
17
19
Sum in ¢’s
Number
of nickels
Sum in ¢’s
2. Arrange the following values from the largest to the smallest
then add them.
b. 639869
a. 513869
93036
9776
1045
10431
504
10429
837372
513896

94. Addition
Exercise B.
Do the following problems two ways.
* Add the following by summing the multiples of 10 first.
* Add by adding in the order.
to find the correct answer.
1. 3 + 5 + 7
2. 8 + 6 + 2
3. 1 + 8 + 9
4. 3 + 5 + 15
5. 9 + 14 + 6
6. 22 + 5 + 8
7. 16 + 5 + 4 + 3
8. 4 + 13 + 5 + 7
9. 19 + 7 + 1 + 3
10. 4 + 5 + 17 + 3
11. 23 + 5 + 17 + 3
12. 22 + 5 + 13 + 28
13. 35 + 6 + 15 + 7 + 14
14. 42 + 5 + 18 + 12
15. 21 + 16 + 19 + 7 + 44
16. 53 + 5 + 18 + 27 + 22
17. 155 + 16 + 25 + 7 + 344 18. 428 + 3 + 32 + 227 + 22