A body travels uniformly a distance of \[\left( {13.8 \pm 0.2} \right)m\;\] in a time \[\left( {4.0 \pm 0.3} \right)s\] . Its velocity with error limits is:
 (A) \[(3.5\; \pm \;0.6)\dfrac{m}{s}\]
(B) \[(3.5\; \pm \;0.3)\dfrac{m}{s}\]
(C) \[(6.1\; \pm \;0.6)\dfrac{m}{s}\]
(D) \[(6.1\; \pm \;0.3)\dfrac{m}{s}\]

Hint: Measurement is the foundation for all experimental science. All the great technological development could not have been possible without ever-increasing levels of accuracy of measurements. The measurement of an amount is based on some international standards which are completely accurate compared with others.
An error may be defined as the difference between the measured value and the actual value. Accuracy is the extent to which a measurement approaches its true value and precision is the degree of refinement. Error is the difference between true value and accurate vale.
Error of a product is given by,
Ex: $z = AB$
\[\dfrac{{\Delta z}}{z} = \dfrac{{\Delta A}}{A} + \dfrac{{\Delta B}}{B}\]

Complete step by step answer:
Given,
Absolute distance travelled, s = 13.8
Error in distance travelled, $\Delta s = \pm 0.2$
Absolute time taken, t= 4 sec
Error in measuring time, $\Delta t = 0.3\sec $
We need to find the velocity of the body in error limit.
The absolute velocity of the body is given by,
$velocity = \dfrac{{dis\tan ce}}{{time}}$
$ \Rightarrow v = \dfrac{s}{t}$
$ \Rightarrow v = \dfrac{{13.8}}{4} = 3.45\dfrac{m}{s}$
The velocity within error limit can be calculated as follows:
The error in velocity is,
$\dfrac{{\Delta v}}{v} = \pm \left\{ {\dfrac{{\Delta s}}{s} + \dfrac{{\Delta t}}{t}} \right\}$
$ \Rightarrow \dfrac{{\Delta v}}{{3.45}} = \pm \left\{ {\dfrac{{0.2}}{{13.8}} + \dfrac{{0.3}}{4}} \right\}$
$ \Rightarrow \Delta v = \pm 0.3\dfrac{m}{s}$
Therefore, the velocity in error limits $(3.45 \pm 0.3)\dfrac{m}{s} \approx (3.5 \pm 0.3)\dfrac{m}{s}$.

Hence, the correct option is B.

Note:
There are different sources of error and are classified as follows:
A). GROSS ERROR: - The human made error is called gross error.
B). SYSTEMATIC ERROR:
The error arises due to fault in the system used in measurement.
These are further classified as instrumental, environmental, observational errors.
1). Instrumental Error- the error due to manufacturing and are inbuilt.it may also be due to misuse of instrument.
2). Environmental Error: - these are the external conditions that affect the measurement of reading for example the pressure. temperature etc.
c). RANDOM ERROR: - these are errors that are caused mostly by nature.