To simplify the expression \( \frac{3}{70} + \frac{3}{126} + \frac{3}{198} + \frac{3}{286} + \frac{3}{390} \), we can factor out the common \( 3 \) from each term:

\[
3\left( \frac{1}{70} + \frac{1}{126} + \frac{1}{198} + \frac{1}{286} + \frac{1}{390} \right)
\]

Now, we will sum the fractions inside the parentheses. To do this, we need a common denominator. The least common multiple (LCM) of the denominators \( 70, 126, 198, 286, 390 \) can be calculated.

1. **Finding the prime factors:**
- \( 70 = 2 \times 5 \times 7 \)
- \( 126 = 2 \times 3^2 \times 7 \)
- \( 198 = 2 \times 3 \times 11 \)
- \( 286 = 2 \times 11 \times 13 \)
- \( 390 = 2 \times 3 \times 5 \times 13 \)

2. **Calculating the LCM:**
- The LCM is found by taking the highest power of each prime number that appears in the factorization.
- From the above, we have:
- For \( 2 \): highest power is \( 2^1 \)
- For \( 3 \): highest power is \( 3^2 \)
- For \( 5 \): highest power is \( 5^1 \)
- For \( 7 \): highest power is \( 7^1 \)
- For \( 11 \): highest power is \( 11^1 \)
- For \( 13 \): highest power is \( 13^1 \)

Thus, the LCM is given by:
\[
LCM = 2^1 \times 3^2 \times 5^1 \times 7^1 \times 11^1 \times 13^1 = 2 \times 9 \times 5 \times 7 \times 11 \times 13
\]
Calculating this step by step,
- \( 2 \times 9 = 18 \)
- \( 18 \times 5 = 90 \)
- \( 90 \times 7 = 630 \)
- \( 630 \times 11 = 6930 \)
- \( 6930 \times 13 = 90090 \)

So, the LCM is \( 90090 \).

3. **Converting each fraction to have the same denominator:**
- Now we convert each fraction:
\[
\frac{1}{70} = \frac{1287}{90090}, \quad \frac{1}{126} = \frac{714}{90090}, \quad \frac{1}{198} = \frac{455}{90090}, \quad \frac{1}{286} = \frac{315}{90090}, \quad \frac{1}{390} = \frac{231}{90090}
\]

Now combine them:
\[
\frac{1287 + 714 + 455 + 315 + 231}{90090} = \frac{3002}{90090}
\]

4. **Putting it all together:**
Finally:
\[
3 \left( \frac{3002}{90090} \right) = \frac{9006}{90090}
\]

5. **Simplifying the fraction:**
To simplify \( \frac{9006}{90090} \), we divide both the numerator and the denominator by their greatest common divisor, which is \( 9006 \):

\[
\frac{9006 \div 9006}{90090 \div 9006} = \frac{1}{10}
\]

Thus, the final answer is:

\[
\boxed{\frac{1}{10}}
\]