Subtracting Mixed Numbers with Different Denominators (Grades 5-6 ... - Free Printable

Problem Overview:

The task involves subtracting mixed numbers with different denominators and reducing the results to their lowest terms. Mixed numbers consist of a whole number part and a fractional part. To solve these problems, we need to:

1. Convert mixed numbers to improper fractions.
2. Find a common denominator for the fractions.
3. Subtract the fractions.
4. Simplify the result (if necessary).
5. Convert the result back to a mixed number (if applicable).

Let's solve each problem step by step.

---

**Problem 1: $ 5 \frac{2}{5} - 4 \frac{1}{7} $

#### Step 1: Convert mixed numbers to improper fractions.
- $ 5 \frac{2}{5} = 5 + \frac{2}{5} = \frac{5 \times 5 + 2}{5} = \frac{27}{5} $
- $ 4 \frac{1}{7} = 4 + \frac{1}{7} = \frac{4 \times 7 + 1}{7} = \frac{29}{7} $

#### Step 2: Find a common denominator.
The denominators are 5 and 7. The least common denominator (LCD) is $ 5 \times 7 = 35 $.

#### Step 3: Rewrite the fractions with the common denominator.
- $ \frac{27}{5} = \frac{27 \times 7}{5 \times 7} = \frac{189}{35} $
- $ \frac{29}{7} = \frac{29 \times 5}{7 \times 5} = \frac{145}{35} $

#### Step 4: Subtract the fractions.
$$
\frac{189}{35} - \frac{145}{35} = \frac{189 - 145}{35} = \frac{44}{35}
$$

#### Step 5: Simplify and convert to a mixed number.
$ \frac{44}{35} $ is already in its simplest form. Convert it to a mixed number:
$$
\frac{44}{35} = 1 \frac{9}{35}
$$

Answer:
$$
\boxed{1 \frac{9}{35}}
$$

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**Problem 2: $ 3 \frac{3}{4} - 2 \frac{1}{2} $

#### Step 1: Convert mixed numbers to improper fractions.
- $ 3 \frac{3}{4} = 3 + \frac{3}{4} = \frac{3 \times 4 + 3}{4} = \frac{15}{4} $
- $ 2 \frac{1}{2} = 2 + \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2} $

#### Step 2: Find a common denominator.
The denominators are 4 and 2. The LCD is 4.

#### Step 3: Rewrite the fractions with the common denominator.
- $ \frac{15}{4} $ remains $ \frac{15}{4} $.
- $ \frac{5}{2} = \frac{5 \times 2}{2 \times 2} = \frac{10}{4} $

#### Step 4: Subtract the fractions.
$$
\frac{15}{4} - \frac{10}{4} = \frac{15 - 10}{4} = \frac{5}{4}
$$

#### Step 5: Simplify and convert to a mixed number.
$ \frac{5}{4} $ is already in its simplest form. Convert it to a mixed number:
$$
\frac{5}{4} = 1 \frac{1}{4}
$$

Answer:
$$
\boxed{1 \frac{1}{4}}
$$

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**Problem 3: $ 2 \frac{1}{2} - 1 \frac{3}{5} $

#### Step 1: Convert mixed numbers to improper fractions.
- $ 2 \frac{1}{2} = 2 + \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2} $
- $ 1 \frac{3}{5} = 1 + \frac{3}{5} = \frac{1 \times 5 + 3}{5} = \frac{8}{5} $

#### Step 2: Find a common denominator.
The denominators are 2 and 5. The LCD is $ 2 \times 5 = 10 $.

#### Step 3: Rewrite the fractions with the common denominator.
- $ \frac{5}{2} = \frac{5 \times 5}{2 \times 5} = \frac{25}{10} $
- $ \frac{8}{5} = \frac{8 \times 2}{5 \times 2} = \frac{16}{10} $

#### Step 4: Subtract the fractions.
$$
\frac{25}{10} - \frac{16}{10} = \frac{25 - 16}{10} = \frac{9}{10}
$$

#### Step 5: Simplify and convert to a mixed number.
$ \frac{9}{10} $ is already in its simplest form and is a proper fraction, so no conversion is needed.

Answer:
$$
\boxed{\frac{9}{10}}
$$

---

**Problem 4: $ 8 \frac{1}{4} - 5 \frac{4}{9} $

#### Step 1: Convert mixed numbers to improper fractions.
- $ 8 \frac{1}{4} = 8 + \frac{1}{4} = \frac{8 \times 4 + 1}{4} = \frac{33}{4} $
- $ 5 \frac{4}{9} = 5 + \frac{4}{9} = \frac{5 \times 9 + 4}{9} = \frac{49}{9} $

#### Step 2: Find a common denominator.
The denominators are 4 and 9. The LCD is $ 4 \times 9 = 36 $.

#### Step 3: Rewrite the fractions with the common denominator.
- $ \frac{33}{4} = \frac{33 \times 9}{4 \times 9} = \frac{297}{36} $
- $ \frac{49}{9} = \frac{49 \times 4}{9 \times 4} = \frac{196}{36} $

#### Step 4: Subtract the fractions.
$$
\frac{297}{36} - \frac{196}{36} = \frac{297 - 196}{36} = \frac{101}{36}
$$

#### Step 5: Simplify and convert to a mixed number.
$ \frac{101}{36} $ is already in its simplest form. Convert it to a mixed number:
$$
\frac{101}{36} = 2 \frac{29}{36}
$$

Answer:
$$
\boxed{2 \frac{29}{36}}
$$

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Final Answers:

1. $ \boxed{1 \frac{9}{35}} $
2. $ \boxed{1 \frac{1}{4}} $
3. $ \boxed{\frac{9}{10}} $
4. $ \boxed{2 \frac{29}{36}} $

(Note: I will continue solving the remaining problems if needed. Let me know!)