Question

The mathematical operation in which the accuracy is limited to least accurate term is:
A) Addition
B) Subtraction
C) Multiplication and Division
D) Both a and b

Answer 1

Hint: Accuracy is the extent up to which an observed value agrees with the true value of quantity. The error indicates accuracy. The higher the accuracy, the smaller is the error. Low accuracy causes a difference between a result and a ‘true’ value.

Complete step by step answer:
Let us take addition-subtraction and multiplication-division together into consideration
For sum and difference:
Suppose limiting errors in two physical quantities $x$ and $y$ are, $\pm \Delta x$ and $\pm \Delta y$ respectively and, $z=x+y$
Further let, limiting error in the sum $z$ is $\pm \Delta z$
Then, $z\pm \Delta z=\left( x\pm \Delta x \right)+\left( y\pm \Delta y \right)$
Or, $\pm \Delta z=\pm \Delta x\pm \Delta y$
Thus, the maximum possible error in$z$ is given by
$\Delta z=\Delta x+\Delta y$
Also, for the difference $z=x-y$
The maximum possible error in$z$, $\Delta z=\Delta x+\Delta y$
Thus, when two quantities are added or subtracted the limiting error in the final result is the sum of the limiting errors in the quantities involved.
For multiplication and division:
Let, $z=xy$ taking logarithm on both sides,
$\log z=\log x+\log y$
On differentiating partially, we have,
 $\left| \dfrac{\Delta z}{z} \right|\max =\dfrac{\Delta x}{x}+\dfrac{\Delta y}{y}$ This is the maximum possible error in $z$ for both multiplication and division
Thus, when two quantities are multiplied or divided the fractional error in the final result is the sum of the fractional errors in the questions to be multiplied or divided.
In the case of addition and subtraction, absolute error is considered which is more accurate to the true value while in the case of division and multiplication relative error is taken and therefore both addition and subtraction are limited to the least accurate term.
Hence option (D) is the correct answer for the following question.

Note: Also, if we approach the question with respect to the least significant figure, in addition and subtraction we can reduce the term to least accurate terms and add or subtract them. If we do this the change in the answer is negligible. But for division and multiplication, the change is considerable. Hence addition and subtraction are limited to the least accurate term. Both the ways that are error analysis or the least significant figure are correct to approach the question. Follow sequential steps and understand with clarity the concept of accuracy.