To solve this problem, we need to translate the given word problem into mathematical language. In this case, the problem gives us two equations based on the conditions provided.
The first condition in the problem tells us that "the denominator of a rational number is greater than its numerator by 3." We can write this as an equation:
```
d = n + 3,
```
where `n` is the numerator and `d` is the denominator of our original rational number.
The second condition tells us that "if the numerator is increased by 7 and the denominator is decreased by 1, the new number becomes 3/2". As a mathematical equation, this can be expressed as:
```
(n + 7) / (d - 1) = 3 / 2.
```
We now have two equations, one from each condition:
```
1) d = n + 3
2) (n + 7) / (d - 1) = 3 / 2
```
We can substitute equation 1) into equation 2) to get an equation with just one variable:
```
(n + 7) / ((n + 3) - 1) = 3 / 2.
```
Solving this equation will give us the value of `n`. Begin by cross-multiplying to get rid of the fractions:
```
2 * (n + 7) = 3 * ((n + 3) - 1),
```
which simplifies to:
```
2n + 14 = 3n + 6.
```
Then, subtract `2n` from both sides of the equation:
```
14 = n + 6.
```
Subtract `6` from both sides of the equation to solve for `n`:
```
n = 8.
```
When `n` equals `8`, we can find the value for `d` by substituting `n = 8` back into equation 1):
```
d = n + 3 = 8 + 3 = 11.
```
Therefore, the original number is `n/d = 8/11`.