Question

If $x = 10.0 \pm 0.1$ and $y = 10.0 \pm 0.1$, then $2x -2y$ is equal to:
$\text{A.}\quad (0.0 \pm 0.1)$
$\text{B.}\quad Zero$
$\text{C.}\quad (0.0 \pm 0.4)$
$\text{D.}\quad (20.0 \pm 0.2)$

Answer 1

The given values of x and y is in the form where the permissible error is also mentioned. Practically there can’t be a measurement where the error becomes totally zero. Every measurement has a certain amount of error which could be human error, instrument error or any other error. But it is a good practice to minimize this error as much as possible.

Complete answer:
First of all, one needs to understand the format of writing a measurement. Suppose we are measuring the length of a thread with a meter scale. The scale can measure accurately only up to millimeter (mm). This is known as the least count of the device or instrument.
Now, upon measuring the length of thread, suppose we came out with the length of 20cm we can’t be exactly sure if the length is 20cm. It can be either less than 20cm or more than 20cm. But the chances of it being exactly 20cm is very rare. Hence, to represent the actual length, we have to write the maximum permissible error in the measurement so that in the experiments, after considering the errors, we can get good and closer results. Hence error is important.
Now, we will represent the length as: $(20\pm0.1)cm$, where 0.1 cm or 1 mm is the instrument with which the length 20 cm is being measured.
Now, as we discussed, the term along with the measured length is the maximum permissible error, thus in addition, subtraction, multiplication and division operation of this length, we must write maximum error.
Thus, given $x = (10.0 \pm 0.1),\quad 2x = (20.0\pm 0.2)$
$y = (10.0 \pm 0.1),\quad 2y = (20.0\pm 0.2)$
Thus $2x - 2y = (20.0 - 20.0) \pm (0.2 + 0.2) = (0 \pm 0.4)$

So, the correct answer is “Option C”.

Note:
One must not confuse here that the error should also be subtracted. Recalling that we’ve discussed that the term along with the measure value must be maximum permissible error, which could be positive or negative, the net error in the result will be maximum only when we will subtract positive and negative value, so that negative and negative becomes positive and get added up to give maximum error.