Question

An approximate value of a number of seconds in a year is $ \pi \times {10^7} $ . The percentage error in this value is:
 $ (A)0.5% $
 $ (B)8% $
 $ (C)4% $
 $ (D)15% $

Answer 1

Hint: We know that one minute has $ 60 $ seconds, One hour has $ 60 $ minutes and one day has $ 24 $ hours. Thus, $ 80 \times 60 \times 24 = 86400 $ seconds in a day. One common calendar year has $ 365 $ days: Thus, $ 365 \times 86400 = 31563000 $ seconds in a year.

Complete answer:
Percent error is the difference between estimated value and the actual value in comparison to the actual value and is expressed as a percentage. In other words, the percent error is the relative error multiplied by 100.. it is used to report the difference between the experimental value to its true or exact value.
An important example is that if we look at a gumball machine and make an estimate of how many gumballs there are and then we actually go ahead and calculate the number of gumballs, then we will be able to measure the percent error we made in our guess.
The formula for percentage error is,
Percentage error $ = \left( {\dfrac{{\text{actual value - experimental value}}}{{\text{actual}}}} \right) \times 100 $
According to question,
Actual value $ = 31563000 $
Experimental value $ (D)15% $
On putting the above values in the formula for percentage error, we get,
Percentage error $ = \left( {\dfrac{{31563000 - 31400000}}{{31563000}}} \right) \times 100 $
 $ = \left( {\dfrac{{163000}}{{31563000}}} \right) \times 100 $
On further simplifying, we get
 $ = \left( {\dfrac{{163}}{{31563}}} \right) \times 100 $
 $ = 0.0051 \times 100 $
 $ = 0.51% $
On taking an approximate value,
 $ \approx 0.5% $
So, the percentage error in this value is $ \approx 0.5% $.

Note:
The sign of the percent error is not considered in most applications except in chemistry and some other sciences where it is customary to keep a negative sign. Percent error is a type of error calculation. Few other types of common error calculations are relative error and absolute error.