Lesson Objectives

- Demonstrate an understanding of how to find the LCM for a group of numbers
- Learn how to add and subtract two or more fractions with the same denominator
- Learn how to find the least common denominator (LCD) for a group of fractions
- Learn how to transform fractions into equivalent fractions where the LCD is the denominator
- Learn how to add and subtract two or more fractions with different denominators

## How to Add & Subtract Fractions

When we add or subtract fractions, we must first obtain a common denominator. This means the denominators must be the same
in order to perform addition or subtraction. The easy scenario occurs when we add or subtract fractions with a common denominator present.

Example 1: Perform each indicated operation: $$\frac{2}{15} + \frac{4}{15}$$ Find the sum of the numerators:

2 + 4 = 6

Place this result (6) over the common denominator (15) $$\frac{2}{15} + \frac{4}{15} = \frac{6}{15}$$ Simplify the fraction: $$\require{cancel}\frac{6}{15} = \frac{\cancel{6}2}{\cancel{15}5} = \frac{2}{5}$$ Example 2: Perform each indicated operation: $$\frac{13}{20} - \frac{3}{20}$$ Find the difference of the numerators:

13 - 3 = 10

Place this result (10) over the common denominator (20) $$\frac{13}{20} - \frac{3}{20} = \frac{10}{20}$$ Simplify the fraction: $$\frac{10}{20} = \frac{\cancel{10}1}{\cancel{20}2} = \frac{1}{2}$$ Example 3: Perform each indicated operation: $$\frac{7}{40} + \frac{11}{40} - \frac{3}{40}$$ Perform the operations with the numerators:

7 + 11 - 3 = 18 - 3 = 15

Place this result (15) over the common denominator (40) $$\frac{7}{40} + \frac{11}{40} - \frac{3}{40} = \frac{15}{40}$$ Simplify the fraction: $$\frac{15}{40} = \frac{\cancel{15}3}{\cancel{40}8} = \frac{3}{8}$$ The more challenging scenario occurs when we add or subtract fractions without a common denominator. In this case, we must first find the LCD or least common denominator. The LCD is the LCM or least common multiple of all denominators. Once we have obtained our LCD, we will transform each fraction into an equivalent fraction where the LCD is its denominator. This can be done by multiplying the numerator and denominator by the same non-zero value needed to acquire the LCD as its denominator. We have previously learned that multiplication by this complex form of 1 does not change the value of the number since multiplication by 1 leaves the number unchanged.

Example 4: Perform each indicated operation: $$\frac{5}{6} + \frac{3}{10}$$ First, let's find the LCD for the fractions. This will be the LCM of the denominators (6, 10).

LCD = LCM(6, 10) = 30

Transform each fraction into an equivalent fraction where the LCD (30) is its denominator. For this part, think about what needs to multiply by each denominator to get a result of 30.

6 x ? = 30 » 5

10 x ? = 30 » 3

To make the process legal, we must multiply the numerator and denominator by the same value. This will be the same as multiplying by 1, which leaves the value unchanged. $$\frac{5}{6} \cdot \frac{5}{5} = \frac{25}{30}$$ $$\frac{3}{10} \cdot \frac{3}{3} = \frac{9}{30}$$ Now we can set up our problem as: $$\frac{25}{30} + \frac{9}{30}$$ Perform the indicated operation (addition) with the numerators only:

25 + 9 = 34

We place the result (34) over the common denominator (30): $$\frac{25}{30} + \frac{9}{30} = \frac{34}{30}$$ Now we can simplify the fraction: $$\frac{34}{30} = \frac{\cancel{34}17}{\cancel{30}15} = \frac{17}{15}$$ $$\frac{5}{6} + \frac{3}{10} = \frac{17}{15}$$ We can also write the result as a mixed number: $$\frac{5}{6} + \frac{3}{10} = 1\frac{2}{15}$$ Example 5: Perform each indicated operation: $$\frac{7}{10} - \frac{9}{22}$$ First, let's find the LCD for the fractions. This will be the LCM of the denominators (10, 22).

LCD = LCM(10, 22) = 110

Transform each fraction into an equivalent fraction where the LCD (110) is its denominator. For this part, think about what needs to multiply by each denominator to get a result of 110.

10 x ? = 110 » 11

22 x ? = 110 » 5

To make the process legal, we must multiply the numerator and denominator by the same value. This will be the same as multiplying by 1, which leaves the value unchanged. $$\frac{7}{10} \cdot \frac{11}{11} = \frac{77}{110}$$ $$\frac{9}{22} \cdot \frac{5}{5} = \frac{45}{110}$$ Now we can set up our problem as: $$\frac{77}{110} - \frac{45}{110}$$ Perform the indicated operation (subtraction) with the numerators only:

77 - 45 = 32

We place the result (32) over the common denominator (110): $$\frac{77}{110} - \frac{45}{110} = \frac{32}{110}$$ Now we can simplify the fraction: $$\frac{32}{110} = \frac{\cancel{32}16}{\cancel{110}55} = \frac{16}{55}$$ $$\frac{7}{10} - \frac{9}{22} = \frac{16}{55}$$ Example 6: Perform each indicated operation: $$\frac{2}{21} + \frac{14}{15} - \frac{9}{10}$$ First, let's find the LCD for the fractions. This will be the LCM of the denominators (21, 15, and 10).

LCD = LCM(10, 15, 21) = 210

Transform each fraction into an equivalent fraction where the LCD (210) is its denominator. For this part, think about what needs to multiply by each denominator to get a result of 210.

10 x ? = 210 » 21

15 x ? = 210 » 14

21 x ? = 210 » 10

To make the process legal, we must multiply the numerator and denominator by the same value. This will be the same as multiplying by 1, which leaves the value unchanged. $$\frac{2}{21} \cdot \frac{10}{10} = \frac{20}{210}$$ $$\frac{14}{15} \cdot \frac{14}{14} = \frac{196}{210}$$ $$\frac{9}{10} \cdot \frac{21}{21} = \frac{189}{210}$$ Now we can set up our problem as: $$\frac{20}{210} + \frac{196}{210} - \frac{189}{210}$$ Perform the indicated operation (addition, and then subtraction) with the numerators only:

20 + 196 - 189 = 216 - 189 = 27

We place the result (27) over the common denominator (210): $$\frac{20}{210} + \frac{196}{210} - \frac{189}{210} = \frac{27}{210}$$ Now we can simplify the fraction: $$\frac{27}{210} = \frac{\cancel{27}9}{\cancel{210}70} = \frac{9}{70}$$ $$\frac{2}{21} + \frac{14}{15} - \frac{9}{10} = \frac{9}{70}$$

### Adding & Subtracting Fractions with a Common Denominator

- Perform the indicated operation (addition or subtraction) with the numerators only
- Place the result over the common denominator
- Simplify the fraction

Example 1: Perform each indicated operation: $$\frac{2}{15} + \frac{4}{15}$$ Find the sum of the numerators:

2 + 4 = 6

Place this result (6) over the common denominator (15) $$\frac{2}{15} + \frac{4}{15} = \frac{6}{15}$$ Simplify the fraction: $$\require{cancel}\frac{6}{15} = \frac{\cancel{6}2}{\cancel{15}5} = \frac{2}{5}$$ Example 2: Perform each indicated operation: $$\frac{13}{20} - \frac{3}{20}$$ Find the difference of the numerators:

13 - 3 = 10

Place this result (10) over the common denominator (20) $$\frac{13}{20} - \frac{3}{20} = \frac{10}{20}$$ Simplify the fraction: $$\frac{10}{20} = \frac{\cancel{10}1}{\cancel{20}2} = \frac{1}{2}$$ Example 3: Perform each indicated operation: $$\frac{7}{40} + \frac{11}{40} - \frac{3}{40}$$ Perform the operations with the numerators:

7 + 11 - 3 = 18 - 3 = 15

Place this result (15) over the common denominator (40) $$\frac{7}{40} + \frac{11}{40} - \frac{3}{40} = \frac{15}{40}$$ Simplify the fraction: $$\frac{15}{40} = \frac{\cancel{15}3}{\cancel{40}8} = \frac{3}{8}$$ The more challenging scenario occurs when we add or subtract fractions without a common denominator. In this case, we must first find the LCD or least common denominator. The LCD is the LCM or least common multiple of all denominators. Once we have obtained our LCD, we will transform each fraction into an equivalent fraction where the LCD is its denominator. This can be done by multiplying the numerator and denominator by the same non-zero value needed to acquire the LCD as its denominator. We have previously learned that multiplication by this complex form of 1 does not change the value of the number since multiplication by 1 leaves the number unchanged.

### Adding & Subtracting Fractions without a Common Denominator

- Find the LCD - this is the LCM of all denominators
- Transform each fraction into an equivalent fraction where the LCD is its denominator
- Perform the indicated operation (addition or subtraction) with the numerators only
- Place the result over the common denominator
- Simplify the fraction

Example 4: Perform each indicated operation: $$\frac{5}{6} + \frac{3}{10}$$ First, let's find the LCD for the fractions. This will be the LCM of the denominators (6, 10).

LCD = LCM(6, 10) = 30

Transform each fraction into an equivalent fraction where the LCD (30) is its denominator. For this part, think about what needs to multiply by each denominator to get a result of 30.

6 x ? = 30 » 5

10 x ? = 30 » 3

To make the process legal, we must multiply the numerator and denominator by the same value. This will be the same as multiplying by 1, which leaves the value unchanged. $$\frac{5}{6} \cdot \frac{5}{5} = \frac{25}{30}$$ $$\frac{3}{10} \cdot \frac{3}{3} = \frac{9}{30}$$ Now we can set up our problem as: $$\frac{25}{30} + \frac{9}{30}$$ Perform the indicated operation (addition) with the numerators only:

25 + 9 = 34

We place the result (34) over the common denominator (30): $$\frac{25}{30} + \frac{9}{30} = \frac{34}{30}$$ Now we can simplify the fraction: $$\frac{34}{30} = \frac{\cancel{34}17}{\cancel{30}15} = \frac{17}{15}$$ $$\frac{5}{6} + \frac{3}{10} = \frac{17}{15}$$ We can also write the result as a mixed number: $$\frac{5}{6} + \frac{3}{10} = 1\frac{2}{15}$$ Example 5: Perform each indicated operation: $$\frac{7}{10} - \frac{9}{22}$$ First, let's find the LCD for the fractions. This will be the LCM of the denominators (10, 22).

LCD = LCM(10, 22) = 110

Transform each fraction into an equivalent fraction where the LCD (110) is its denominator. For this part, think about what needs to multiply by each denominator to get a result of 110.

10 x ? = 110 » 11

22 x ? = 110 » 5

To make the process legal, we must multiply the numerator and denominator by the same value. This will be the same as multiplying by 1, which leaves the value unchanged. $$\frac{7}{10} \cdot \frac{11}{11} = \frac{77}{110}$$ $$\frac{9}{22} \cdot \frac{5}{5} = \frac{45}{110}$$ Now we can set up our problem as: $$\frac{77}{110} - \frac{45}{110}$$ Perform the indicated operation (subtraction) with the numerators only:

77 - 45 = 32

We place the result (32) over the common denominator (110): $$\frac{77}{110} - \frac{45}{110} = \frac{32}{110}$$ Now we can simplify the fraction: $$\frac{32}{110} = \frac{\cancel{32}16}{\cancel{110}55} = \frac{16}{55}$$ $$\frac{7}{10} - \frac{9}{22} = \frac{16}{55}$$ Example 6: Perform each indicated operation: $$\frac{2}{21} + \frac{14}{15} - \frac{9}{10}$$ First, let's find the LCD for the fractions. This will be the LCM of the denominators (21, 15, and 10).

LCD = LCM(10, 15, 21) = 210

Transform each fraction into an equivalent fraction where the LCD (210) is its denominator. For this part, think about what needs to multiply by each denominator to get a result of 210.

10 x ? = 210 » 21

15 x ? = 210 » 14

21 x ? = 210 » 10

To make the process legal, we must multiply the numerator and denominator by the same value. This will be the same as multiplying by 1, which leaves the value unchanged. $$\frac{2}{21} \cdot \frac{10}{10} = \frac{20}{210}$$ $$\frac{14}{15} \cdot \frac{14}{14} = \frac{196}{210}$$ $$\frac{9}{10} \cdot \frac{21}{21} = \frac{189}{210}$$ Now we can set up our problem as: $$\frac{20}{210} + \frac{196}{210} - \frac{189}{210}$$ Perform the indicated operation (addition, and then subtraction) with the numerators only:

20 + 196 - 189 = 216 - 189 = 27

We place the result (27) over the common denominator (210): $$\frac{20}{210} + \frac{196}{210} - \frac{189}{210} = \frac{27}{210}$$ Now we can simplify the fraction: $$\frac{27}{210} = \frac{\cancel{27}9}{\cancel{210}70} = \frac{9}{70}$$ $$\frac{2}{21} + \frac{14}{15} - \frac{9}{10} = \frac{9}{70}$$

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