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Units And Measurement

Question
CBSEENPH11016789

Discuss the process of combination of errors when two physical quantities are multiplied.

Solution

Combination error when two quantities are multiplied, is given by
Let X = AB 
Let ∆A, ∆B and ∆X be the absolute errors in A, B and X respectively.
Therefore the above equation with error can be written as, 

space space space space space space space space space space space space space space space space straight X plus-or-minus straight capital delta equals left parenthesis straight A plus-or-minus ΔA right parenthesis left parenthesis straight B plus-or-minus ΔB right parenthesis space

space space space space rightwards double arrow space space space space straight X open parentheses 1 plus-or-minus ΔX over straight X close parentheses equals AB open parentheses 1 plus-or-minus ΔA over straight A close parentheses open parentheses 1 plus-or-minus ΔB over straight B close parentheses
space space space space space 

space space space space space space space space space space space space
rightwards double arrow space space space space space space straight X open parentheses 1 plus-or-minus ΔX over straight X close parentheses equals AB open parentheses 1 plus-or-minus ΔA over straight A plus-or-minus ΔB over straight B plus-or-minus ΔA over straight A ΔB over straight B close parentheses

rightwards double arrow space space space space space open parentheses 1 plus-or-minus ΔX over straight X close parentheses equals open parentheses 1 plus-or-minus ΔX over straight X plus-or-minus ΔB over straight B plus-or-minus ΔA over straight A ΔB over straight B close parentheses
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#6 {main}</pre> are small quantities.

Therefore the product ΔA over straight A. ΔB over straight B will be very small and hence can be neglected.
Thus the above equation reduce to,
space space space space
space space space plus-or-minus ΔX over straight X equals plus-or-minus ΔA over straight A plus-or-minus ΔB over straight B 
Maximum value of fractional error in X is 
ΔX over straight X equals ΔA over straight A plus ΔB over straight B
Multiplying both sides by 100
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#6 {main}</pre> 
i.e. Maximum possible %age error in X = Maximum possibe %age error in A + Maximum possible %age error in B.