# 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% $

Answer

Verified

176.4k+ views

**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.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Change the following sentences into negative and interrogative class 10 english CBSE

A Short Paragraph on our Country India

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

What is the difference between anaerobic aerobic respiration class 10 biology CBSE