Write four more rational number in each of the patterns:
`(-3)/(5), (-6)/(10), (-9)/(15), (-12)/(20)`

Answer

To find four more rational numbers that follow the given pattern of \((-3/5), (-6/10), (-9/15), (-12/20)\), we will analyze the numerators and denominators separately to identify the pattern.

### Step-by-Step Solution:

1. **Identify the Pattern in the Numerators:**
- The numerators of the given rational numbers are: -3, -6, -9, -12.
- We can see that each numerator decreases by 3:
- \(-3\) to \(-6\) is a decrease of 3.
- \(-6\) to \(-9\) is a decrease of 3.
- \(-9\) to \(-12\) is a decrease of 3.
- Continuing this pattern, the next four numerators will be:
- \(-12 - 3 = -15\)
- \(-15 - 3 = -18\)
- \(-18 - 3 = -21\)
- \(-21 - 3 = -24\)

2. **Identify the Pattern in the Denominators:**
- The denominators of the given rational numbers are: 5, 10, 15, 20.
- We can see that each denominator increases by 5:
- \(5\) to \(10\) is an increase of 5.
- \(10\) to \(15\) is an increase of 5.
- \(15\) to \(20\) is an increase of 5.
- Continuing this pattern, the next four denominators will be:
- \(20 + 5 = 25\)
- \(25 + 5 = 30\)
- \(30 + 5 = 35\)
- \(35 + 5 = 40\)

3. **Combine the New Numerators and Denominators:**
- Now we can form the new rational numbers using the numerators and denominators we found:
- The first new rational number is \(\frac{-15}{25}\).
- The second new rational number is \(\frac{-18}{30}\).
- The third new rational number is \(\frac{-21}{35}\).
- The fourth new rational number is \(\frac{-24}{40}\).

### Final Answer:
The four more rational numbers in the pattern are:
- \(\frac{-15}{25}\)
- \(\frac{-18}{30}\)
- \(\frac{-21}{35}\)
- \(\frac{-24}{40}\)