A line is divided into three parts. The first part is two thirds the length of the second part. The third part is \[\dfrac{1}{2}cm\] shorter than the first part and 2 cm shorter than the second part. Find the length of the line?
A. 12cm
B. 10cm
C. 9cm
D. 8cm

Hint: As the line is divided into three parts, consider x, y and z be the three parts and as per the data given form the equations and find out the values of x, y and z and to find the length of the line add all the values of x, y, z which we have find out by solving the equations.

Complete step by step solution:
Let the length of the three parts be x, y, z as per the given data. As mentioned, first part is two third the length of the second part i.e.,
\[x = \dfrac{2}{3}y\] ……………………. 1
As per the data, the third part is \[\dfrac{1}{2}cm\] shorter than the first part i.e.,
\[z = x - \dfrac{1}{2}\]
And also, it is 2 cm shorter than the second part
\[z = y - 2\]
Hence, it implies that
\[x - \dfrac{1}{2} = y - 2\]
\[ \Rightarrow \] \[2y = 2x + 3\] …………………….. 2
From equation 1 and equation 2, substituting the expression of x from equation 1 in equation 2 as
\[2y = 2 \times \dfrac{2}{3}y + 3\]
Hence, after simplifying the terms the value of y is
\[y = \dfrac{9}{2}\]
Now, to get the value of x, substitute the value of y in equation 1 as
\[x = \dfrac{2}{3}y\]
\[\Rightarrow x = \dfrac{2}{3} \times \dfrac{9}{2}\]
\[\Rightarrow x = \dfrac{{18}}{6} = 3\]
We know that
\[z = y - 2\]
Hence, substituting the value of y we get
\[z = \dfrac{9}{2} - 2\]
\[\Rightarrow z = \dfrac{5}{2}\]
Now, to find the length of the wire consider all the three parts i.e., x, y and z as the values are:
\[x = 3\], \[y = \dfrac{9}{2}\] and \[z = \dfrac{5}{2}\]
Therefore, the length of the wire with respect to the three parts is
\[x + y + z = 3 + \dfrac{9}{2} + \dfrac{5}{2}\]
\[\therefore x + y + z = 10cm\]

Hence, option B is the right answer.

Note: The algebraic expression should be any one of the forms such as addition, subtraction, multiplication and division. To find the length of the line we need to know that the line is divided into how many parts i.e., considering constant variables and form the equations based on the data given and hence, we can find out the length.