**Math Problem: 1 3/4 divided by 1/2 - (1 1/2)3**

In this article, we will solve a complex math problem involving mixed numbers, division, and multiplication. The problem is:

**1 3/4 ÷ 1/2 - (1 1/2)3**

To solve this problem, we need to follow the order of operations (PEMDAS) and break it down into smaller parts.

**Step 1: Convert mixed numbers to improper fractions**

Let's convert the mixed numbers to improper fractions:

- 1 3/4 = 7/4
- 1/2 = 1/2
- 1 1/2 = 3/2

Now, the problem becomes:

**7/4 ÷ 1/2 - (3/2)3**

**Step 2: Divide 7/4 by 1/2**

To divide 7/4 by 1/2, we can multiply 7/4 by the reciprocal of 1/2, which is 2/1:

**(7/4) × (2/1) = 14/4**

**Step 3: Simplify the fraction**

We can simplify the fraction 14/4 by dividing both numerator and denominator by their greatest common divisor, which is 2:

**14/4 = 7/2**

So, the result of the division is 7/2.

**Step 4: Calculate (3/2)3**

To calculate (3/2)3, we need to multiply 3/2 by itself three times:

**(3/2) × (3/2) × (3/2) = (3 × 3 × 3) / (2 × 2 × 2) = 27/8**

**Step 5: Subtract (3/2)3 from 7/2**

Finally, we subtract 27/8 from 7/2:

**7/2 - 27/8**

To subtract these fractions, we need to find a common denominator, which is 8. We can convert 7/2 to 28/8 and then subtract:

**28/8 - 27/8 = 1/8**

Therefore, the final answer is:

**1 3/4 ÷ 1/2 - (1 1/2)3 = 1/8**