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. Let's begin with the easy scenario where we add or subtract fractions with a common denominator present. Recall that the denominator of a fraction shows the number of equal parts in the whole. For example, suppose Molly purchases a pizza that is cut up into 4 equal slices. Let's suppose that Molly eats 1/4 of the pizza immediately. After a few hours, Molly is hungry again. She then eats another 1/4 of the pizza. At this point, how much pizza has Molly eaten? She first ate 1/4 of the pizza and then a few hours later, another 1/4 of the same pizza. This means she ate 2/4 of the pizza, which is the same as saying she ate 1/2 of the pizza. We can achieve the same result by performing the following operation: $$\frac{1}{4}+ \frac{1}{4}=\frac{1 + 1}{4}=\frac{2}{4}=\frac{1}{2}$$

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. Let's use another example with pizzas. Jennifer and Megan order two identical pizzas. Jennifer's pizza is cut up into 4 equal pieces and Megan's pizza is cut up into 8 equal pieces. If both girls ate exactly 1 piece of their pizza, how much total pizza was eaten? Jennifer ate 1/4 of her pizza since it is 1 out of 4 slices available in the whole amount. Megan, ate 1/8 of her pizza since it is 1 out of 8 slices available in the whole amount. Both girls ate 1 slice of pizza, but since the pizzas were cut up differently, each slice is a different size. Jennifer's slice is actually larger since it is 1 out of 4 pieces, whereas Megan's slice is only 1 out of 8 pieces. In order to add these two amounts together, we need to think about how much pizza each girl ate if the pizzas had been cut up in the same way. Since the LCM of 4 and 8 is 8, we can think about each pizza being cut up into 8 slices. Megan's piece of pizza will remain the same but Jennifer's piece of pizza will turn into two slices. Now that the pizzas are cut up in the same way, we can find the sum. In other words, we are adding like parts of a whole. One slice of Jennifer's pizza is now the same as one slice of Megan's pizza. Since Jennifer ate 2/8 of a pizza and Megan ate 1/8 of a pizza, the two girls ate 3/8 of a pizza. We can achieve the same result by performing the following operations: $$\frac{1}{4}+ \frac{1}{8}$$ Convert 1/4 into an equivalent fraction where 8 is the denominator: $$\frac{1}{4}\cdot \frac{2}{2}+ \frac{1}{8}=\frac{2}{8}+ \frac{1}{8}$$ Note: 2/2 is 1 and multiplying by 1 does not change a number.

Since we have a common denominator, we can perform the addition with the numerators and place the result over the common denominator. $$\frac{2}{8}+ \frac{1}{8}=\frac{1 + 2}{8}=\frac{3}{8}$$ When we add or subtract fractions that do not have a common denominator, we will 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 number 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. Let's use another example with pizzas. Jennifer and Megan order two identical pizzas. Jennifer's pizza is cut up into 4 equal pieces and Megan's pizza is cut up into 8 equal pieces.

Since we have a common denominator, we can perform the addition with the numerators and place the result over the common denominator. $$\frac{2}{8}+ \frac{1}{8}=\frac{1 + 2}{8}=\frac{3}{8}$$ When we add or subtract fractions that do not have a common denominator, we will 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 number 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: $$\frac{27}{210}=\frac{\cancel{27}9}{\cancel{210}70}=\frac{9}{70}$$ $$\frac{2}{21}+ \frac{14}{15}- \frac{9}{10}=\frac{9}{70}$$

#### Skills Check:

Example #1

Perform each indicated operation. $$\frac{19}{30}- \frac{7}{22}$$

Please choose the best answer.

A

$$\frac{52}{165}$$

B

$$\frac{14}{47}$$

C

$$\frac{5}{13}$$

D

$$\frac{4}{3}$$

E

$$\frac{31}{85}$$

Example #2

Perform each indicated operation. $$-\frac{7}{25}+ \left(-\frac{3}{20}\right)$$

Please choose the best answer.

A

$$-\frac{17}{50}$$

B

$$-\frac{13}{25}$$

C

$$-\frac{43}{100}$$

D

$$\frac{27}{31}$$

E

$$\frac{27}{29}$$

Example #3

Perform each indicated operation. $$\frac{9}{32}+ \frac{19}{36}- \frac{3}{24}$$

Please choose the best answer.

A

$$\frac{99}{505}$$

B

$$\frac{197}{288}$$

C

$$-\frac{151}{84}$$

D

$$-\frac{17}{4}$$

E

$$\frac{3}{2}$$

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