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Question
If the measured values of the two quantities are A ± ΔA and B ± ΔB, ΔA and ΔB being the mean absolute errors. What is the maximum possible error in A ± B? Show that if Z = `"A"/"B"`
`(Delta "Z")/"Z" = (Delta "A")/"A" + (Delta "B")/"B"`
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Solution
Suppose, Z = `"A"/"B"` and measured values of A and B are (A ± ΔA) and (B ± ΔB) then,
`"Z" +- Delta"Z" = ("A" +- Delta "A")/("B" +- Delta"B")`
∴ Z`(1 +- (Delta"Z")/"Z") = "A"/"B"([1 +- (Delta "A"//"A")]/[1 +- (Delta "B"// "B")])`
`= "A"/"B" xx [1 +- (Delta "A"//"A")]/[1 +- (Delta "B"// "B")]`
As, `(Delta "B")/"B"`<< 1, expanding using Binomial theorem,
`"Z"(1 +- (Delta"Z")/"Z") = "Z" xx (1 +- (Delta "A")/"A") xx (1 -+ (Delta"B")/"B") ...(because "A"/"B" = "Z")`
∴ `1 +- (Delta"Z")/"Z" = 1 +- (Delta "A")/"A"-+ (Delta"B")/"B" +- (Delta"A")/"A" xx (Delta"B")/"B"`
Ignoring term `(Delta "A")/"A" xx (Delta "B")/"B", (Delta "Z")/"A" = +- (Delta "A")/"A" = (Delta "B")/"B"`
This gives four possible values of `(Delta "Z")/"Z"` as `(+ (Delta "A")/"A" - (Delta "B")/"B"), (+ (Delta "A")/"A" + (Delta "B")/"B"), (- (Delta "A")/"A" - (Delta"B")/"B")` and `(- (Delta "A")/"A" + (Delta "B")/"B")`
∴ Maximum relative error of `(Delta "Z")/"Z" = +- ((Delta "A")/"A" + (Delta "B")/"B")`
Thus, when two quantities are divided, the maximum relative error in the result is the sum of relative errors in each quantity.
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