Question

The percentage error in quantities P, Q, R and S are point 5%, 1%, 3% and 1.5% respectively in the measurement of a physical quantities $\text{A}\ \text{= }\dfrac{{{\text{P}}^{\text{3}}}{{\text{Q}}^{\text{2}}}}{\sqrt{\text{R}}\text{S}}\text{ }$. The maximum percentage error in the value of ‘A’ will be
a) 8.5% b) 6.0% c) 7.5% d) 6.5%

Answer 1

Hint In this question, we are asked to find the maximum percentage error in the value of 'A' , which we know is given by the formula:-
$\dfrac{\left( \text{estimated no}-\text{actual no} \right)}{\left[ \text{actual number} \right]}\text{ }\times \ 100$.
Thus, here in this question, the Percentage error will be calculated by following the formula given below:-
$\dfrac{\Delta \text{A}}{\text{A}}\ \times \ 100=3\dfrac{\Delta \text{P}}{\text{P}}\ \times \ 100\ +\ \left( \dfrac{2\Delta \text{Q}}{\text{Q}}\ +\ \dfrac{1}{2}\dfrac{\Delta \text{R}}{\text{R}} \right)\text{ }\times 100+\left( \dfrac{\Delta \text{S}}{\text{S}}\ \times \ 100 \right)$

Complete step-by-step solution
The above question deals with the percentage error in quantities.
Percentage error is the difference between the estimated number and the actual number when compared to the actual number expressed in percentage form at. The formula looks like:-
Percentage error=$\dfrac{\left( \text{estimated no}-\text{actual no} \right)}{\left[ \text{actual number} \right]}\text{ }\times \ 100$
In other words you take the difference between the real answer and the guessed answer divide, by the real answer and then turn it into a percent.
So, the Percentage error :- $\dfrac{\Delta \text{A}}{\text{A}}\ \times \ 100=3\dfrac{\Delta \text{P}}{\text{P}}\ \times \ 100\ +\ \left( \dfrac{2\Delta \text{Q}}{\text{Q}}\ +\ \dfrac{1}{2}\dfrac{\Delta \text{R}}{\text{R}} \right)\text{ }\times 100+\left( \dfrac{\Delta \text{S}}{\text{S}}\ \times \ 100 \right)$

The percentage error in P is 5% and is given by $=\ \left( \dfrac{\Delta \text{P}}{\text{P}}\ \times \ 100 \right)$
The percentage error in Q is 1% and is given by$=\ \left( \dfrac{\Delta \text{Q}}{\text{Q}}\ \times \ 100 \right)$
The percentage error in R is 3% and is given by$=\ \left( \dfrac{\Delta \text{R}}{\text{R}}\ \times \ 100 \right)$
The percentage error in S is 1.5% and is given by $=\ \left( \dfrac{\Delta \text{S}}{\text{S}}\ \times \ 100 \right)$
So, $\dfrac{\Delta \text{A}}{\text{A}}\times 100=\left( 3\ \times \ 0.5 \right)\ +\ \left( 2\ \times \ 1\ +\ 0.5\ \times \ 3 \right)\ +\ 1.5$
$=\ 1.5\ +\ \left( 2\ +\ 1.5 \right)\ +\ 1.5=65$

Hence, option d gives the correct answer for this question

Note The purpose of calculating the percent error is to analyse how close the measured value is to an actual value. It is part of a comprehensive error analysis. In most of the fields, percent error is always expressed as a positive number whereas in others, it is correct to have either a positive or negative value.
For the percentage error, we can use the following formula here:-
$\left( \dfrac{\Delta \text{A}}{\text{A}}\ \times \ 100 \right)=a\left( \dfrac{\Delta x}{x}\times 100 \right)\text{ +}\ b\text{ }\left( \dfrac{\Delta y}{y}\times 100 \right)\text{ }+\text{ }c\text{ }\left( \dfrac{\Delta z}{z}\text{ }\times 100 \right)$