Chapter 1.3, Problem 1.5WE

Perform the following arithmetic operations and report the result to the proper number of significant figures: (a) 317.5 mL + 0.675 mL, (b) 47.80 L − 2.075 L, (c) 13.5 g ÷ 45.18 L, (d) 6.25 cm × 1.175 cm, (e) 5.46 × 10²g + 4.991 × 10³ g.

Interpretation Introduction

Interpretation:

For the given set of calculations, the significant figures and their correct arithmetic operations has to be given.

Concept introduction:

Significant figures: The digits having a meaning in a given number are called as significant figures.

Significant figures –calculation rules

• Significant figure is a non-zero number (4443.2 has five significant figures)
• Zeros placement in between non-zero number are significant figures (40005 has five significant figures).
• Zeros before the non-zero number are not significant number (0.00040005 has five significant figures).
• Zeros after the non-zero number are significant in case of that number contains decimal point (4.5000 has five significant figures).
• Zeros after the non-zero number may or may not be a significant figure in case of that number does not have a decimal point (500 may have 1, 2 or 3 significant figures).  In this cases use scientific notation to avoid ambiguity.

To determine: Arithmetic operation is calculated with significant figure.

Answer to Problem 1.5WE

The following arithmetic operation, and report for the resulting the possible number of significant figure is four with answer $\text{318}\text{.2mL}$.

Explanation of Solution

Condition for significant figure:

The answer is obtained on the basis of the given number having smallest number (in digits) to the right side of the decimal point

All the non-zero is significant.

There are some conditions for nonzero and significant:

Zero is significant when they appear between non zero digits.

It appears after non zero digit in a number that contain a decimal point.

Zero may or mayn’t be significant if they appear a non-zero to right of the last non zero digit in a number that doesn’t contain a decimal point.

The following arithmetic operation $\text{317}\text{.5mL + 0}\text{.675mL}$

We can find by adding both the arithmetic value.

$\begin{array}{l}\text{317}\text{.5mL}\\ \underset{\_}{\text{+0}\text{.675mL}}\\ \text{318}\text{.175mL}\end{array}$

Round off the resulting data, we get $\text{318}\text{.175mL}\stackrel{}{\leftarrow }\text{round to 318}\text{.2mL}$

For the resulting the possible number of significant figure is four with answer $\text{318}\text{.2mL}$.

The answer should contain one digit to the right of the decimal point to match 317.5, which has the fewest digits to the right of the decimal point.

Interpretation Introduction

Interpretation:

For the given set of calculations, the significant figures and their correct arithmetic operations has to be given.

Concept introduction:

Significant figures: The digits having a meaning in a given number are called as significant figures.

Significant figures –calculation rules

• Significant figure is a non-zero number (4443.2 has five significant figures)
• Zeros placement in between non-zero number are significant figures (40005 has five significant figures).
• Zeros before the non-zero number are not significant number (0.00040005 has five significant figures).
• Zeros after the non-zero number are significant in case of that number contains decimal point (4.5000 has five significant figures).
• Zeros after the non-zero number may or may not be a significant figure in case of that number does not have a decimal point (500 may have 1, 2 or 3 significant figures).  In this cases use scientific notation to avoid ambiguity.

To determine: Arithmetic operation is calculated with significant figure

Answer to Problem 1.5WE

The following arithmetic operation, and report for the resulting the possible number of significant figure is four with answer $\text{45}\text{.73L}$.

Explanation of Solution

Condition for significant figure:

The answer is obtained on the basis of the given number having smallest number (in digits) to the right side of the decimal point

All the non-zero is significant.

There are some conditions for nonzero and significant:

Zero is significant when they appear between non zero digits.

It appears after non zero digit in a number that contain a decimal point.

Zero may or mayn’t be significant if they appear a non-zero to right of the last non zero digit in a number that doesn’t contain a decimal point.

The following arithmetic operation $\text{47}\text{.80L+2}\text{.075L }$.

We can find by adding both the arithmetic value.

$\begin{array}{l}\text{47}\text{.80L }\\ \underset{\_}{\text{+2}\text{.075L}}\\ \text{45}\text{.725L}\end{array}$

Round off the resulting data, we get $\text{45}\text{.725L}\stackrel{}{\leftarrow }\text{round to 45}\text{.73L}$

The answer must contain two digits to the right of the decimal point to match 47.80.

Interpretation Introduction

Interpretation:

For the given set of calculations, the significant figures and their correct arithmetic operations has to be given.

Concept introduction:

Significant figures: The digits having a meaning in a given number are called as significant figures.

Significant figures –calculation rules

• Significant figure is a non-zero number (4443.2 has five significant figures)
• Zeros placement in between non-zero number are significant figures (40005 has five significant figures).
• Zeros before the non-zero number are not significant number (0.00040005 has five significant figures).
• Zeros after the non-zero number are significant in case of that number contains decimal point (4.5000 has five significant figures).
• Zeros after the non-zero number may or may not be a significant figure in case of that number does not have a decimal point (500 may have 1, 2 or 3 significant figures).  In this cases use scientific notation to avoid ambiguity.

To determine: Arithmetic operation is calculated with significant figure

Answer to Problem 1.5WE

The following arithmetic operation, and report for the resulting the possible number of significant figure is three with answer $\text{0}\text{.299g/L}$.

Explanation of Solution

Condition for significant figure:

The answer is obtained on the basis of the given number having smallest number (in digits) to the right side of the decimal point

All the non-zero is significant.

There are some conditions for nonzero and significant:

Zero is significant when they appear between non zero digits.

It appears after non zero digit in a number that contain a decimal point.

Zero may or mayn’t be significant if they appear a non-zero to right of the last non zero digit in a number that doesn’t contain a decimal point.

The following arithmetic operation $\text{13}\text{.5g }÷\text{ 45}\text{.18L =0}\text{.298804781g/L}$

We can find by adding both the arithmetic value.

$\text{13}\text{.5g }÷\text{ 45}\text{.18L =0}\text{.298804781g/L}$ $\text{=0}\text{.298804781g/L}$

Round off the resulting data, we get $\text{0}\text{.298804781g/L}\stackrel{}{\leftarrow }\text{round to 0}\text{.299g/L}$.

The answer will contain three significant figures to match 13.5, which has the few number of significant figures in the calculation.

Interpretation Introduction

Interpretation:

For the given set of calculations, the significant figures and their correct arithmetic operations has to be given.

Concept introduction:

Significant figures: The digits having a meaning in a given number are called as significant figures.

Significant figures –calculation rules

• Significant figure is a non-zero number (4443.2 has five significant figures)
• Zeros placement in between non-zero number are significant figures (40005 has five significant figures).
• Zeros before the non-zero number are not significant number (0.00040005 has five significant figures).
• Zeros after the non-zero number are significant in case of that number contains decimal point (4.5000 has five significant figures).
• Zeros after the non-zero number may or may not be a significant figure in case of that number does not have a decimal point (500 may have 1, 2 or 3 significant figures).  In this cases use scientific notation to avoid ambiguity.

To determine: Arithmetic operation is calculated with significant figure

Answer to Problem 1.5WE

The following arithmetic operation, and report for the resulting the possible number of significant figure is three with answer $\text{7}{\text{.34cm}}^2$.

Explanation of Solution

Condition for significant figure:

The answer is obtained on the basis of the given number having smallest number (in digits) to the right side of the decimal point

All the non-zero is significant.

There are some conditions for nonzero and significant:

Zero is significant when they appear between non zero digits.

It appears after non zero digit in a number that contain a decimal point.

Zero may or mayn’t be significant if they appear a non-zero to right of the last non zero digit in a number that doesn’t contain a decimal point.

The following arithmetic operation $\text{6}\text{.25cm×1}\text{.175cm}$

We can find by multiplying both the arithmetic value.

$\text{6}\text{.25cm×1}\text{.175cm = 7}{\text{.34375cm}}^{\text{2}}$

Round off the resulting data, we get $\text{7}{\text{.34375cm}}^{\text{2}}\stackrel{}{\leftarrow }\text{round to 7}{\text{.34cm}}^2$.

The answer contains three significant figures to match 6.25.

Interpretation Introduction

Interpretation:

For the given set of calculations, the significant figures and their correct arithmetic operations has to be given.

Concept introduction:

Significant figures: The digits having a meaning in a given number are called as significant figures.

Significant figures –calculation rules

• Significant figure is a non-zero number (4443.2 has five significant figures)
• Zeros placement in between non-zero number are significant figures (40005 has five significant figures).
• Zeros before the non-zero number are not significant number (0.00040005 has five significant figures).
• Zeros after the non-zero number are significant in case of that number contains decimal point (4.5000 has five significant figures).
• Zeros after the non-zero number may or may not be a significant figure in case of that number does not have a decimal point (500 may have 1, 2 or 3 significant figures).  In this cases use scientific notation to avoid ambiguity.

To determine: Arithmetic operation is calculated with significant figure

Answer to Problem 1.5WE

The following arithmetic operation, and report for the resulting the possible number of significant figure is four with answer $\text{5}{\text{.537×10}}^{\text{3}}\text{g}$.

Explanation of Solution

Condition for significant figure:

The answer is obtained on the basis of the given number having smallest number (in digits) to the right side of the decimal point

All the non-zero is significant.

There are some conditions for nonzero and significant:

Zero is significant when they appear between non zero digits.

It appears after non zero digit in a number that contain a decimal point.

Zero may or mayn’t be significant if they appear a non-zero to right of the last non zero digit in a number that doesn’t contain a decimal point.

The following arithmetic operation, $\text{ 5}{\text{.46×10}}^{\text{2}}\text{g+49}{\text{.91×10}}^{\text{2}}\text{g}$.

We can find by adding both the arithmetic value.

$\text{ 5}{\text{.46×10}}^{\text{2}}\text{g+49}{\text{.91×10}}^{\text{2}}\text{g = 55}{\text{.37×10}}^{\text{2}}\text{g }$

Round off the resulting data, we get $\text{ 55}{\text{.37×10}}^{\text{2}}\text{g = 5}{\text{.537×10}}^{\text{3}}\text{g}$.

Firstly, we have to add number to the same power of 10.  That is, $\text{4}{\text{.991×10}}^{\text{3}}\text{=49}{\text{.91×10}}^{\text{2}}$.  So, the answer will contain two digit ,to the right of the decimal point to match both $\text{5}\text{.46 and 49}\text{.91}$.