$frac{1}{15}$ +$frac{1}{35}$ +$frac{1}{63}$ +$frac{1}{99}$ +$frac{1}{143}$

To solve the expression $\frac{1}{15} + \frac{1}{35} + \frac{1}{63} + \frac{1}{99} + \frac{1}{143}$, we must first find a common denominator.

The denominators are 15, 35, 63, 99, and 143. We will factor each of these numbers:

- 15 = 3 × 5
- 35 = 5 × 7
- 63 = 3 × 21 = 3 × 3 × 7 = 3² × 7
- 99 = 9 × 11 = 3² × 11
- 143 = 11 × 13

Next, we find the least common multiple (LCM) of these denominators. The LCM is found by taking the highest powers of all prime factors that appear in any of the denominators:

- From 3: the highest power is 3² from 63 and 99.
- From 5: the highest power is 5¹ from 15 and 35.
- From 7: the highest power is 7¹ from 35 and 63.
- From 11: the highest power is 11¹ from 99 and 143.
- From 13: the highest power is 13¹ from 143.

Thus, the least common multiple (LCM) is:

LCM = 3² × 5¹ × 7¹ × 11¹ × 13¹ = 9 × 5 × 7 × 11 × 13

Calculating the LCM step by step:

1. 9 × 5 = 45
2. 45 × 7 = 315
3. 315 × 11 = 3465
4. 3465 × 13 = 45045

So, the common denominator is 45045.

Next, we convert each fraction to have this common denominator:

1. $\frac{1}{15} = \frac{45045 / 15}{45045} = \frac{3003}{45045}$
2. $\frac{1}{35} = \frac{45045 / 35}{45045} = \frac{1287}{45045}$
3. $\frac{1}{63} = \frac{45045 / 63}{45045} = \frac{715}{45045}$
4. $\frac{1}{99} = \frac{45045 / 99}{45045} = \frac{455}{45045}$
5. $\frac{1}{143} = \frac{45045 / 143}{45045} = \frac{315}{45045}$

Now we can add these fractions together:

$\frac{3003 + 1287 + 715 + 455 + 315}{45045} = \frac{5760}{45045}$

Now we will simplify the result. We can divide both the numerator and denominator by their greatest common divisor (GCD).

Calculating the GCD of 5760 and 45045 can be done using prime factorization or the Euclidean algorithm, but for this purpose, we note that:

5760 = 2⁷ × 3² × 5 × 11 = 5760
45045 = 3² × 5 × 7 × 11 × 13 = 45045

The GCD is 3² × 5 × 11 = 165

Now we can simplify:

$\frac{5760 / 165}{45045 / 165} = \frac{34.909\ldots}{273} = \frac{34}{273}$

Thus the final result can be expressed as follows:

$\frac{64}{819}$.

Therefore, the final answer is $\frac{64}{819}$.