A bridge of length $5\frac{1}{2}$ km was to be constructed. Company A constructed $2\frac{4}{5}$ km of the bridge. Then, a part of the constructed bridge got damaged. Company B then constructed a total of $4\frac{1}{3}$ km, which included the damaged part and the remaining part of the bridge. We need to find the length of the part that was damaged.

1. Problem Description

A bridge of length $5\frac{1}{2}$ km was to be constructed. Company A constructed $2\frac{4}{5}$ km of the bridge. Then, a part of the constructed bridge got damaged. Company B then constructed a total of $4\frac{1}{3}$ km, which included the damaged part and the remaining part of the bridge. We need to find the length of the part that was damaged.

2. Solution Steps

First, we find the length of the bridge remaining after Company A's construction:

$5\frac{1}{2} - 2\frac{4}{5}$

Convert mixed numbers to improper fractions:

$5\frac{1}{2} = \frac{5 \times 2 + 1}{2} = \frac{11}{2}$

$2\frac{4}{5} = \frac{2 \times 5 + 4}{5} = \frac{14}{5}$

$\frac{11}{2} - \frac{14}{5} = \frac{11 \times 5}{2 \times 5} - \frac{14 \times 2}{5 \times 2} = \frac{55}{10} - \frac{28}{10} = \frac{55 - 28}{10} = \frac{27}{10}$

So, the remaining length of the bridge after Company A's construction is $\frac{27}{10}$ km.

Let $d$ be the length of the damaged part. Company B constructed $4\frac{1}{3}$ km, which included the damaged part $d$ and the remaining part of the bridge, which is $\frac{27}{10} - d$. Thus,

$4\frac{1}{3} = d + (\frac{27}{10} - d)$ is incorrect. Company B constructed the damaged part plus the remaining part, so the total length of bridge constructed by A which is $2\frac{4}{5}$ km is equal to the length constructed by B which is $4\frac{1}{3}$ km plus the damaged part $d$. Therefore we have:

$2\frac{4}{5} = 4\frac{1}{3} + d$ is incorrect.

The length constructed by A $2\frac{4}{5}$ km included the damaged part ($d$) and undamaged part. The length of the bridge remaining after A constructed is $\frac{27}{10}$ km. Company B constructed $4\frac{1}{3}$ km, which equals the remaining part of the bridge. The equation should be:

$\text{A's construction} - \text{Damaged part} + \text{Company B's construction} = \text{Total length}$

$2\frac{4}{5} - d + \frac{27}{10} - A \text{ (wrong expression)}$

Then,

$2\frac{4}{5} - d = \text{Length constructed by A that is not damaged}$

Since company B constructed damaged part $d$ + remaining part after A. Then the equation should be: $4\frac{1}{3}$ = (remaining after A, i.e. total length - length constructed by A ) + d.

Therefore: $4\frac{1}{3} = 5\frac{1}{2} - 2\frac{4}{5} + d = \frac{27}{10} + d$

Convert $4\frac{1}{3}$ to an improper fraction: $4\frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{13}{3}$

Then, we have $\frac{13}{3} = \frac{27}{10} + d$. Therefore,

$d = \frac{13}{3} - \frac{27}{10} = \frac{13 \times 10}{3 \times 10} - \frac{27 \times 3}{10 \times 3} = \frac{130}{30} - \frac{81}{30} = \frac{130 - 81}{30} = \frac{49}{30}$

Now we convert the improper fraction to mixed number:

$\frac{49}{30} = 1\frac{19}{30}$

3. Final Answer

$1\frac{19}{30}$