Question

A quantity P is related as \(X^{-2}Y^{-3/2}Z^{2/5}\) where X,Y,Z are independent parameters which have fractional errors of \(0.1,0.2\) and \(0.5\) respectively in measurement. The maximum fractional error in P is

• A. \(0.7\)
• B. \(0.1\)
• C. \(0.8\)
• D. \(0.6\)

Hint:

Maximum fractional error is the sum of (power times fractional error) for each quantity.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
When a quantity is \(P=X^aY^bZ^c\), the maximum fractional error is \(\dfrac{\Delta P}{P}=|a|\dfrac{\Delta X}{X}+|b|\dfrac{\Delta Y}{Y}+|c|\dfrac{\Delta Z}{Z}\). Errors always add in the worst case, even for negative powers.

Step 2: Read off the powers:
Here \(a=-2\), \(b=-\dfrac32\), \(c=\dfrac25\). The fractional errors are \(0.1\), \(0.2\), \(0.5\).

Step 3: Add the contributions:
\[ \dfrac{\Delta P}{P}=2(0.1)+\dfrac32(0.2)+\dfrac25(0.5)=0.2+0.3+0.2=0.7 \]

Step 4: Why the other options are wrong.
0.1 is just one of the inputs. 0.8 and 0.6 come from taking the powers with wrong magnitudes or from subtracting the terms for negative powers. Subtraction is never allowed for maximum error.

Final Answer:
The maximum fractional error in P is 0.7. \[ \boxed{\text{(A) }0.7} \]