Question

Find limits to the error in taking $ \dfrac{{222}}{{203}} $ yards as equivalent to a meter, given that a meter is equal to $ 1.0936\,yards $.

Answer 1

Hint: An error is defined as the mistake in doing an observation or calculating something. It can also be defined as an inaccurate or incorrect value. To remove the errors, it is important to find out the extent of the error. Error is equal to the ratio of the difference between the actual value and the experimental value and the actual value. Error percentage is calculated by multiplying the above quantity with 100, using this information, we can find out the correct answer.

Complete step-by-step answer:
Actual value of 1 meter = $ 1.0936\,yards $
But here in the question, we are given that the value of 1 meter
$ = \dfrac{{222}}{{203}}\,yards = 1.09359606\,yards $
So, the error can be calculated as,
$ error = \dfrac{{actual\,value - calculated\,value}}{{actual\,value}} \times 100 $
$
  error = \dfrac{{1.0936 - 1.09359606}}{{1.0936}} \times 100 \\
   \Rightarrow error = \dfrac{{3.94 \times {{10}^{ - 6}}}}{{1.0936}} \times 100 = 3.60277981 \times {10^{ - 6}} \times 100 \\
   \Rightarrow error = 3.60277981 \times {10^{ - 4}} \;
  $
This value is rounded off.
Therefore, error in taking $ \dfrac{{222}}{{203}} $ yards as equivalent to a meter, given that a meter is equal to $ 1.0936\,yards $ is $ 3.6 \times {10^{ - 4}}\% $ .
So, the correct answer is “ $ 3.6 \times {10^{ - 4}}\% $ ”.

Note: Both meters and yards are the units for calculating the length of an object or the distance covered by an object. While experimenting there can be various errors that are the obtained value can deviate from the actual value. In the given question, a similar error has happened; the calculated value of yards in one meter comes out to be different than the actual value that’s why we have to find the error in the calculations. If an answer comes out with more than 2 digits after the decimal then we round off the answer to one or two digits to express the answer conveniently.