Question

A physical quantity P is related to four observables a, b, c and d as follows:
\[P=\dfrac{{{a}^{3}}{{b}^{2}}}{\sqrt{c}d}\]
The percentage errors of measurement in a, b, c and d are \[\mathbf{1}%,\text{ }\mathbf{3}%,\text{ }\mathbf{4}%\text{ }\mathbf{and}\text{ }\mathbf{2}%,\]\[\mathbf{1}%,\text{ }\mathbf{3}%,\text{ }\mathbf{4}%\text{ }\mathbf{and}\text{ }\mathbf{2}%,\], respectively. What is the percentage error in the quantity P? If the value of P calculated using the above relation turns out to be 3.763, to what value should you round off the result?

Answer 1

Hint:First of all, we will express the percentage errors into its respective expressions. After that we will expand the expression as given in the question. The powers raised to the observables will get multiplied while finding the percentage error of the whole expression.

Formula used:
In this question we have to find the percentage error of the quantity P, so you know the percentage error expression.
Relative error is the ratio of the mean absolute error to the mean value.
Relative Error =\[\dfrac{\Delta x}{x}\] (Multiply by 100 to get percentage error)
Percentage Error \[=\dfrac{\Delta x}{x}\times 100=\] Relative error \[\times 100\]

Complete step by step solution:
Given Data
Percentages Errors of a
\[\dfrac{\Delta a}{a}\times 100=1%\]
Percentages Errors of b
\[\dfrac{\Delta b}{b}\times 100=3%\]
Percentages Errors of c
\[\dfrac{\Delta c}{c}\times 100=4%\]
Percentages Errors of d
\[\dfrac{\Delta d}{d}\times 100=2%\]

Given Equations
\[P=\dfrac{{{a}^{3}}{{b}^{2}}}{\sqrt{c}d}\]
Expression of Error in Multiplication
If \[x=ab\]
Then the percentage error of x is
\[\dfrac{\Delta x}{x}\times 100=\left( \dfrac{\Delta a}{a}\times 100 \right)+\left( \dfrac{\Delta b}{b}\times 100 \right)\]
Expression of Error in Division
\[x=\dfrac{a}{b}\]
Then the percentage error of x is
\[\dfrac{\Delta x}{x}\times 100=\left( \dfrac{\Delta a}{a}\times 100 \right)+\left( \dfrac{\Delta b}{b}\times 100 \right)\]
Expression of Errors in exponent
\[x=a{{b}^{n}}\]
Then the percentage error of x is
\[\dfrac{\Delta x}{x}\times 100=\left( \dfrac{\Delta a}{a}\times 100 \right)+n\left( \dfrac{\Delta b}{b}\times 100 \right)\]

To find out the percentages error in the quantity of P is
\[
\dfrac{\Delta p}{p}\times 100=3\left( \dfrac{\Delta a}{a}\times 100 \right)+2\left( \dfrac{\Delta b}{b}\times 100 \right)+\dfrac{1}{2}\left( \dfrac{\Delta c}{c}\times 100 \right)+\left( \dfrac{\Delta d}{d}\times 100 \right) \\
\Rightarrow\dfrac{\Delta p}{p} \times 100=3 (1\%)+2(3\%)+\dfrac{1}{2}(4\%)+2\% \\
\therefore\dfrac{\Delta p}{p}\times 100=13\% \\
\]
The percentage error in the quantity of P is 13%.The value of P is given in question is 3.763. By rounding this value to the first decimal place, we get 3.8.

Note:While solving this problem, most of the students tend to make mistakes while finding the overall percentage of the given expression. In most of the cases it is seen that the students forget to multiply the powers raised to the observables. Again, it is important to note that while rounding off, if the succeeding digit is less than 5 then its preceding digit remains the same.