Errors in Measurement PYQ Solved Fast

 

❓ Question

A quantity $Q$ is formulated as

$$Q=X^{-2}Y^{3/2}Z^{-2/5}$$

where $X$, $Y$, and $Z$ are independent parameters having fractional errors:

$$\frac{\Delta X}{X}=0.1,\qquad \frac{\Delta Y}{Y}=0.2,\qquad \frac{\Delta Z}{Z}=0.5$$

Find the maximum fractional error in $Q$.

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✍️ Solution

For a quantity of the form

$$Q=X^aY^bZ^c$$

the maximum fractional error is

$$\frac{\Delta Q}{Q} = |a|\frac{\Delta X}{X} + |b|\frac{\Delta Y}{Y} + |c|\frac{\Delta Z}{Z}$$

Here,

$$a=-2,\qquad b=\frac32,\qquad c=-\frac25$$

Therefore,

$$\frac{\Delta Q}{Q} = |-2|(0.1) + \left|\frac32\right|(0.2) + \left|-\frac25\right|(0.5)$$
$$= 2(0.1) + \frac32(0.2) + \frac25(0.5)$$
$$= 0.2+0.3+0.2$$
$$\boxed{\frac{\Delta Q}{Q}=0.7}$$

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✅ Final Answer

$$\boxed{0.7}$$

or equivalently,

$$\boxed{70\%}$$

Key Idea

For maximum error, always add the absolute contributions of all independent variables. Hence negative powers such as $-2$ and $-\frac25$ also contribute positively to the maximum fractional error.