Addition and Subtraction of fractions.

To add two fractions with equal denominators, it is necessary to add their numerators, and to leave without modifications a denominator:

a	+	b	=	a + b
c		c		c

Find the sum of two fractions with equal denominator:

1	+	2	=	1 + 2	=	3
5		5		5		5

Find the sum of two fractions with equal denominator:

3	+	2	=	3 + 2	=	5
7		7		7		7

To add two fractions, it is necessary:

• to reduce fraction to a least common denominator;
• add the numerators and put the answer over the same denominator;
• simplify the fraction;
• to convert improper fractions to mixed numbers if needed.

Find the sum of two fractions:

1	+	1	=	1·2	+	1	=	2	+	1	=	2 + 1	=	3	=	3	=	1
3		6		3·2		6		6		6		6		6		3·2		2

Find the sum of two fractions:

29	+	44	=	29·3	+	44·2	=	87	+	88	=	87 + 88	=
30		45		30·3		45·2		90		90		90

=	175	=	35·5	=	35	=	18 + 17	= 1	17
	90		18·5		18		18		18

To add two mixed numbers, it is necessary:

• to reduce fraction to a least common denominator;
• add the numerators and put the answer over the same denominator;
• simplify the fraction;
• if this fraction is improper then convert fraction to a mixed number;
• add the integer portions of the two mixed numbers;
• if adding the fractional parts created a mixed number then add its integer portion to the sum.

Find the sum of two mixed numbers:

2	+	1	1	=	2·2	+	1	1·3	=	4	+	1	3	=	1 +	4 + 3	=
3			2		3·2			2·3		6			6			6

=	1 +	7	=	1 +	6 + 1	=	1 + 1	1	= 2	1
		6			6			6		6

Find the sum of two mixed numbers:

1	5	+	2	3	=	1	5·4	+	2	3·3	=	1	20	+	2	9	=	3 +	20 + 9	=
	6			8			6·4			8·3			24			24			24

=	3 +	29	=	3 +	24 + 5	=	3 + 1	5	= 4	5
		24			24			24		24

To receive a difference of fractions with equal denominatorsTo receive a difference of fractions with equal denominators, it is necessary to subtract their numerators , and to leave without modifications a denominator:

a	-	b	=	a - b
c		c		c

Find the difference between two fractions with the same denominators:

3	-	1	=	3 - 1	=	2
5		5		5		5

Find the difference between two fractions with the same denominators:

8	-	5	=	8 - 5	=	3
41		41		41		41

To subtract two fractions, it is necessary:

• to reduce fraction to a least common denominator;
• subtract the numerators and put the answer over the same denominator;
• simplify the fraction.

Find the difference between two fractions:

5	-	1	=	5	-	1·3	=	5	-	3	=	5 - 3	=	2	=	2	=	1
6		2		6		2·3		6		6		6		6		2·3		3

Find the difference between two fractions:

3	-	1	=	3·3	-	1·5	=	9	-	5	=	9 - 5	=	4	=	2·2	=	2
10		6		10·3		6·5		30		30		30		30		15·2		15

To subtract two mixed numbers, it is necessary:

• to reduce fraction to a least common denominator;
• make the first numerator larger than the second if it is not;
• subtract the second numerator from the first;
• place that difference over the common denominator;
• Subtract the integer portions of the two mixed numbers;
• simplify the fraction.

Find the difference between two mixed numbers:

2	1	-	1	1	=	2	1·3	-	1	1·2	=	(2 - 1)	+	3	-	2	=
	2			3			2·3			3·2				6		6

=	1	+	3 -2	=	1	+	1	=	1	1
			6				6			6

Find the difference between two mixed numbers:

3	1	-	1	3	=	3	1·4	-	1	3·3	=	3	4	-	1	9	=
	6			8			6·4			8·3			24			24

=	2	24 + 4	-	1	9	=	1 +	28 - 9	=	1 +	19	= 1	19
		24			24			24			24		24

Find the difference between two mixed numbers:

1	1	-	3	2	=	1	1	-	3	2·2	=	1	1	-	3	4	=	(1-3)	+	1 - 4	=
	6			3			6			3·2			6			6				6

= -2	-	3	=	-2	-	3	=	-2	-	1	=	-2	1
		6				2·3				2			2