A physical quantity *P *is related to four observables *a, b, c *and *d *as follows:

`P=(a^3b^2)/((sqrtcd))`

The percentage errors of measurement in *a*, *b*, *c *and *d* are 1%, 3%, 4% and 2%, respectively. What is the percentage error in the quantity *P*? If the value of *P *calculated using the above relation turns out to be 3.763, to what value should you round off the result?

#### Solution 1

`P = (a^3b^2)/(sqrtcd)`

`(triangleP)/P = (3trianglea)/a + (2triangleb)/b + 1/2 (trianglec)/c + (triangled)/d`

`((triangleP)/P xx100)%=(3xx(trianglea)/axx10+2xx(triangleb)/bxx100+1/2xx(trianglec)/c xx 100 + (triangled)/d xx 100)%`

= `3xx1+2xx3+1/2xx4+2`

=3+6+2+2 = 13%

Percentage error in *P* = 13 %

Value of *P *is given as 3.763.

By rounding off the given value to the first decimal place, we get *P* = 3.8.

#### Solution 2

As `P = (a^3b^2)/(sqrtcd) = a^3b^2 c^((-1)/2) d^(-1)`

∴Maximum fractional error in the measurement

`(triangleP)/P = 3 (trianglea)/a + 2 (triangleb)/b + 1/2 (trianglec)/c + (triangled)/d`

As `(trianglea)/a` = 1%, `(triangleb)/b` = 3%, `(trianglec)/c` = 4% and `(triangled)/d` = 2%

∴Maximum fractional error in the measurement

`(triangleP)/P = 3xx1%+2xx3%+ 1/2xx4% + 2%`

= 3% + 6% + 2% + 2% = 13%

if P = 3.763, then traingleP = 13% of P

=`(13P)/100 = (13xx3.763)/100 = 0.489`

As the error lies in first decimal place, the answer should be rounded off to first decimal place. Hence, we shall express the value of P after rounding it off as P = 3.8.