The area of a rectangle is `78(2)/(9)` square metres. If its length is `10 (2)/(3)` m, find its breadth.

Answer

To find the breadth of the rectangle given its area and length, we can follow these steps:

### Step 1: Convert the mixed fractions to improper fractions.
- **Area**: \( 78 \frac{2}{9} \)
- Convert to improper fraction:
\[ 78 \frac{2}{9} = \frac{78 \times 9 + 2}{9} = \frac{702 + 2}{9} = \frac{704}{9} \]

- **Length**: \( 10 \frac{2}{3} \)
- Convert to improper fraction:
\[ 10 \frac{2}{3} = \frac{10 \times 3 + 2}{3} = \frac{30 + 2}{3} = \frac{32}{3} \]

### Step 2: Use the formula for the area of a rectangle.
The area \( A \) of a rectangle is given by:
\[ A = \text{length} \times \text{breadth} \]
We can rearrange this to find the breadth:
\[ \text{breadth} = \frac{A}{\text{length}} \]

### Step 3: Substitute the values into the formula.
Substituting the values we found:
\[ \text{breadth} = \frac{\frac{704}{9}}{\frac{32}{3}} \]

### Step 4: Simplify the division of fractions.
To divide by a fraction, we multiply by its reciprocal:
\[ \text{breadth} = \frac{704}{9} \times \frac{3}{32} \]

### Step 5: Simplify the expression.
- First, simplify \( \frac{3}{9} \):
\[ \frac{3}{9} = \frac{1}{3} \]
So,
\[ \text{breadth} = \frac{704}{1} \times \frac{1}{3 \times 32} = \frac{704}{96} \]

### Step 6: Simplify \( \frac{704}{96} \).
To simplify \( \frac{704}{96} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 32:
\[ \frac{704 \div 32}{96 \div 32} = \frac{22}{3} \]

### Final Answer:
The breadth of the rectangle is:
\[ \text{breadth} = \frac{22}{3} \text{ metres} \]

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