Question

In the expression \[A=\frac{xy^{3}}{z^{2}}\] the percentage error is given by ______.

Options

• \[\left(\frac{\Delta\mathrm{x}}{\mathrm{x}}+3\frac{\Delta\mathrm{y}}{\mathrm{y}}-2\frac{\Delta\mathrm{z}}{\mathrm{z}}\right)\times100\%\]

• \[\left(\frac{\Delta\mathrm{x}}{\mathrm{x}}+\frac{3\Delta\mathrm{y}}{\mathrm{y}}+\frac{2\Delta\mathrm{z}}{\mathrm{z}}\right)\times100\%\]

• \[\left(\frac{\Delta\mathbf{x}}{\mathbf{x}}-\frac{3\Delta\mathbf{y}}{\mathbf{y}}-\frac{2\Delta\mathbf{z}}{\mathbf{z}}\right)\times100\%\]

• \[\left(\frac{\Delta x}{x}-3\frac{\Delta y}{y}+2\frac{\Delta z}{z}\right)\times100\%\]

Answer 1

In the expression \[A=\frac{xy^{3}}{z^{2}}\] the percentage error is given by \[\left(\frac{\Delta\mathrm{x}}{\mathrm{x}}+\frac{3\Delta\mathrm{y}}{\mathrm{y}}+\frac{2\Delta\mathrm{z}}{\mathrm{z}}\right)\times100\%\].

Explanation:

For \[A=\frac{xy^3}{z^2}\]​:

• Error in multiplication/division → add relative errors
• Powers become coefficients

So:

• \[x\to\frac{\Delta x}{x}\]
• \[y^3\to3\frac{\Delta y}{y}\]
• \[z^2\to2\frac{\Delta z}{z}\]

All are added to get total percentage error.