Chapter 1, Problem 68P

(a)

To determine

The calculation of 186.300+0.0030.

Answer to Problem 68P

The calculation of 186.300+0.0030 is $186.303$.

Explanation of Solution

Calculate the given operation

    $\begin{array}{c}186.300+0.0030=186.3030\\ =186.303\end{array}$

Conclusion:

Therefore, the calculation of 186.300+0.0030 is $186.303$.

(b)

To determine

The calculation of 186.300-0.0030.

Answer to Problem 68P

The calculation of 186.300-0.0030 is $186.297$.

Explanation of Solution

Calculate the given operation

    $\begin{array}{c}186.300-0.0030=186.2970\\ =186.297\end{array}$

Conclusion:

Therefore, the calculation of 186.300+0.0030 is $186.297$.

(c)

To determine

The calculation of 186.300×0.0030.

Answer to Problem 68P

The calculation of 186.300×0.0030 is $0.5589$.

Explanation of Solution

Calculate the given operation

    $186.300\times 0.0030=0.5589$

Conclusion:

Therefore, the calculation of 186.300+0.0030 is $0.5589$.

(d)

To determine

The calculation of 186.300/0.0030.

Answer to Problem 68P

The calculation of 186.300/0.0030 is $62100$.

Explanation of Solution

Calculate the given operation

    $186.300/0.0030=62100$

Conclusion:

Therefore, the calculation of 186.300+0.0030 is $62100$.

(e)

To determine

The percentage error in part (a) and (b) if 0.0030 is ignored.

Answer to Problem 68P

The percentage error in part(a) and (b) is $1.61\times {10}^{-3}\%$ which is very small and can be neglected whereas in part (c) and (d) the percentage error is too large to be neglected.

Explanation of Solution

Write the formula for percentage error

    $\text{\%error}=\left|\frac{x_{\text{app}}-x_{\text{cor}}}{x_{\text{cor}}}\right|\times 100$        (I)

Here, $x_{\text{app}}$ is the approximate value and $x_{\text{cor}}$ is the exact value.

Conclusion:

(a) Substitute $186.3030$ for $x_{\text{cor}}$ and $186.300$ for $x_{\text{app}}$ in equation (I) to find $\text{error}$

    $\begin{array}{c}\text{\%error}=\left|\frac{186.300-186.3030}{186.3030}\right|\times 100\\ =1.61\times {10}^{-3}\%\end{array}$

 (b) Substitute $186.297$ for $x_{\text{cor}}$ and $186.300$ for $x_{\text{app}}$ in equation (I) to find $\text{error}$

    $\begin{array}{c}\text{\%error}=\left|\frac{186.300-186.297}{186.297}\right|\times 100\\ =1.61\times {10}^{-3}\%\end{array}$

(c) Substitute $0.5589$ for $x_{\text{cor}}$ and $186.300$ for $x_{\text{app}}$ in equation (I) to find $\text{error}$

    $\begin{array}{c}\text{\%error}=\left|\frac{186.300-0.5589}{0.5589}\right|\times 100\\ =3.3233\times {10}^4\%\end{array}$

(d) Substitute $62100$ for $x_{\text{cor}}$ and $186.300$ for $x_{\text{app}}$ in equation (I) to find $\text{error}$

    $\begin{array}{c}\text{\%error}=\left|\frac{186.300-62100}{62100}\right|\times 100\\ =99.7\%\end{array}$

Therefore, the percentage error in part(a) and (b) is $1.61\times {10}^{-3}\%$ which is very small and can be neglected whereas in part (c) and (d) the percentage error is too large to be neglected.

(f)

To determine

The rule to be followed while ignoring smaller numbers.

Answer to Problem 68P

Smaller numbers can be ignored during subtraction and multiplication whereas it can’t be neglected during multiplication and division.

Explanation of Solution

Adding or subtracting a smaller number doesn’t make much of a difference to the calculation as they result in a smaller percentage error. Hence they can be neglected.

While multiplying and dividing, smaller numbers need to be taken into account as they make a significant difference in the answer. Their percentage error is too large to be neglected.

Conclusion:

(a) Substitute $186.3030$ for $x_{\text{cor}}$ and $186.300$ for $x_{\text{app}}$ in equation (I) to find $\text{error}$

    $\begin{array}{c}\text{\%error}=\left|\frac{186.300-186.3030}{186.3030}\right|\times 100\\ =1.61\times {10}^{-3}\%\end{array}$

(a) Substitute $186.297$ for $x_{\text{cor}}$ and $186.300$ for $x_{\text{app}}$ in equation (I) to find $\text{error}$

    $\begin{array}{c}\text{\%error}=\left|\frac{186.300-186.297}{186.297}\right|\times 100\\ =1.61\times {10}^{-3}\%\end{array}$

Therefore, smaller numbers can be ignored during subtraction and multiplication whereas it can’t be neglected during multiplication and division.