If Y=a b, the maximum percentage error in the measurement of Y will be ?

Question

Answer

Maximum Percentage Error in the Measurement of Y

When measuring a certain quantity, it is inevitable to encounter errors. These errors can be classified into two types, systematic and random errors. Systematic errors are those that consistently occur in the same direction, while random errors occur unpredictably and randomly. In measuring a product of two variables, the maximum percentage error can be calculated using the following formula:

Percentage Error = (Absolute Error / Measurement) x 100%

Deriving the Formula for Maximum Percentage Error

Suppose we have two variables, A and B, and we want to measure their product Y = AB. Let's say that the true values of A and B are A0 and B0, respectively, and the measured values are A and B. Using the formula for the product of two variables, we can express the true value of Y as:

Y0 = A0B0

and the measured value of Y as:

Y = AB

The absolute error in measuring A and B can be expressed as:

Absolute Error in A = |A - A0|

Absolute Error in B = |B - B0|

Using the formula for the product of two variables, we can express the absolute error in measuring Y as:

Absolute Error in Y = |Y - Y0| = |AB - A0B0| = |A(B - B0) + B0(A - A0)|

By applying the triangle inequality, we can express the absolute error in Y as:

Absolute Error in Y ≤ |A(B - B0)| + |B0(A - A0)|

Dividing both sides by |Y0| and multiplying by 100%, we get:

Percentage Error in Y ≤ (|A(B - B0)| + |B0(A - A0)|) / |A0B0| x 100%

Since A0B0 = Y0, we can simplify the formula to:

Percentage Error in Y ≤ (|A(B - B0)| / Y0 + |B0(A - A0)| / Y0) x 100%

Using the fact that |A(B - B0)| / Y0 and |B0(A - A0)| / Y0 are both less than or equal to 1, we can simplify the formula further to:

Percentage Error in Y ≤ (|A(B - B0)| / Y0 + |B0(A - A0)| / Y0) x 100% ≤ (|B - B0| / |B0| + |A - A0| / |A0|) x 100%

Finally, we arrive at the formula for the maximum percentage error in measuring Y:

Maximum Percentage Error in Y = (|B - B0| / |B0| + |A - A0| / |A0|) x 100%

Interpreting the Formula for Maximum Percentage Error

The formula for maximum percentage

Answer

0 if a and b has 0 percentage error . if a and b has some percentage error then add both of them multiplied by their respective powers which are one here .
Hope this would help .
Have a nice day .

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