In an experiment four quantities a, b, c, d are measured with percentage error $ 1%,2%,3%,4% $ respectively quantity P is calculated as followed:
$ P=\dfrac{{{a}^{3}}{{b}^{2}}}{cd} $
$ % $ Errors in P are:
(A) $ 14% $
(B) $ 10% $
(C) $ 7% $
(D) $ 4% $
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Hint: Percentage error is defined as ratio of absolute error to the true value multiplied by 100. When the result involves the multiplication or quotient of two observed quantities, the maximum possible percentage error in the result is equal to the sum of percentage errors in the observed quantities.
Complete step by step solution
We are given that:
$ P=\dfrac{{{a}^{3}}{{b}^{2}}}{cd} $
Percentage error is given by :
$ \dfrac{\Delta P}{P}=3\left( \dfrac{\Delta a}{a}\times 100 \right)+2\left( \dfrac{\Delta b}{b}\times 100 \right)+1\left( \dfrac{\Delta c}{c}\times 100 \right)+\left( \dfrac{\Delta d}{d}\times 100 \right) $
$ =3\times 1+2\times 2+1\times 3+1\times 4 $
$ =3+4+3+4 $
$ =14% $
Additional information
We can derive the formula for calculating percentage error when the result involves the product of powers of observed quantities.
Suppose $ X={{a}^{l}}{{b}^{m}}{{c}^{-n}} $
Let $ \Delta a $ , $ \Delta b $ , $ \Delta c $ are the absolute errors in the measurements of quantities a, b and c $ \Delta X $ is the absolute error in the result.
So,
$ X\pm \Delta X={{\left( a\pm \Delta a \right)}^{l}}{{\left( b\pm \Delta b \right)}^{m}}{{\left( c\pm \Delta c \right)}^{-n}} $
$\Rightarrow X\pm \Delta X={{a}^{l}}{{\left( 1\pm \dfrac{\Delta a}{a} \right)}^{l}}{{b}^{m}}{{\left( 1\pm \dfrac{\Delta b}{b} \right)}^{m}}{{c}^{-n}}{{\left( 1\pm \dfrac{\Delta c}{c} \right)}^{-n}} $
$\Rightarrow X\pm \Delta X={{a}^{l}}{{b}^{m}}{{c}^{-n}}{{\left( 1\pm \dfrac{\Delta a}{a} \right)}^{l}}{{\left( 1\pm \dfrac{\Delta b}{b} \right)}^{m}}{{\left( 1\pm \dfrac{\Delta c}{c} \right)}^{-n}} $
$\Rightarrow \dfrac{X\pm \Delta X}{X}=\dfrac{{{a}^{l}}{{b}^{m}}{{c}^{-n}}}{{{a}^{l}}{{b}^{m}}{{c}^{-n}}}\left( 1\pm \dfrac{l\Delta a}{a} \right)\left( 1\pm \dfrac{m\Delta b}{b} \right)\left( 1\pm \dfrac{n\Delta c}{c} \right) $
$\Rightarrow 1\pm \dfrac{\Delta X}{X}=1\pm \dfrac{m\Delta b}{b}\pm \dfrac{l\Delta a}{a}\pm \dfrac{n\Delta c}{c}+ $ neglected other terms
$ \therefore $ $ \dfrac{\Delta X}{X}=\dfrac{l\Delta a}{a}+\dfrac{m\Delta b}{b}+\dfrac{n\Delta c}{c} $
This is the maximum percentage error that can be calculated by multiplying both sides by 100.
Therefore, maximum percentage error in X
$ \dfrac{\Delta X}{X}\times 100=l\dfrac{\Delta a}{a}\times 100+m\dfrac{\Delta b}{b}\times 100+n\dfrac{\Delta c}{c}\times 100 $
This is the required equation.
Note
Suppose we have
$ X=\dfrac{{{a}^{l}}{{b}^{m}}}{{{c}^{n}}} $
Then maximum percentage error in x axis:
$ \dfrac{\Delta x}{x}\times 100=l\dfrac{\Delta a}{a}\times 100+m\dfrac{\Delta b}{b}\times 100+n\dfrac{\Delta c}{c}\times 100 $
That is, maximum percentage error in X
$ =l $ time maximum percentage error in $ a+m $ times maximum percentage error in $ b+n $ times maximum percentage error in c.
Complete step by step solution
We are given that:
$ P=\dfrac{{{a}^{3}}{{b}^{2}}}{cd} $
Percentage error is given by :
$ \dfrac{\Delta P}{P}=3\left( \dfrac{\Delta a}{a}\times 100 \right)+2\left( \dfrac{\Delta b}{b}\times 100 \right)+1\left( \dfrac{\Delta c}{c}\times 100 \right)+\left( \dfrac{\Delta d}{d}\times 100 \right) $
$ =3\times 1+2\times 2+1\times 3+1\times 4 $
$ =3+4+3+4 $
$ =14% $
Additional information
We can derive the formula for calculating percentage error when the result involves the product of powers of observed quantities.
Suppose $ X={{a}^{l}}{{b}^{m}}{{c}^{-n}} $
Let $ \Delta a $ , $ \Delta b $ , $ \Delta c $ are the absolute errors in the measurements of quantities a, b and c $ \Delta X $ is the absolute error in the result.
So,
$ X\pm \Delta X={{\left( a\pm \Delta a \right)}^{l}}{{\left( b\pm \Delta b \right)}^{m}}{{\left( c\pm \Delta c \right)}^{-n}} $
$\Rightarrow X\pm \Delta X={{a}^{l}}{{\left( 1\pm \dfrac{\Delta a}{a} \right)}^{l}}{{b}^{m}}{{\left( 1\pm \dfrac{\Delta b}{b} \right)}^{m}}{{c}^{-n}}{{\left( 1\pm \dfrac{\Delta c}{c} \right)}^{-n}} $
$\Rightarrow X\pm \Delta X={{a}^{l}}{{b}^{m}}{{c}^{-n}}{{\left( 1\pm \dfrac{\Delta a}{a} \right)}^{l}}{{\left( 1\pm \dfrac{\Delta b}{b} \right)}^{m}}{{\left( 1\pm \dfrac{\Delta c}{c} \right)}^{-n}} $
$\Rightarrow \dfrac{X\pm \Delta X}{X}=\dfrac{{{a}^{l}}{{b}^{m}}{{c}^{-n}}}{{{a}^{l}}{{b}^{m}}{{c}^{-n}}}\left( 1\pm \dfrac{l\Delta a}{a} \right)\left( 1\pm \dfrac{m\Delta b}{b} \right)\left( 1\pm \dfrac{n\Delta c}{c} \right) $
$\Rightarrow 1\pm \dfrac{\Delta X}{X}=1\pm \dfrac{m\Delta b}{b}\pm \dfrac{l\Delta a}{a}\pm \dfrac{n\Delta c}{c}+ $ neglected other terms
$ \therefore $ $ \dfrac{\Delta X}{X}=\dfrac{l\Delta a}{a}+\dfrac{m\Delta b}{b}+\dfrac{n\Delta c}{c} $
This is the maximum percentage error that can be calculated by multiplying both sides by 100.
Therefore, maximum percentage error in X
$ \dfrac{\Delta X}{X}\times 100=l\dfrac{\Delta a}{a}\times 100+m\dfrac{\Delta b}{b}\times 100+n\dfrac{\Delta c}{c}\times 100 $
This is the required equation.
Note
Suppose we have
$ X=\dfrac{{{a}^{l}}{{b}^{m}}}{{{c}^{n}}} $
Then maximum percentage error in x axis:
$ \dfrac{\Delta x}{x}\times 100=l\dfrac{\Delta a}{a}\times 100+m\dfrac{\Delta b}{b}\times 100+n\dfrac{\Delta c}{c}\times 100 $
That is, maximum percentage error in X
$ =l $ time maximum percentage error in $ a+m $ times maximum percentage error in $ b+n $ times maximum percentage error in c.