Question

A quantity $X = {a^2}{b^3}{c^{\dfrac{5}{2}}}{d^{ - 2}}$ is related to four measurable quantities $a,b,c$ and $d$. Given that, the percentage error in the measurement of $a,b,c$ and $d$ are $1\%, 2\%, 2\% $ and $4\% $ respectively. Find out the percentage error in quantity $X$.

Answer 1

Hint: We have been given that the quantity $X = {a^2}{b^3}{c^{\dfrac{5}{2}}}{d^{ - 2}}$ then, we will use the expression:
$\dfrac{{\Delta X}}{X} \times 100 = \left[ {2\dfrac{{\Delta a}}{a} + 3\dfrac{{\Delta b}}{b} + \dfrac{5}{2}\dfrac{{\Delta c}}{c} + 2\dfrac{{\Delta d}}{d}} \right] \times 100$
Now, find out the value of $\dfrac{{\Delta X}}{X} \times 100$ to get the percentage error in quantity $X$.

Complete step-by-step answer:
The difference between estimated value and actual value when compared to actual value and is expressed in percentage then the value we get is called the percentage error. In other words, we can say that the percentage error is the relative error which is multiplied by 100.
The formula for percentage error is given below:
If a quantity is given $A = {b^c}$ and is related to a quantity $b$having percentage error of $2\% $.
Then, percentage error in quantity $A$ is
$\dfrac{{\Delta A}}{A} \times 100 = c\dfrac{{\Delta b}}{b} \times 100$
Now, according to question, it is given that
$X = {a^2}{b^3}{c^{\dfrac{5}{2}}}{d^{ - 2}}$
$
  \dfrac{{\Delta a}}{a} = 1\% \\
  \dfrac{{\Delta b}}{b} = 2\% \\
  \dfrac{{\Delta c}}{c} = 2\% \\
  \dfrac{{\Delta d}}{d} = 4\% \\
 $
Now, putting the values n the formula we get
$\dfrac{{\Delta X}}{X} \times 100 = \left[ {2\dfrac{{\Delta a}}{a} + 3\dfrac{{\Delta b}}{b} + \dfrac{5}{2}\dfrac{{\Delta c}}{c} + 2\dfrac{{\Delta d}}{d}} \right] \times 100$
$
  \dfrac{{\Delta X}}{X} = 2 \times 1 + 3 \times 2 + \dfrac{5}{2} \times 2 + 2 \times 4 \\
  \dfrac{{\Delta X}}{X} = 21\% \\
 $

Therefore, we got the percentage error for $X$ which is $21\% $

Note: Percentage error mean can be defined as the average of all the percent errors. It is also called the Mean percentage error. The formula for this is given below:
$MP = \dfrac{{100\% }}{n}\sum\limits_{i = 1}^n {\dfrac{{\left| {{T_i} - {E_i}} \right|}}{{{T_i}}}} $
Here, ${T_i}$ is the actual value
${E_i}$ is the estimated value
$n$ is the number of quantities in the model
But when the actual value is zero, the value of mean percentage error becomes undefined. This is the main disadvantage of this formula.