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Addition and Subtraction of Fractions
Suppose a pizza is cut into 4 equal slices
Addition and Subtraction of Fractions
Suppose a pizza is cut into 4 equal slices
Addition and Subtraction of Fractions
Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza,
Addition and Subtraction of Fraction...
Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the ...
Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the ...
Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the ...
Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the ...
Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the ...
Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the ...
Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the ...
Suppose a pizza is cut into 4 equal slices and Joe takes
one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the ...
Example A.
a. 7
12
+
Addition and Subtraction of Fractions
11
12
Example A.
a. 7
12
+ =
7 + 11
12
Addition and Subtraction of Fractions
11
12 =
Example A.
a. 7
12
+ =
7 + 11
12
18
12
Addition and Subtraction of Fractions
11
12 =
Example A.
a. 7
12
+ =
7 + 11
12
18
12
= 18/6
12/6
=
Addition and Subtraction of Fractions
11
12 =
Example A.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
Addition and Subtraction of Fractions
11
12 =
Example A.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =b.
Addition and Subtraction of Fractions
8
15
4
15
– 2
15
11
1...
Example A.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
=b.
Addition and Subtraction of Fractions
8
15
4...
Example A.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
=b.
Addition and Subtraction of Fractions
8
15
4...
Example A.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
...
Example A.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
...
Example A.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
...
Example A.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
...
Example A.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
...
Example A.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
...
Example A.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
...
Example A.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
...
Example A.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
...
Example A.
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
...
Addition and Subtraction of Fractions
c. There were 8 pizzas on the table and Bolo ate pizzas,
5
2
3
how much pizzas are l...
Addition and Subtraction of Fractions
c. There were 8 pizzas on the table and Bolo ate pizzas,
5
2
3
how much pizzas are l...
Addition and Subtraction of Fractions
c. There were 8 pizzas on the table and Bolo ate pizzas,
5
2
3
how much pizzas are l...
Addition and Subtraction of Fractions
c. There were 8 pizzas on the table and Bolo ate pizzas,
5
2
3
how much pizzas are l...
Addition and Subtraction of Fractions
c. There were 8 pizzas on the table and Bolo ate pizzas,
5
2
3
how much pizzas are l...
Addition and Subtraction of Fractions
c. There were 8 pizzas on the table and Bolo ate pizzas,
5
2
3
how much pizzas are l...
Addition and Subtraction of Fractions
c. There were 8 pizzas on the table and Bolo ate pizzas,
5
2
3
how much pizzas are l...
Addition and Subtraction of Fractions
c. There were 8 pizzas on the table and Bolo ate pizzas,
5
2
3
how much pizzas are l...
Addition and Subtraction of Fractions
c. There were 8 pizzas on the table and Bolo ate pizzas,
5
2
3
how much pizzas are l...
Addition and Subtraction of Fractions
c.
5
4
35
2
8 –
Addition and Subtraction of Fractions
c.
5
4
35
2
= 3
8 –
5
4
35
2
8 –
5
2
8 – –
5
4
Addition and Subtraction of Fractions
c.
5
4
35
2
= 3
8 –
5
4
35
2
8 –
5
2
8 – – =
5
4
5
2
5 –
5
4
Addition and Subtraction of Fractions
c.
5
4
35
2
= 3
8 –
5
4
35
2
8 –
5
2
8 – – =
5
4
5
2
5 –
5
4
=
5
21 –
5
44 +
Borrow ...
Addition and Subtraction of Fractions
c.
5
4
35
2
= 3
8 –
5
4
35
2
8 –
5
2
8 – – =
5
4
5
2
5 –
5
4
=
5
21 –
5
44 + = 5
–
5...
Addition and Subtraction of Fractions
Fractions with different denominators can’t be added directly
since the “size” of th...
Addition and Subtraction of Fractions
Fractions with different denominators can’t be added directly
since the “size” of th...
Addition and Subtraction of Fractions
Fractions with different denominators can’t be added directly
since the “size” of th...
Addition and Subtraction of Fractions
Fractions with different denominators can’t be added directly
since the “size” of th...
Addition and Subtraction of Fractions
To add them, first find the LCD of ½ and 1/3, which is 6.
We then cut each pizza int...
Addition and Subtraction of Fractions
To add them, first find the LCD of ½ and 1/3, which is 6.
We then cut each pizza int...
Addition and Subtraction of Fractions
Fractions with different denominators can’t be added directly
since the “size” of th...
We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
Addition ...
We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easie...
We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easie...
We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easie...
We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easie...
We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easie...
We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easie...
We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easie...
We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easie...
We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easie...
We need to convert fractions of different denominators to
a common denominator in order to add or subtract them.
The easie...
Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator...
Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator...
Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator...
Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator...
Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator...
Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator...
Addition and Subtraction of Fractions
Step 2: Convert each fraction to have 24 as the denominator.
For , the new numerator...
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
Addi...
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This...
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This...
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This...
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This...
We introduce the following Multiplier-Method to add or subtract
fractions to reduce the amount of repetitive copying.
This...
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
We introduce the following ...
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)
To add or subtract fraction...
Example E. a.
The LCD is 24.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions...
Example E. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
Addition and Subtraction of Fractions
Multipli...
Example E. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24
Addition and Subtraction...
Example E. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24
Addition and Subtraction...
Example E. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24
4
Addition and Subtracti...
Example E. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24
4 3
Addition and Subtrac...
Example E. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24
4 3
29
24
Addition and S...
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtracti...
( ) * 48 / 48
7
12
5
8+ – 16
9
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the resul...
( ) * 48 / 48
7
12
5
8+ – 16
94
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the resu...
( ) * 48 / 48
67
12
5
8+ – 16
94
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the res...
( ) * 48 / 48
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the r...
( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multipli...
( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 4...
( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
=
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by...
Exercise A. Calculate and simplify the answers.
1
2
3
2
+1. 2. 3. 4.5
3
1
3+
5
4
3
4
+
5
2
3
2
+
5
5
3
5–5. 6. 7. 8.6
6
5
...
Addition and Subtraction of FractionsC. Addition and Subtraction of Fractions
6
1
4
5
25. 26. 27. 28.
29. 30. 31. 32.
33....
Addition and Subtraction of Fractions
7
18
443. + 15
7
12
11
+44.
18
19
24
7
– 15
5
4
541. – 6
4
9
5
– 6
3
8
5
+ 42.
12
+
...
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1 f5 addition and subtraction of fractions

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1 f5 addition and subtraction of fractions

  1. 1. Addition and Subtraction of Fractions
  2. 2. Suppose a pizza is cut into 4 equal slices Addition and Subtraction of Fractions
  3. 3. Suppose a pizza is cut into 4 equal slices Addition and Subtraction of Fractions
  4. 4. Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Addition and Subtraction of Fractions 1 4
  5. 5. Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, Addition and Subtraction of Fractions 1 4 2 4
  6. 6. Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take + Addition and Subtraction of Fractions 1 4 2 4 1 4 2 4
  7. 7. Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take + Addition and Subtraction of Fractions 1 4 2 4 = 3 4 of the entire pizza. 1 4 2 4
  8. 8. Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take + Addition and Subtraction of Fractions 1 4 2 4 = 3 4 of the entire pizza. In picture: + = 1 4 2 4 3 4
  9. 9. Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take + Addition and Subtraction of Fractions Addition and Subtraction of Fractions With the Same Denominator 1 4 2 4 = 3 4 of the entire pizza. In picture: + = 1 4 2 4 3 4
  10. 10. Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take + Addition and Subtraction of Fractions Addition and Subtraction of Fractions With the Same Denominator To add or subtract fractions of the same denominator, keep the same denominator, add or subtract the numerators 1 4 2 4 = 3 4 of the entire pizza. In picture: ± a d b d = + = 1 4 2 4 3 4
  11. 11. Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take + Addition and Subtraction of Fractions Addition and Subtraction of Fractions With the Same Denominator To add or subtract fractions of the same denominator, keep the same denominator, add or subtract the numerators 1 4 2 4 = 3 4 of the entire pizza. In picture: ± a d b d = d + = 1 4 2 4 3 4
  12. 12. Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take + Addition and Subtraction of Fractions Addition and Subtraction of Fractions With the Same Denominator To add or subtract fractions of the same denominator, keep the same denominator, add or subtract the numerators then simplify the result. 1 4 2 4 = 3 4 of the entire pizza. In picture: ± a d b d = a ± b d + = 1 4 2 4 3 4
  13. 13. Example A. a. 7 12 + Addition and Subtraction of Fractions 11 12
  14. 14. Example A. a. 7 12 + = 7 + 11 12 Addition and Subtraction of Fractions 11 12 =
  15. 15. Example A. a. 7 12 + = 7 + 11 12 18 12 Addition and Subtraction of Fractions 11 12 =
  16. 16. Example A. a. 7 12 + = 7 + 11 12 18 12 = 18/6 12/6 = Addition and Subtraction of Fractions 11 12 =
  17. 17. Example A. a. 7 12 + = 7 + 11 12 18 12 = 3 2 18/6 12/6 = Addition and Subtraction of Fractions 11 12 =
  18. 18. Example A. a. 7 12 + = 7 + 11 12 18 12 = 3 2 18/6 12/6 = + =b. Addition and Subtraction of Fractions 8 15 4 15 – 2 15 11 12 =
  19. 19. Example A. a. 7 12 + = 7 + 11 12 18 12 = 3 2 18/6 12/6 = + = 8 + 4 – 2 15 =b. Addition and Subtraction of Fractions 8 15 4 15 – 2 15 11 12 =
  20. 20. Example A. a. 7 12 + = 7 + 11 12 18 12 = 3 2 18/6 12/6 = + = 8 + 4 – 2 15 =b. Addition and Subtraction of Fractions 8 15 4 15 – 2 15 10 15 11 12 =
  21. 21. Example A. a. 7 12 + = 7 + 11 12 18 12 = 3 2 18/6 12/6 = + = 8 + 4 – 2 15 = 2 3 =b. Addition and Subtraction of Fractions 8 15 4 15 – 2 15 10 15 11 12 =
  22. 22. Example A. a. 7 12 + = 7 + 11 12 18 12 = 3 2 18/6 12/6 = + = 8 + 4 – 2 15 = 2 3 =b. Addition and Subtraction of Fractions 8 15 4 15 – 2 15 10 15 11 12 = Subtraction of Whole Numbers with Fractions
  23. 23. Example A. a. 7 12 + = 7 + 11 12 18 12 = 3 2 18/6 12/6 = + = 8 + 4 – 2 15 = 2 3 =b. Addition and Subtraction of Fractions 8 15 4 15 – 2 15 10 15 11 12 = Subtraction of Whole Numbers with Fractions Example B. a. Bolo ate of 1 pizza, what’s left? 8 5
  24. 24. Example A. a. 7 12 + = 7 + 11 12 18 12 = 3 2 18/6 12/6 = + = 8 + 4 – 2 15 = 2 3 =b. Addition and Subtraction of Fractions 8 15 4 15 – 2 15 10 15 11 12 = Subtraction of Whole Numbers with Fractions Example B. a. Bolo ate of 1 pizza, what’s left? 8 5 5 8Treating 1 as 8 8, what’s left is: = 8 8 – 5 81 –
  25. 25. Example A. a. 7 12 + = 7 + 11 12 18 12 = 3 2 18/6 12/6 = + = 8 + 4 – 2 15 = 2 3 =b. Addition and Subtraction of Fractions 8 15 4 15 – 2 15 10 15 11 12 = Subtraction of Whole Numbers with Fractions Example B. a. Bolo ate of 1 pizza, what’s left? 8 5 5 8Treating 1 as 8 8, what’s left is: = 8 8 – 5 8 = 3 8.1 –
  26. 26. Example A. a. 7 12 + = 7 + 11 12 18 12 = 3 2 18/6 12/6 = + = 8 + 4 – 2 15 = 2 3 =b. Addition and Subtraction of Fractions 8 15 4 15 – 2 15 10 15 11 12 = Subtraction of Whole Numbers with Fractions Example B. a. Bolo ate of 1 pizza, what’s left? 8 5 5 8Treating 1 as 8 8, what’s left is: = 8 8 – 5 8 = 3 8.1 –
  27. 27. Example A. a. 7 12 + = 7 + 11 12 18 12 = 3 2 18/6 12/6 = + = 8 + 4 – 2 15 = 2 3 =b. Addition and Subtraction of Fractions 8 15 4 15 – 2 15 10 15 11 12 = Subtraction of Whole Numbers with Fractions Example B. a. Bolo ate of 1 pizza, what’s left? 8 5 5 8Treating 1 as 8 8, what’s left is: = 8 8 – 5 8 = 3 8.1 – b. There were 3 pizzas on the table and Bolo ate of 1 pizza, what’s left? 9 7
  28. 28. Example A. a. 7 12 + = 7 + 11 12 18 12 = 3 2 18/6 12/6 = + = 8 + 4 – 2 15 = 2 3 =b. Addition and Subtraction of Fractions 8 15 4 15 – 2 15 10 15 11 12 = Subtraction of Whole Numbers with Fractions Example B. a. Bolo ate of 1 pizza, what’s left? 8 5 5 8Treating 1 as 8 8, what’s left is: = 8 8 – 5 8 = 3 8.1 – Of the 3 pizzas, 2/9 is left of the eaten one, and 2 pizzas are untouched. b. There were 3 pizzas on the table and Bolo ate of 1 pizza, what’s left? 9 7
  29. 29. Example A. a. 7 12 + = 7 + 11 12 18 12 = 3 2 18/6 12/6 = + = 8 + 4 – 2 15 = 2 3 =b. Addition and Subtraction of Fractions 8 15 4 15 – 2 15 10 15 11 12 = Subtraction of Whole Numbers with Fractions Example B. a. Bolo ate of 1 pizza, what’s left? 8 5 5 8Treating 1 as 8 8, what’s left is: = 8 8 – 5 8 = 3 8.1 – Of the 3 pizzas, 2/9 is left of the eaten one, and 2 pizzas + 29 7 = 1 – 9 7are untouched. So 3 – b. There were 3 pizzas on the table and Bolo ate of 1 pizza, what’s left? 9 7
  30. 30. Example A. a. 7 12 + = 7 + 11 12 18 12 = 3 2 18/6 12/6 = + = 8 + 4 – 2 15 = 2 3 =b. Addition and Subtraction of Fractions 8 15 4 15 – 2 15 10 15 11 12 = Subtraction of Whole Numbers with Fractions Example B. a. Bolo ate of 1 pizza, what’s left? 8 5 5 8Treating 1 as 8 8, what’s left is: = 8 8 – 5 8 = 3 8.1 – Of the 3 pizzas, 2/9 is left of the eaten one, and 2 pizzas + 2 =9 7 = 1 – 9 7 9 2 2are untouched. So 3 – pizzas are left. b. There were 3 pizzas on the table and Bolo ate of 1 pizza, what’s left? 9 7
  31. 31. Addition and Subtraction of Fractions c. There were 8 pizzas on the table and Bolo ate pizzas, 5 2 3 how much pizzas are left?
  32. 32. Addition and Subtraction of Fractions c. There were 8 pizzas on the table and Bolo ate pizzas, 5 2 3 how much pizzas are left? Of the 8 pizzas, 3 are eaten with 5 left, then another of one is eaten. 5 2
  33. 33. Addition and Subtraction of Fractions c. There were 8 pizzas on the table and Bolo ate pizzas, 5 2 3 how much pizzas are left? Of the 8 pizzas, 3 are eaten with 5 left, then another of one 5 2= 8 – 3 –is eaten. 5 2 5 2 3 = 5 – 5 2So 8 –
  34. 34. Addition and Subtraction of Fractions c. There were 8 pizzas on the table and Bolo ate pizzas, 5 2 3 how much pizzas are left? Of the 8 pizzas, 3 are eaten with 5 left, then another of one 5 2= 8 – 3 –is eaten. 5 2 5 2 3 = 5 – 5 2 = 4 5 3So 8 – are left.
  35. 35. Addition and Subtraction of Fractions c. There were 8 pizzas on the table and Bolo ate pizzas, 5 2 3 how much pizzas are left? Of the 8 pizzas, 3 are eaten with 5 left, then another of one 5 2= 8 – 3 –is eaten. 5 2 5 2 3 = 5 – 5 2 = 4 5 3So 8 – are left. To add/subtract (±) mixed fractions of the same denominator, (±) the whole number parts first, then (±) the fractional parts of which it might be necessary to carry or borrow.
  36. 36. Addition and Subtraction of Fractions c. There were 8 pizzas on the table and Bolo ate pizzas, 5 2 3 how much pizzas are left? Of the 8 pizzas, 3 are eaten with 5 left, then another of one 5 2= 8 – 3 –is eaten. 5 2 5 2 3 = 5 – 5 2 = 4 5 3So 8 – are left. To add/subtract (±) mixed fractions of the same denominator, (±) the whole number parts first, then (±) the fractional parts of which it might be necessary to carry or borrow. + Example C. Calculate. a. 8 5 4 8 8 7
  37. 37. Addition and Subtraction of Fractions c. There were 8 pizzas on the table and Bolo ate pizzas, 5 2 3 how much pizzas are left? Of the 8 pizzas, 3 are eaten with 5 left, then another of one 5 2= 8 – 3 –is eaten. 5 2 5 2 3 = 5 – 5 2 = 4 5 3So 8 – are left. To add/subtract (±) mixed fractions of the same denominator, (±) the whole number parts first, then (±) the fractional parts of which it might be necessary to carry or borrow. + Example C. Calculate. a. 8 5 4 8 8 7 = 4 + 8 + 8 5 8 7 +
  38. 38. Addition and Subtraction of Fractions c. There were 8 pizzas on the table and Bolo ate pizzas, 5 2 3 how much pizzas are left? Of the 8 pizzas, 3 are eaten with 5 left, then another of one 5 2= 8 – 3 –is eaten. 5 2 5 2 3 = 5 – 5 2 = 4 5 3So 8 – are left. To add/subtract (±) mixed fractions of the same denominator, (±) the whole number parts first, then (±) the fractional parts of which it might be necessary to carry or borrow. + Example C. Calculate. a. 8 5 4 8 8 7 = 4 + 8 + 8 5 8 7 + = 12 + 8 12
  39. 39. Addition and Subtraction of Fractions c. There were 8 pizzas on the table and Bolo ate pizzas, 5 2 3 how much pizzas are left? Of the 8 pizzas, 3 are eaten with 5 left, then another of one 5 2= 8 – 3 –is eaten. 5 2 5 2 3 = 5 – 5 2 = 4 5 3So 8 – are left. To add/subtract (±) mixed fractions of the same denominator, (±) the whole number parts first, then (±) the fractional parts of which it might be necessary to carry or borrow. + Example C. Calculate. a. 8 5 4 8 8 7 = 4 + 8 + 8 5 8 7 + = 12 + 8 12 = 12 + 2 1 1 = 13 2 1
  40. 40. Addition and Subtraction of Fractions c. 5 4 35 2 8 –
  41. 41. Addition and Subtraction of Fractions c. 5 4 35 2 = 3 8 – 5 4 35 2 8 – 5 2 8 – – 5 4
  42. 42. Addition and Subtraction of Fractions c. 5 4 35 2 = 3 8 – 5 4 35 2 8 – 5 2 8 – – = 5 4 5 2 5 – 5 4
  43. 43. Addition and Subtraction of Fractions c. 5 4 35 2 = 3 8 – 5 4 35 2 8 – 5 2 8 – – = 5 4 5 2 5 – 5 4 = 5 21 – 5 44 + Borrow 1 to subtract 4/5
  44. 44. Addition and Subtraction of Fractions c. 5 4 35 2 = 3 8 – 5 4 35 2 8 – 5 2 8 – – = 5 4 5 2 5 – 5 4 = 5 21 – 5 44 + = 5 – 5 4 + 7 4 = 4 5 3 Borrow 1 to subtract 4/5
  45. 45. Addition and Subtraction of Fractions Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. c. 5 4 35 2 = 3 8 – 5 4 35 2 8 – 5 2 8 – – = 5 4 5 2 5 – 5 4 = 5 21 – 5 44 + = 5 – 5 4 + 7 4 = 4 5 3 Borrow 1 to subtract 4/5
  46. 46. Addition and Subtraction of Fractions Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example + 1 2 1 3 = ? ? c. 5 4 35 2 = 3 8 – 5 4 35 2 8 – 5 2 8 – – = 5 4 5 2 5 – 5 4 = 5 21 – 5 44 + = 5 – 5 4 + 7 4 = 4 5 3 Borrow 1 to subtract 4/5
  47. 47. Addition and Subtraction of Fractions Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example + 1 2 1 3 = ? ? To add them, first find the LCD of ½ and 1/3, which is 6. c. 5 4 35 2 = 3 8 – 5 4 35 2 8 – 5 2 8 – – = 5 4 5 2 5 – 5 4 = 5 21 – 5 44 + = 5 – 5 4 + 7 4 = 4 5 3 Borrow 1 to subtract 4/5
  48. 48. Addition and Subtraction of Fractions Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example To add them, first find the LCD of ½ and 1/3, which is 6. We then cut each pizza into 6 slices. + = ? ? 1 3 1 2 c. 5 4 35 2 = 3 8 – 5 4 35 2 8 – 5 2 8 – – = 5 4 5 2 5 – 5 4 = 5 21 – 5 44 + = 5 – 5 4 + 7 4 = 4 5 3 Borrow 1 to subtract 4/5
  49. 49. Addition and Subtraction of Fractions To add them, first find the LCD of ½ and 1/3, which is 6. We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. + = Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example ? ? 1 3 1 2 c. 5 4 35 2 = 3 8 – 5 4 35 2 8 – 5 2 8 – – = 5 4 5 2 5 – 5 4 = 5 21 – 5 44 + = 5 – 5 4 + 7 4 = 4 5 3 Borrow 1 to subtract 4/5
  50. 50. Addition and Subtraction of Fractions To add them, first find the LCD of ½ and 1/3, which is 6. We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically, 1 2 = 3 6 1 3 = 2 6 + = Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example ? ? 1 3 1 2 c. 5 4 35 2 = 3 8 – 5 4 35 2 8 – 5 2 8 – – = 5 4 5 2 5 – 5 4 = 5 21 – 5 44 + = 5 – 5 4 + 7 4 = 4 5 3 Borrow 1 to subtract 4/5
  51. 51. Addition and Subtraction of Fractions Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example To add them, first find the LCD of ½ and 1/3, which is 6. We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically, 1 2 = 3 6 1 3 = 2 6 Hence, 1 2 + 1 3 = 3 6 + 2 6 = 5 6 + 3 6 2 6 = 5 6 c. 5 4 35 2 = 3 8 – 5 4 35 2 8 – 5 2 8 – – = 5 4 5 2 5 – 5 4 = 5 21 – 5 44 + = 5 – 5 4 + 7 4 = 4 5 3 Borrow 1 to subtract 4/5
  52. 52. We need to convert fractions of different denominators to a common denominator in order to add or subtract them. Addition and Subtraction of Fractions
  53. 53. We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. Addition and Subtraction of Fractions
  54. 54. We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions
  55. 55. We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions Addition and Subtraction of Fractions With the Different Denominator (The Traditional Method)
  56. 56. We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions Addition and Subtraction of Fractions With the Different Denominator (The Traditional Method) 1. Find their LCD
  57. 57. We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions Addition and Subtraction of Fractions With the Different Denominator (The Traditional Method) 1. Find their LCD 2. Convert all the different-denominator-fractions to the have the LCD as the denominator.
  58. 58. We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions Addition and Subtraction of Fractions With the Different Denominator (The Traditional Method) 1. Find their LCD 2. Convert all the different-denominator-fractions to the have the LCD as the denominator. 3. Add and subtract the adjusted fractions then simplify the result.
  59. 59. We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions Addition and Subtraction of Fractions With the Different Denominator (The Traditional Method) 1. Find their LCD 2. Convert all the different-denominator-fractions to the have the LCD as the denominator. 3. Add and subtract the adjusted fractions then simplify the result. Example D. 5 6 3 8 +a.
  60. 60. We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions Addition and Subtraction of Fractions With the Different Denominator (The Traditional Method) 1. Find their LCD 2. Convert all the different-denominator-fractions to the have the LCD as the denominator. 3. Add and subtract the adjusted fractions then simplify the result. Example D. 5 6 3 8 +a. Step 1: To find the LCD, list the multiples of 8
  61. 61. We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions Addition and Subtraction of Fractions With the Different Denominator (The Traditional Method) 1. Find their LCD 2. Convert all the different-denominator-fractions to the have the LCD as the denominator. 3. Add and subtract the adjusted fractions then simplify the result. Example D. 5 6 3 8 +a. Step 1: To find the LCD, list the multiples of 8 which are 8, 16, 24, ..
  62. 62. We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions Addition and Subtraction of Fractions With the Different Denominator (The Traditional Method) 1. Find their LCD 2. Convert all the different-denominator-fractions to the have the LCD as the denominator. 3. Add and subtract the adjusted fractions then simplify the result. Example D. 5 6 3 8 +a. Step 1: To find the LCD, list the multiples of 8 which are 8, 16, 24, .. we see that the LCD is 24.
  63. 63. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator.
  64. 64. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. For , the new numerator is 24 * = 20, 5 6 5 6
  65. 65. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. For , the new numerator is 24 * = 20, 5 6 5 6 5 6 = 20 24 so
  66. 66. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. For , the new numerator is 24 * = 20, 5 6 5 6 5 6 = 20 24 For , the new numerator is 24 * = 9, 3 8 3 8 so
  67. 67. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. For , the new numerator is 24 * = 20, 5 6 5 6 5 6 = 20 24 For , the new numerator is 24 * = 9, 3 8 3 8 so 3 8 = 9 24so
  68. 68. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. For , the new numerator is 24 * = 20, 5 6 5 6 5 6 = 20 24 For , the new numerator is 24 * = 9, 3 8 3 8 Step 3: Add the converted fractions. so 3 8 = 9 24so
  69. 69. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. For , the new numerator is 24 * = 20, 5 6 5 6 5 6 = 20 24 For , the new numerator is 24 * = 9, 3 8 3 8 Step 3: Add the converted fractions. 5 6 3 8 + = 20 24 + 9 24 So so 3 8 = 9 24so
  70. 70. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. For , the new numerator is 24 * = 20, 5 6 5 6 5 6 = 20 24 For , the new numerator is 24 * = 9, 3 8 3 8 Step 3: Add the converted fractions. 5 6 3 8 + = 20 24 + 9 24 = 29 24 So so 3 8 = 9 24so (It’s reduced.)
  71. 71. We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. Addition and Subtraction of Fractions
  72. 72. We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. Addition and Subtraction of Fractions
  73. 73. We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2 * 5 / 5 Addition and Subtraction of Fractions
  74. 74. We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2 * 5 / 5 = 10/5 = 2, Addition and Subtraction of Fractions
  75. 75. We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = Addition and Subtraction of Fractions
  76. 76. We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3. Addition and Subtraction of Fractions
  77. 77. Addition and Subtraction of Fractions Multiplier Method (for adding and subtracting fractions) We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3. Example E. a. 5 6 3 8 +
  78. 78. Addition and Subtraction of Fractions Multiplier Method (for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD, distributive to find the numerator over the LCD for the answer. We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3. Example E. a. 5 6 3 8 +
  79. 79. Example E. a. The LCD is 24. Addition and Subtraction of Fractions Multiplier Method (for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD, distributive to find the numerator over the LCD for the answer. 5 6 3 8 + We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
  80. 80. Example E. a. The LCD is 24. Multiply the problem by 24, then divide by 24. Addition and Subtraction of Fractions Multiplier Method (for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD, distributive to find the numerator over the LCD for the answer. 5 6 3 8 + We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
  81. 81. Example E. a. The LCD is 24. Multiply the problem by 24, then divide by 24. 5 6 3 8+( ) * 24 / 24 Addition and Subtraction of Fractions Multiplier Method (for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD, distributive to find the numerator over the LCD for the answer. 5 6 3 8 + We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3.
  82. 82. Example E. a. The LCD is 24. Multiply the problem by 24, then divide by 24. 5 6 3 8+( ) * 24 / 24 Addition and Subtraction of Fractions Multiplier Method (for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD, distributive to find the numerator over the LCD for the answer. 5 6 3 8 + We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3. distribute the multiplication
  83. 83. Example E. a. The LCD is 24. Multiply the problem by 24, then divide by 24. 5 6 3 8+( ) * 24 / 24 4 Addition and Subtraction of Fractions Multiplier Method (for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD, distributive to find the numerator over the LCD for the answer. 5 6 3 8 + We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3. distribute the multiplication
  84. 84. Example E. a. The LCD is 24. Multiply the problem by 24, then divide by 24. 5 6 3 8+( ) * 24 / 24 4 3 Addition and Subtraction of Fractions Multiplier Method (for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD, distributive to find the numerator over the LCD for the answer. 5 6 3 8 + We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3. distribute the multiplication
  85. 85. Example E. a. The LCD is 24. Multiply the problem by 24, then divide by 24. 5 6 3 8+( ) * 24 / 24 4 3 29 24 Addition and Subtraction of Fractions Multiplier Method (for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD, distributive to find the numerator over the LCD for the answer. 5 6 3 8 + We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 = 24/8 = 3. = (4*5 + 3*3) / 24 = distribute the multiplication
  86. 86. Addition and Subtraction of Fractions b. 7 12 5 8 + – 16 9
  87. 87. The LCD is 48. Multiply the problem by 48, expand the multiplication, then divide the result by 48. Addition and Subtraction of Fractions b. 7 12 5 8 + – 16 9
  88. 88. ( ) * 48 / 48 7 12 5 8+ – 16 9 The LCD is 48. Multiply the problem by 48, expand the multiplication, then divide the result by 48. Addition and Subtraction of Fractions b. 7 12 5 8 + – 16 9
  89. 89. ( ) * 48 / 48 7 12 5 8+ – 16 94 The LCD is 48. Multiply the problem by 48, expand the multiplication, then divide the result by 48. Addition and Subtraction of Fractions b. 7 12 5 8 + – 16 9
  90. 90. ( ) * 48 / 48 67 12 5 8+ – 16 94 The LCD is 48. Multiply the problem by 48, expand the multiplication, then divide the result by 48. Addition and Subtraction of Fractions b. 7 12 5 8 + – 16 9
  91. 91. ( ) * 48 / 48 67 12 5 8+ – 16 94 3 The LCD is 48. Multiply the problem by 48, expand the multiplication, then divide the result by 48. Addition and Subtraction of Fractions b. 7 12 5 8 + – 16 9
  92. 92. ( ) * 48 / 48 = (4*7 + 6*5 – 3*9) / 48 67 12 5 8+ – 16 94 3 The LCD is 48. Multiply the problem by 48, expand the multiplication, then divide the result by 48. Addition and Subtraction of Fractions b. 7 12 5 8 + – 16 9
  93. 93. ( ) * 48 / 48 = (4*7 + 6*5 – 3*9) / 48 = (28 + 30 – 27) / 48 67 12 5 8+ – 16 94 3 The LCD is 48. Multiply the problem by 48, expand the multiplication, then divide the result by 48. Addition and Subtraction of Fractions b. 7 12 5 8 + – 16 9
  94. 94. ( ) * 48 / 48 = (4*7 + 6*5 – 3*9) / 48 = (28 + 30 – 27) / 48 = 67 12 5 8+ – 16 94 3 The LCD is 48. Multiply the problem by 48, expand the multiplication, then divide the result by 48. Addition and Subtraction of Fractions b. 7 12 5 8 + – 16 9 48 31 The Multiplier–Method would be the main method used for adding/subtracting fractional numbers and formulas through out this and following courses.
  95. 95. Exercise A. Calculate and simplify the answers. 1 2 3 2 +1. 2. 3. 4.5 3 1 3+ 5 4 3 4 + 5 2 3 2 + 5 5 3 5–5. 6. 7. 8.6 6 5 6 – 9 9 4 9– 1 4 7– B. Calculate by the Multiplier Method and simplify the answers. 1 2 1 3 +17. 18. 19. 20.1 2 1 3 – 2 3 3 2 + 3 4 2 5 + 5 6 4 7–21. 22. 23. 24.7 10 2 5 – 5 11 3 4+ 5 9 7 15– Addition and Subtraction of Fractions 9. 1 2 9 – 10. 1 3 8 – 11. 4 3 4– 12. 8 3 8 – 13. 11 3 5– 14. 9 3 8– 15.14 1 6 – 16. 21 9 11–5 6 81 8 5 11
  96. 96. Addition and Subtraction of FractionsC. Addition and Subtraction of Fractions 6 1 4 5 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 38.37. 39. 40. 6 5 4 7  12 7 9 5  12 5 8 3  16 5 24 7  18 5 12 7  20 3 24 11  15 7 18 5  9 4 6 1 4 3  10 7 6 1 4 5  12 5 6 1 8 3  12 1 9 5 8 7  9 2 16 1 24 5  18 7 12 1 4 5  12 7 16 1 18 5  10 7 18 5 24 7 
  97. 97. Addition and Subtraction of Fractions 7 18 443. + 15 7 12 11 +44. 18 19 24 7 – 15 5 4 541. – 6 4 9 5 – 6 3 8 5 + 42. 12 + – 5 745. – 12 7 9 1 – 6 5 8 5 + 46. 8 –3

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