Question

If $X = a - b$, then the maximum percentage error in the measurement of $X$ will be:
A. $\left( {\dfrac{{\Delta a}}{a} + \dfrac{{\Delta b}}{b}} \right) \times 100\%$
B. $\left( {\dfrac{{\Delta a}}{a} - \dfrac{{\Delta b}}{b}} \right) \times 100\%$
C. $\left( {\dfrac{{\Delta a}}{{a - b}} + \dfrac{{\Delta b}}{{a - b}}} \right) \times 100\%$
D. $\left( {\dfrac{{\Delta a}}{{a - b}} - \dfrac{{\Delta b}}{{a - b}}} \right) \times 100\%$

Answer 1

First determine the absolute error in the measurement of $X$. Use the formula for the percentage error. This formula gives the relation between the absolute error and actual error. Substitute the values of absolute error and actual error in this formula and determine the maximum percentage error in the measurement of $X$.

Formula used:
The percentage error in the measurement of a quantity is given by
${\text{Percentage error}} = \dfrac{{{\text{Absolute error}}}}{{{\text{Actual error}}}} \times 100$ …… (1)

Complete step by step solution:
We have given that $X = a - b$. First we should find the absolute error in the measurement of $X$.The absolute error in the percentage of $X$ is $\Delta X$. Hence, the absolute error in the measurement of a and b will be,
$X \pm \Delta X = a \pm \Delta a - b \pm \Delta b$
$\Rightarrow X \pm \Delta X = a - b \pm \Delta a \pm \Delta b$
Substitute $X$ for $a - b$ in the above equation.
$X \pm \Delta X = X \pm \Delta a \pm \Delta b$
$\Rightarrow \pm \Delta X = \pm \Delta a \pm \Delta b$
We can neglect the negative sign from the above equation as the errors are always added.
$\Rightarrow \Delta X = \Delta a + \Delta b$
Now let use determine the maximum percentage error in the measurement of $X$.
Substitute $\Delta X$ for ${\text{Absolute error}}$ and $X$ for ${\text{Actual error}}$ in equation (1).
${\text{Percentage error}} = \dfrac{{\Delta X}}{X} \times 100$
Substitute $\Delta a + \Delta b$ for $\Delta X$ and $a - b$ for $X$ in the above equation.
${\text{Percentage error}} = \dfrac{{\Delta a + \Delta b}}{{a - b}} \times 100$
$\therefore {\text{Percentage error}} = \left( {\dfrac{{\Delta a}}{{a - b}} + \dfrac{{\Delta b}}{{a - b}}} \right) \times 100$
Therefore, the percentage error in the measurement of $X$ is $\left( {\dfrac{{\Delta a}}{{a - b}} + \dfrac{{\Delta b}}{{a - b}}} \right) \times 100\%$.

Hence, the correct option is C.

Additional information:
When the same physical quantity is measured two times, there is a slight difference in the measurement of that physical quantity. This difference in the measurement of the physical quantity is called the error in the measurement of physical quantity.There are different types of errors such as observational error, instrumental error, environmental error, systematic error, etc.

Note: One can also use the negative sign in the measurement of the absolute error. If we use negative signs in the measurement of the absolute error, all the negative signs with the three terms of X, a and b will get cancelled and we obtain the same value of the absolute error in the measurement of $X$.