A train travels a distance of $60\text{km}$ at a speed of $80\text{km/h}$, $100\text{km}$ at a speed of $90\text{km/h}$, and another $140\text{km}$ at a speed of $120\text{km/h}$. Find the average speed for the entire journey. (Round off to the nearest integer.)

To find the average speed of the entire journey, we use the formula:
$$\text{Average Speed}=\frac{\text{Total Distance}}{\text{Total Time}}$$

1. Calculate Total Distance:

$$\text{Total Distance}=60\text{km}+100\text{km}+140\text{km}=300\text{km}$$

2. Calculate Total Time Taken:
* Time taken for the first part ($t_1$) $=\frac{60}{80}=\frac{3}{4}\text{hours}$
* Time taken for the second part ($t_2$) $=\frac{100}{90}=\frac{10}{9}\text{hours}$
* Time taken for the third part ($t_3$) $=\frac{140}{120}=\frac{7}{6}\text{hours}$

$$\text{Total Time}=\frac{3}{4}+\frac{10}{9}+\frac{7}{6}$$

Taking LCM of the denominators ($4,9,6$), which is $36$:

$$\text{Total Time}=\frac{3\times 9+10\times 4+7\times 6}{36}=\frac{27+40+42}{36}=\frac{109}{36}\text{hours}$$

3. Calculate Average Speed:

$$\text{Average Speed}=\frac{300}{\frac{109}{36}}=\frac{300\times 36}{109}=\frac{10800}{109}\approx 99.08\text{km/h}$$

Rounding off to the nearest integer gives $99\text{km/h}$.