Question

In an experiment four quantities a, b, c and d are measured with percentage error 1%, 2%, 3% and 4% respectively. Quantity P is calculated as follows: P= $\frac{a^3 b^2}{cd}$ % error in P is

• A. 14%
• B. 10%
• C. 7%
• D. 4%

Answer 1

The Correct Option is A

To determine the percentage error in the calculated quantity \( P \), we need to understand the propagation of errors in the expression:

P = \frac{a^3 b^2}{cd}

The general rule for the propagation of errors in a product or a quotient is as follows: if \( Q = \frac{x^p y^q}{z^r} \), then the percentage error in \( Q \) is given by:

\Delta Q\% = p \Delta x\% + q \Delta y\% + r \Delta z\%

Applying this rule to the expression for \( P \), we have:

• The exponent of \( a \) is 3, so the error contribution from \( a \) is 3 \times 1\%.
• The exponent of \( b \) is 2, so the error contribution from \( b \) is 2 \times 2\%.
• The exponent of \( c \) is 1, so the error contribution from \( c \) is 1 \times 3\%.
• The exponent of \( d \) is 1, so the error contribution from \( d \) is 1 \times 4\%.

Now, summing up all these contributions gives:

3 \times 1\% + 2 \times 2\% + 1 \times 3\% + 1 \times 4\% = 3\% + 4\% + 3\% + 4\% = 14\%

Therefore, the percentage error in \( P \) is 14%.