**What is a mixed number?**

A mixed number is a number expressed as the combination of a whole number and fraction. In general, it denotes a value between two whole numbers.

Consider the mixed number $2\frac{3}{5}$.

In this example,

- 2 is the whole number
- 3 is the numerator of the fraction; and
- 5 is the denominator of the fraction.

**How to convert an improper fraction to a mixed number?**

An improper fraction is a fraction that has a numerator greater than the denominator. Improper fractions can be converted into mixed numbers.

Consider the fraction $\frac{21}{8}$.

Since the numerator is greater than the denominator, it is considered as an improper fraction.

To convert improper fractions to mixed numbers:

- Divide the numerator by the denominator.
- Write the whole number.
- Write the remainder as the numerator.
- Write the same denominator.

**Example #1**

Convert $\frac{11}{2}$ to a mixed number.

**Solution**

Process | Step-by-step Explanation |

11÷2 = 5 with a remainder of 1 | Divide the numerator by the denominator. In this case, the numerator is 11 and the denominator is 2. |

$5\frac{1}{}$ | Write the whole number as the whole number of the mixed number and write the remainder as the numerator of the fraction. |

$5\frac{1}{2}$ | Copy the denominator of the given improper fraction. |

Therefore, $\frac{11}{2}$ will become $5\frac{1}{2}$ if converted as a mixed number. |

**Example #2**

Convert $\frac{27}{7}$ to a mixed number.

**Solution**

Process | Step-by-step Explanation |

27÷7 = 3 with a remainder of 6 | Divide the numerator by the denominator. In this case, the numerator is 27 and the denominator is 7. |

$3\frac{6}{}$ | Write the whole number as the whole number of the mixed number and write the remainder as the numerator of the fraction. |

$3\frac{6}{7}$ | Copy the denominator of the given improper fraction. |

Therefore, $\frac{27}{7}$ will become $3\frac{6}{7}$ if converted as a mixed number. |

**How to convert a mixed number to an improper fraction?**

Converting mixed numbers to an improper fraction is one step that you need to know in adding and subtracting mixed numbers.

To convert mixed numbers to improper fractions:

- Multiply the denominator with the whole number.
- Add the numerator to the product of the denominator and whole number.
- Write the sum on the numerator of the improper fraction.
- Copy the same denominator.

**Example #1**

Convert $6\frac{6}{13}$ to an improper fraction.

**Solution**

Process | Step-by-step Explanation |

6 x 13 = 78 | Multiply the denominator to the given whole number of the mixed number. |

78 + 6 = 84 | Add the product to the numerator of the mixed number. |

$\frac{84}{}$ | Write the sum as the numerator of the improper fraction. |

$\frac{84}{13}$ | Copy the denominator of the given mixed number. |

Therefore, $6\frac{6}{13}$ will become $\frac{84}{13}$ if converted as a mixed number. |

**Example #2**

Convert $2\frac{7}{11}$ to an improper fraction.

**Solution**

Process | Step-by-step Explanation |

2 x 11 = 22 | Multiply the denominator to the given whole number of the mixed number. |

22 + 7 = 29 | Add the product to the numerator of the mixed number. |

$\frac{29}{}$ | Write the sum as the numerator of the improper fraction. |

$\frac{29}{11}$ | Copy the denominator of the given mixed number. |

Therefore, $2\frac{7}{11}$ will become $\frac{29}{11}$ if converted as a mixed number. |

**How to add mixed numbers?**

When adding mixed numbers, we need to consider the addends and the denominators of the addends.

### Adding whole number and mixed number

To add a whole number and a mixed number, follow these steps:

- Add the given whole number and the whole number of the mixed number.
- Copy the fraction part of the mixed number on the final answer.

**Example #1**

What is the sum of 2 and $3\frac{1}{4}$?

**Solution**

Addition Process | Step-by-step Explanation |

$2+3\frac{1}{4} =$ | Set up the addition process. |

$2+3\frac{1}{4} =5$ | Add the whole numbers 2 and 3. |

$2+3\frac{1}{4} =5\frac{1}{4}$ | Copy the fraction part of the given in mixed numbers. |

Therefore, the sum of 2 and $3\frac{1}{4}\:is\:5\frac{1}{4}$. |

**Example #2**

Determine the sum of $10\frac{4}{7}$ and 6.

**Solution**

Addition Process | Step-by-step Explanation |

$10\frac{4}{7} + 6$ | Set up the addition process. |

$10\frac{4}{7} + 6 = 16$ | Add the whole numbers 10 and 36. |

$10\frac{4}{7} + 6 = 16\frac{4}{7}$ | Copy the fraction part of the given in mixed numbers. |

Therefore, the sum of $10\frac{4}{7}$ and 6 is $16\frac{4}{7}$. |

### Adding mixed numbers with the same denominator

To add mixed numbers with the same denominators, follow these steps:

- Convert the mixed number to an improper fraction.
- Add the numerators.
- Copy the denominator.
- Convert the improper fraction to a mixed number.

**Example #1**

What is the sum of $3\frac{2}{5}$ and $7\frac{4}{5}$?

**Solution**

Addition Process | Step-by-step Explanation |

$3\frac{2}{5} + 7\frac{4}{5} =$ | Set up the addition process. |

$3\frac{2}{5}$ ⟹ $\frac{17}{5}$ | Convert $3\frac{2}{5}$ to an improper fraction. |

$7\frac{4}{5}$ ⟹ $\frac{39}{5}$ | Convert $7\frac{4}{5}$ to an improper fraction. |

$\frac{17}{5} + \frac{39}{5} =$ | Set up the addition process. |

$\frac{17}{5} + \frac{39}{5} =\frac{56}{}$ | Add the numerators. |

$\frac{17}{5} + \frac{39}{5} =\frac{56}{5}$ | Copy the denominator. |

$\frac{56}{5}$ ⟹$11\frac{1}{5}$ | Convert $\frac{56}{5}$ to a mixed number. |

Therefore, the sum of $ 3\frac{2}{5}$ and $7\frac{4}{5}$ is $11\frac{1}{5}$. |

**Example #2**

Determine the sum of $6\frac{5}{7}$ and $3\frac{3}{7}$.

**Solution**

Addition Process | Step-by-step Explanation |

$6\frac{5}{7} + 3\frac{3}{7} = $ | Set up the addition process. |

$6\frac{5}{7}$ ⟹ $\frac{47}{7}$ | Convert $6\frac{5}{7}$ to an improper fraction. |

$3\frac{3}{7}$ ⟹ $\frac{24}{7}$ | Convert $3\frac{3}{7}$ to an improper fraction. |

$\frac{47}{7} + \frac{24}{7} =$ | Set up the addition process. |

$\frac{47}{7} + \frac{24}{7} = \frac{71}{}$ | Add the numerators. |

$\frac{47}{7} + \frac{24}{7} = \frac{71}{7}$ | Copy the denominator. |

$\frac{71}{7}$⟹ $10\frac{1}{7}$ | Convert $\frac{71}{7}$ to a mixed number. |

Therefore, the result of adding $6\frac{5}{7}$ and $3\frac{3}{7}$ is $10\frac{1}{7}$. |

**Example #3**

Find the result of $13\frac{2}{3} + 6\frac{1}{3}$.

**Solution**

Addition Process | Step-by-step Explanation |

$13\frac{2}{3} + 6\frac{1}{3} =$ | Set up the addition process. |

$13\frac{2}{3}$ ⟹ $\frac{41}{3}$ | Convert $13\frac{2}{3}$ to an improper fraction. |

$6\frac{1}{3}$ ⟹ $\frac{19}{3}$ | Convert $6\frac{1}{3}$ to an improper fraction. |

$\frac{41}{3} + \frac{19}{3} =$ | Set up the addition process. |

$\frac{41}{3} + \frac{19}{3} = \frac{60}{}$ | Add the numerators. |

$\frac{41}{3} + \frac{19}{3} = \frac{60}{3}$ | Copy the denominator. |

$\frac{60}{3}$ ⟹ 20 | Convert 60 3 to a mixed number. |

Therefore, the sum of $13\frac{2}{3}$ and $6\frac{1}{3}$ is 20. |

**Example #4**

What is the sum of $5\frac{1}{2}$ and $\frac{7}{2}$?

**Solution**

Addition Process | Step-by-step Explanation |

$5\frac{1}{2} + \frac{7}{2} =$ | Set up the addition process. |

$5\frac{1}{2}$ ⟹ $\frac{11}{2}$ | Convert $5\frac{1}{2}$ to an improper fraction. |

$\frac{11}{2} + \frac{7}{2} = $ | Set up the addition process. |

$\frac{11}{2} + \frac{7}{2} = \frac{18}{}$ | Add the numerators. |

$\frac{11}{2} + \frac{7}{2} = \frac{18}{2}$ | Copy the denominator. |

$\frac{18}{2}$ ⟹ 9 | Convert $\frac{18}{2}$ to a mixed number. |

Therefore, the sum of $5\frac{1}{2}\:and\:\frac{7}{2}$ is 9. |

### Adding mixed numbers with different denominators

To add mixed numbers with different denominators, follow these steps:

- Convert the mixed number to an improper fraction.
- Find the least common denominator (LCD).
- Divide the LCD by the original denominators of the given mixed numbers.
- Multiply the quotient to the denominator of the improper fraction.
- Write the fractions based on the LCD.
- Add the numerators.
- Copy the LCD.
- Convert the improper fraction to a mixed number.

**Example #1**

What is the sum of $4\frac{1}{2}$ and $\frac{3}{4}$?

**Solution**

Addition Process | Step-by-step Explanation |

$4\frac{1}{2} + \frac{3}{4} =$ | Set up the addition process. |

$4\frac{1}{2}$ ⟹ $\frac{9}{2}$ | Convert $4\frac{1}{2}$ to an improper fraction. |

$\frac{9}{2} + \frac{3}{4} =$ | Set up the addition process. |

$\frac{9}{2}$ ⟹ $\frac{18}{4}$ $\frac{3}{4}$ ⟹ $\frac{3}{4}$ | Find the least common denominator of 2 and 4. By prime factorization, the least common multiple of 2 and 4 is 4. Hence, we will divide the LCD to the original denominator and multiply the quotient to the numerator. |

$\frac{18}{4} + \frac{3}{4} =$ | Set up the addition process. |

$\frac{18}{4} + \frac{3}{4} =\frac{21}{}$ | Add the numerators. |

$\frac{18}{4} + \frac{3}{4} =\frac{21}{4}$ | Copy th4e denominator. |

$\frac{21}{4}$⟹ $5\frac{1}{4}$ | Convert \frac{21}{4}$ to a mixed number. |

Therefore, the sum of $4\frac{1}{2}$ and $\frac{3}{4}$ is $5\frac{1}{4}$. |

**Example #2**

Determine the result of adding $7\frac{2}{5}$ and $6\frac{1}{3}$.

**Solution**

Addition Process | Step-by-step Explanation |

$7\frac{2}{5} + 6\frac{1}{3} =$ | Set up the addition process. |

$7\frac{2}{5}$ ⟹ $\frac{37}{5}$ | Convert $7\frac{2}{5}$ to an improper fraction. |

$6\frac{1}{3}$ ⟹ $\frac{19}{3}$ | Convert $6\frac{1}{3}$ to an improper fraction. |

$\frac{37}{5} + \frac{19}{3} =$ | Set up the addition process. |

$\frac{37}{5}$ ⟹ $\frac{111}{15}$ $\frac{19}{3}$ ⟹ $\frac{95}{15}$ | Find the least common denominator of 3 and 5. By prime factorization, the least common multiple of 3 and 5 is 15. Thus, we will divide the LCD to the original denominator and multiply the quotient to the numerator. |

$\frac{111}{15}$ ⟹ $\frac{95}{15} =$ | Set up the addition process. |

$\frac{111}{15}$ ⟹ $\frac{95}{15} =\frac{206}{}$ | Add the numerators. |

$\frac{111}{15}$ ⟹ $\frac{95}{15} =\frac{206}{15}$ | Copy the denominator. |

$\frac{206}{15}$ ⟹ $13\frac{11}{15}$ | Convert $\frac{206}{15}$ to a mixed number. |

Therefore, the result of adding $7\frac{2}{5}$ and $6\frac{1}{3}$ is $13\frac{11}{15}$. |

**How to subtract mixed numbers?**

Subtraction is simply the opposite of addition. When subtracting mixed numbers, we always need to consider the minuends and the subtrahends and the denominators of the two.

### Subtracting whole number and mixed number

We need to consider two cases if we are given a subtraction problem given a whole number and a mixed number.

**Case #1: If the minuend is a mixed number**

To subtract a whole number from a minuend, follow these steps:

- Subtract the given whole number from the whole number of the mixed number.
- Copy the fraction part of the mixed number on the final answer.

**Example**

What is the difference between $5\frac{6}{7}$ and 3?

**Solution**

Subtraction Process | Step-by-step Explanation |

$5\frac{6}{7} – 3 =$ | Set up the subtraction process. |

$5\frac{6}{7} – 3 =2$ | Subtract the whole numbers. Hence, 5 – 3 = 2 |

$5\frac{6}{7} – 3 = 2\frac{6}{7}$ | Copy the fraction part of the given in mixed numbers. |

Therefore, the difference between $5\frac{6}{7}$ and 3 is $2\frac{6}{7}$. |

**Case #2: If the subtrahend is a mixed number**

To subtract a mixed number from a whole number, follow these steps.

- Convert the whole number to a fraction having the same denominator as the given mixed number.
- Convert the mixed number to an improper fraction.
- Subtract the numerators.
- Copy the denominator.
- Convert your final answer to a mixed number

**Example**

Find the result of subtracting $1\frac{1}{3}$ from 4.

**Solution**

Subtraction Process | Step-by-step Explanation |

$4 – 1\frac{1}{3} =$ | Set up the subtraction process. |

4 ⟹ $\frac{12}{3}$ | Convert 4 to an improper fraction. |

$1\frac{1}{3}$ ⟹ $\frac{4}{3}$ | Convert $1\frac{1}{3}$ to an improper fraction. |

$\frac{12}{3} – \frac{4}{3} = \frac{8}{}$ | Subtract the numerators. |

$\frac{12}{3} – \frac{4}{3} = \frac{8}{3}$ | Copy the denominator. |

$\frac{8}{3}$ ⟹ $2\frac{2}{3}$ | Convert $\frac{8}{3}$ to a mixed number. |

Therefore, the result of subtracting $1\frac{1}{3}$ from 4 is $2\frac{2}{3}$. |

### Subtracting mixed numbers with the same denominator

To subtract mixed numbers with the same denominators, follow these steps:

- Convert the mixed number to an improper fraction.
- Subtract the numerators.
- Copy the denominator.
- Convert the improper fraction to a mixed number.

**Example #1**

What is the difference between $3\frac{3}{8}$ and $\frac{5}{8}$?

**Solution**

Subtraction Process | Step-by-step Explanation |

$3\frac{3}{8} – \frac{5}{8} =$ | Set up the subtraction process. |

$3\frac{3}{8}$ ⟹ $\frac{27}{8}$ | Convert $3\frac{3}{8}$ to an improper fraction. |

$\frac{27}{8} – \frac{5}{8} = \frac{22}{}$ | Subtract the numerators. |

$\frac{27}{8} – \frac{5}{8} = \frac{22}{8}$ | Copy the denominator. |

$\frac{22}{8}$ ⟹ $2\frac{6}{8}$ | Convert $\frac{22}{8}$ to a mixed number. |

$2\frac{6}{8}$ ⟹ $2\frac{3}{4}$ | Get the lowest term of $\frac{6}{8}$. |

Therefore, the difference is $2\frac{3}{4}$. |

**Example #2**

Find the difference of subtracting $3\frac{2}{9}$ from $5\frac{4}{9}$.

**Solution**

Subtraction Process | Step-by-step Explanation |

$5\frac{4}{9} – 3\frac{2}{9} =$ | Set up the subtraction process. |

$5\frac{4}{9}$ ⟹ $\frac{49}{9}$ | Convert $5\frac{4}{9}$ to an improper fraction. |

$3\frac{2}{9}$ ⟹ $\frac{29}{9}$ | Convert $3\frac{2}{9}$ to an improper fraction. |

$\frac{49}{9} – \frac{29}{9} = \frac{20}{}$ | Subtract the numerators. |

$\frac{49}{9} – \frac{29}{9} = \frac{20}{9}$ | Copy the denominator. |

$\frac{20}{9}⟹ 2\frac{2}{9}$ | Convert $\frac{20}{9}$ to a mixed number. |

Therefore, the difference is $2\frac{2}{9}$. |

**Example #3**

What is the result of subtracting $\frac{23}{11}$ from $6\frac{2}{11}$?

**Solution**

Subtraction Process | Step-by-step Explanation |

$6\frac{2}{11} – \frac{23}{11} =$ | Set up the subtraction process. |

$6\frac{2}{11}$ ⟹ $\frac{68}{11}$ | Convert $6\frac{2}{11}$ to an improper fraction. |

$\frac{68}{11} – \frac{23}{11} = \frac{45}{}$ | Subtract the numerators. |

$\frac{68}{11} – \frac{23}{11} = \frac{45}{11}$ | Copy the denominator. |

$\frac{45}{11}$ ⟹ $4\frac{1}{11}$ | Convert $\frac{45}{11}$ to a mixed number. |

Therefore, the result of subtracting $\frac{23}{11}$ from $6\frac{2}{11}$ is $4\frac{1}{11}$. |

### Subtracting mixed numbers with different denominators

To subtract mixed numbers with different denominators, follow these steps:

- Convert the mixed number to an improper fraction.
- Find the least common denominator (LCD).
- Divide the LCD by the original denominators of the given mixed numbers.
- Multiply the quotient to the denominator of the improper fraction.
- Write the fractions based on the LCD.
- Subtract the numerators.
- Copy the LCD.
- Convert the improper fraction to a mixed number.

**Example #1**

What is the difference between $9\frac{3}{4}$ and $\frac{2}{3}$?

**Solution**

Subtraction Process | Step-by-step Explanation |

$9\frac{3}{4} – \frac{2}{3} =$ | Set up the subtraction process |

$9\frac{3}{4}$ ⟹ $\frac{39}{4}$ | Convert $9\frac{3}{4}$ to an improper fraction. |

$\frac{39}{4} – \frac{2}{3} =$ | Set up subtraction process. |

$\frac{39}{4}$ ⟹ $\frac{117}{12}$ $\frac{2}{3}$ ⟹ $\frac{8}{12}$ | Find the least common denominator of 4 and 3. By prime factorization, the least common multiple of 4 and 3 is 12. Hence, we will divide the LCD to the original denominator and multiply the quotient to the numerator. |

$\frac{117}{12} – \frac{8}{12} =$ | Set up the subtraction process. |

$\frac{117}{12} – \frac{8}{12} = \frac{109}{}$ | Subtract the numerators. |

$\frac{117}{12} – \frac{8}{12} = \frac{109}{12}$ | Copy the denominator. |

$\frac{109}{12}$ ⟹ $9\frac{11}{2}$ | Convert $\frac{109}{12}$ to a mixed number. |

Therefore, the difference is $9\frac{1}{12}$. |

**Example #2**

Determine the result of subtracting $3\frac{1}{10}$ from $8\frac{2}{5}$.

**Solution**

Subtraction Process | Step-by-step Explanation |

$8\frac{2}{5} – 3\frac{1}{10}$ | Set up the subtraction process |

$8\frac{2}{5}$ ⟹ 425 | Convert $8\frac{2}{5}$ to an improper fraction. |

$3\frac{1}{10}$ ⟹ $\frac{31}{10}$ | Convert $3\frac{1}{10}$ to an improper fraction. |

$\frac{42}{5} – \frac{31}{10} =$ | Set up subtraction process. |

$\frac{42}{5}$ ⟹ $\frac{84}{10}$ $\frac{31}{10}$ ⟹ $\frac{31}{10}$ | Find the least common denominator of 5 and 10. By prime factorization, the least common multiple of 5 and 10 is 10. Hence, we will divide the LCD to the original denominator and multiply the quotient to the numerator. |

$\frac{84}{10} – \frac{31}{10} =$ | Set up the subtraction process. |

$\frac{84}{10} – \frac{31}{10} =\frac{53}{}$ | Subtract the numerators. |

$\frac{84}{10} – \frac{31}{10} =\frac{53}{10}$ | Copy the denominator. |

$\frac{53}{10}$ ⟹ $5\frac{3}{10}$ | Convert $\frac{53}{10}$ to a mixed number. |

Therefore, the difference is $5\frac{3}{10}$. |

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