Simplify the following :
`(1""7/8 div 1""1/2) times (8""1/3 div 1""1/2)`

Answer

To simplify the expression \( (1 \frac{7}{8} \div 1 \frac{1}{2}) \times (8 \frac{1}{3} \div 1 \frac{1}{2}) \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \( 1 \frac{7}{8} \) to an improper fraction:
\[ 1 \frac{7}{8} = \frac{1 \times 8 + 7}{8} = \frac{15}{8} \]

2. Convert \( 1 \frac{1}{2} \) to an improper fraction:
\[ 1 \frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2} \]

3. Convert \( 8 \frac{1}{3} \) to an improper fraction:
\[ 8 \frac{1}{3} = \frac{8 \times 3 + 1}{3} = \frac{25}{3} \]

### Step 2: Rewrite the Expression
Now, we can rewrite the original expression using the improper fractions:
\[ \left( \frac{15}{8} \div \frac{3}{2} \right) \times \left( \frac{25}{3} \div \frac{3}{2} \right) \]

### Step 3: Change Division to Multiplication
To divide by a fraction, we multiply by its reciprocal:
1. For \( \frac{15}{8} \div \frac{3}{2} \):
\[ \frac{15}{8} \times \frac{2}{3} \]

2. For \( \frac{25}{3} \div \frac{3}{2} \):
\[ \frac{25}{3} \times \frac{2}{3} \]

### Step 4: Simplify Each Part
1. Simplifying \( \frac{15}{8} \times \frac{2}{3} \):
\[ = \frac{15 \times 2}{8 \times 3} = \frac{30}{24} = \frac{5}{4} \]

2. Simplifying \( \frac{25}{3} \times \frac{2}{3} \):
\[ = \frac{25 \times 2}{3 \times 3} = \frac{50}{9} \]

### Step 5: Multiply the Results
Now we multiply the two results:
\[ \frac{5}{4} \times \frac{50}{9} = \frac{5 \times 50}{4 \times 9} = \frac{250}{36} \]

### Step 6: Simplify the Final Fraction
Now we simplify \( \frac{250}{36} \):
1. Find the GCD of 250 and 36, which is 2.
2. Divide both the numerator and denominator by 2:
\[ \frac{250 \div 2}{36 \div 2} = \frac{125}{18} \]

### Step 7: Convert to Mixed Number (if needed)
To convert \( \frac{125}{18} \) to a mixed number:
1. Divide 125 by 18, which gives 6 with a remainder of 17.
2. Thus, \( \frac{125}{18} = 6 \frac{17}{18} \).

### Final Answer
The simplified result is:
\[ 6 \frac{17}{18} \]

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