The following calculation is to be solved to the correct number of significant figures. 2.420 g + 15.6 g 4.8 g Concept introduction: Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit. The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement. To determine the number of significant figures in a quantity following steps is followed. 1. The quantity must has a decimal point. 2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures. 3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures. 4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as 4.30 × 10 3 or 4.300 × 10 3 . The number of significant figures in 4.30 × 10 3 and 4.300 × 10 3 is 3 and 4 respectively. 5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures. Rules to determine significant figures in calculations are as follows: (1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures. (2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places. (3) Exact numbers do not affect the number of significant digits in the final answer. In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows: (1) If the last dropped digit is greater than 5 then increase the preceding digit by 1. (2) If the last dropped digit is less than 5 then the preceding digit does not change. (3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

Question

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Answer 1

Question

Chapter 1, Problem 1.59P

(a)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$2.420 g+15.6 g4.8 g$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit. The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(a)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $3.8 g$ .

Explanation of Solution

The given expression is,

$2.420 g+15.6 g4.8 g$

The least significant digit in each number is underlined as follows:

$2.420_ g+15.6_ g4.8_ g$

Add $2.420_ g$ and $15.6_ g$ in the numerator as:

$2.420_ g+15.6_ g=18.0_2 g$

The result of the addition operation must carry the same number of decimal places as the measurement with fewest decimal places. Measurement $15.6_ g$ has one digit after the decimal point. Therefore, the least significant figure in the result is 0.

Divide $18.02 g$ by $4.8 g$ as:

$18.02 g4.8 g=3.7_54 g=3.8 g$

The result of the division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $18.02 g$ has 4 significant figures and $4.8 g$ has 2 significant figures. The result must have 2 significant figures. Therefore, the final answer is rounded off to the 2 significant figures.

The last digit to be dropped is 5 and the preceding digit is odd thus it is increased by 1.

Conclusion

The final answer is rounded off to 2 significant figures. The answer of the calculation to a correct number of significant figures is $3.8 g$ .

(b)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$7.87 mL16.1 mL−8.44 mL$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit.

The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(b)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $1.0$ .

Explanation of Solution

The given expression is,

$7.87 mL16.1 mL−8.44 mL$

The least significant digit in each number is underlined as follows:

$7.87_ mL16.1_ mL−8.44_ mL$

Subtract $8.44 mL$ from $16.1 mL$ in the denominator as,

$16.1_ mL−8.44_ mL=7.6_6 mL$

The result of the subtraction operation must carry the same number of decimal places as the measurement with fewest decimal places. Measurement $16.1 mL$ has one digit after the decimal point. Therefore, the result must also contain only one digit after the decimal point. The least significant figure in the result is 6.

Divide $7.87 mL$ by $7.66 mL$ .

$7.87_ mL7.6_6 mL=1.0_274=1.0$

The result of the division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $7.6_6 mL$ has 2 significant figures and $7.87_ mL$ has 3 significant figures. The result must have 2 significant figures. Therefore, the final answer must be rounded off to the 2 significant figures. The last digit to be dropped is 2. 2 is less than 5, therefore, the preceding digit remains the same.

Conclusion

The final answer is rounded off to 2 significant figures. The answer of the calculation to a correct number of significant figures is $1.0$ .

(c)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$V=πr2h$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit.

The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(c)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $565 cm3$ .

Explanation of Solution

The value of $r$ is $6.23 cm$ and the value of $h$ is $4.630 cm$ .

The formula to calculate the volume is,

$V=πr2h$

Substitute $6.23 cm$ for $r$ , $4.630 cm$ for $h$ and $3.14159$ for $π$ in the above equation.

$V=(3.14159_)(6.23_ cm)2(4.630_ cm)=564_.556 cm3=565 cm3$

The result of the multiplication and division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $6.23 cm$ has 3 significant figures, $4.630 cm$ has 4 significant figures and $3.14159$ has 6 significant figures. The result must have 3 significant figures. Therefore, the final answer must be rounded off to the 3 significant figures. The last digit to be dropped is 5 thus the preceding digit is increased by 1.

Conclusion

The final answer is rounded off to 3 significant figures. The answer of the calculation to a correct number of significant figures is $565 cm3$ .

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Answer 2

Question

Chapter 1, Problem 1.59P

(a)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$2.420 g+15.6 g4.8 g$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit. The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(a)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $3.8 g$ .

Explanation of Solution

The given expression is,

$2.420 g+15.6 g4.8 g$

The least significant digit in each number is underlined as follows:

$2.420_ g+15.6_ g4.8_ g$

Add $2.420_ g$ and $15.6_ g$ in the numerator as:

$2.420_ g+15.6_ g=18.0_2 g$

The result of the addition operation must carry the same number of decimal places as the measurement with fewest decimal places. Measurement $15.6_ g$ has one digit after the decimal point. Therefore, the least significant figure in the result is 0.

Divide $18.02 g$ by $4.8 g$ as:

$18.02 g4.8 g=3.7_54 g=3.8 g$

The result of the division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $18.02 g$ has 4 significant figures and $4.8 g$ has 2 significant figures. The result must have 2 significant figures. Therefore, the final answer is rounded off to the 2 significant figures.

The last digit to be dropped is 5 and the preceding digit is odd thus it is increased by 1.

Conclusion

The final answer is rounded off to 2 significant figures. The answer of the calculation to a correct number of significant figures is $3.8 g$ .

(b)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$7.87 mL16.1 mL−8.44 mL$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit.

The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(b)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $1.0$ .

Explanation of Solution

The given expression is,

$7.87 mL16.1 mL−8.44 mL$

The least significant digit in each number is underlined as follows:

$7.87_ mL16.1_ mL−8.44_ mL$

Subtract $8.44 mL$ from $16.1 mL$ in the denominator as,

$16.1_ mL−8.44_ mL=7.6_6 mL$

The result of the subtraction operation must carry the same number of decimal places as the measurement with fewest decimal places. Measurement $16.1 mL$ has one digit after the decimal point. Therefore, the result must also contain only one digit after the decimal point. The least significant figure in the result is 6.

Divide $7.87 mL$ by $7.66 mL$ .

$7.87_ mL7.6_6 mL=1.0_274=1.0$

The result of the division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $7.6_6 mL$ has 2 significant figures and $7.87_ mL$ has 3 significant figures. The result must have 2 significant figures. Therefore, the final answer must be rounded off to the 2 significant figures. The last digit to be dropped is 2. 2 is less than 5, therefore, the preceding digit remains the same.

Conclusion

The final answer is rounded off to 2 significant figures. The answer of the calculation to a correct number of significant figures is $1.0$ .

(c)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$V=πr2h$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit.

The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(c)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $565 cm3$ .

Explanation of Solution

The value of $r$ is $6.23 cm$ and the value of $h$ is $4.630 cm$ .

The formula to calculate the volume is,

$V=πr2h$

Substitute $6.23 cm$ for $r$ , $4.630 cm$ for $h$ and $3.14159$ for $π$ in the above equation.

$V=(3.14159_)(6.23_ cm)2(4.630_ cm)=564_.556 cm3=565 cm3$

The result of the multiplication and division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $6.23 cm$ has 3 significant figures, $4.630 cm$ has 4 significant figures and $3.14159$ has 6 significant figures. The result must have 3 significant figures. Therefore, the final answer must be rounded off to the 3 significant figures. The last digit to be dropped is 5 thus the preceding digit is increased by 1.

Conclusion

The final answer is rounded off to 3 significant figures. The answer of the calculation to a correct number of significant figures is $565 cm3$ .

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 3

Question

Chapter 1, Problem 1.59P

(a)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$2.420 g+15.6 g4.8 g$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit. The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(a)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $3.8 g$ .

Explanation of Solution

The given expression is,

$2.420 g+15.6 g4.8 g$

The least significant digit in each number is underlined as follows:

$2.420_ g+15.6_ g4.8_ g$

Add $2.420_ g$ and $15.6_ g$ in the numerator as:

$2.420_ g+15.6_ g=18.0_2 g$

The result of the addition operation must carry the same number of decimal places as the measurement with fewest decimal places. Measurement $15.6_ g$ has one digit after the decimal point. Therefore, the least significant figure in the result is 0.

Divide $18.02 g$ by $4.8 g$ as:

$18.02 g4.8 g=3.7_54 g=3.8 g$

The result of the division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $18.02 g$ has 4 significant figures and $4.8 g$ has 2 significant figures. The result must have 2 significant figures. Therefore, the final answer is rounded off to the 2 significant figures.

The last digit to be dropped is 5 and the preceding digit is odd thus it is increased by 1.

Conclusion

The final answer is rounded off to 2 significant figures. The answer of the calculation to a correct number of significant figures is $3.8 g$ .

(b)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$7.87 mL16.1 mL−8.44 mL$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit.

The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(b)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $1.0$ .

Explanation of Solution

The given expression is,

$7.87 mL16.1 mL−8.44 mL$

The least significant digit in each number is underlined as follows:

$7.87_ mL16.1_ mL−8.44_ mL$

Subtract $8.44 mL$ from $16.1 mL$ in the denominator as,

$16.1_ mL−8.44_ mL=7.6_6 mL$

The result of the subtraction operation must carry the same number of decimal places as the measurement with fewest decimal places. Measurement $16.1 mL$ has one digit after the decimal point. Therefore, the result must also contain only one digit after the decimal point. The least significant figure in the result is 6.

Divide $7.87 mL$ by $7.66 mL$ .

$7.87_ mL7.6_6 mL=1.0_274=1.0$

The result of the division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $7.6_6 mL$ has 2 significant figures and $7.87_ mL$ has 3 significant figures. The result must have 2 significant figures. Therefore, the final answer must be rounded off to the 2 significant figures. The last digit to be dropped is 2. 2 is less than 5, therefore, the preceding digit remains the same.

Conclusion

The final answer is rounded off to 2 significant figures. The answer of the calculation to a correct number of significant figures is $1.0$ .

(c)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$V=πr2h$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit.

The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(c)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $565 cm3$ .

Explanation of Solution

The value of $r$ is $6.23 cm$ and the value of $h$ is $4.630 cm$ .

The formula to calculate the volume is,

$V=πr2h$

Substitute $6.23 cm$ for $r$ , $4.630 cm$ for $h$ and $3.14159$ for $π$ in the above equation.

$V=(3.14159_)(6.23_ cm)2(4.630_ cm)=564_.556 cm3=565 cm3$

The result of the multiplication and division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $6.23 cm$ has 3 significant figures, $4.630 cm$ has 4 significant figures and $3.14159$ has 6 significant figures. The result must have 3 significant figures. Therefore, the final answer must be rounded off to the 3 significant figures. The last digit to be dropped is 5 thus the preceding digit is increased by 1.

Conclusion

The final answer is rounded off to 3 significant figures. The answer of the calculation to a correct number of significant figures is $565 cm3$ .

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Answer 4

Question

Chapter 1, Problem 1.59P

(a)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$2.420 g+15.6 g4.8 g$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit. The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(a)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $3.8 g$ .

Explanation of Solution

The given expression is,

$2.420 g+15.6 g4.8 g$

The least significant digit in each number is underlined as follows:

$2.420_ g+15.6_ g4.8_ g$

Add $2.420_ g$ and $15.6_ g$ in the numerator as:

$2.420_ g+15.6_ g=18.0_2 g$

The result of the addition operation must carry the same number of decimal places as the measurement with fewest decimal places. Measurement $15.6_ g$ has one digit after the decimal point. Therefore, the least significant figure in the result is 0.

Divide $18.02 g$ by $4.8 g$ as:

$18.02 g4.8 g=3.7_54 g=3.8 g$

The result of the division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $18.02 g$ has 4 significant figures and $4.8 g$ has 2 significant figures. The result must have 2 significant figures. Therefore, the final answer is rounded off to the 2 significant figures.

The last digit to be dropped is 5 and the preceding digit is odd thus it is increased by 1.

Conclusion

The final answer is rounded off to 2 significant figures. The answer of the calculation to a correct number of significant figures is $3.8 g$ .

(b)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$7.87 mL16.1 mL−8.44 mL$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit.

The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(b)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $1.0$ .

Explanation of Solution

The given expression is,

$7.87 mL16.1 mL−8.44 mL$

The least significant digit in each number is underlined as follows:

$7.87_ mL16.1_ mL−8.44_ mL$

Subtract $8.44 mL$ from $16.1 mL$ in the denominator as,

$16.1_ mL−8.44_ mL=7.6_6 mL$

The result of the subtraction operation must carry the same number of decimal places as the measurement with fewest decimal places. Measurement $16.1 mL$ has one digit after the decimal point. Therefore, the result must also contain only one digit after the decimal point. The least significant figure in the result is 6.

Divide $7.87 mL$ by $7.66 mL$ .

$7.87_ mL7.6_6 mL=1.0_274=1.0$

The result of the division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $7.6_6 mL$ has 2 significant figures and $7.87_ mL$ has 3 significant figures. The result must have 2 significant figures. Therefore, the final answer must be rounded off to the 2 significant figures. The last digit to be dropped is 2. 2 is less than 5, therefore, the preceding digit remains the same.

Conclusion

The final answer is rounded off to 2 significant figures. The answer of the calculation to a correct number of significant figures is $1.0$ .

(c)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$V=πr2h$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit.

The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(c)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $565 cm3$ .

Explanation of Solution

The value of $r$ is $6.23 cm$ and the value of $h$ is $4.630 cm$ .

The formula to calculate the volume is,

$V=πr2h$

Substitute $6.23 cm$ for $r$ , $4.630 cm$ for $h$ and $3.14159$ for $π$ in the above equation.

$V=(3.14159_)(6.23_ cm)2(4.630_ cm)=564_.556 cm3=565 cm3$

The result of the multiplication and division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $6.23 cm$ has 3 significant figures, $4.630 cm$ has 4 significant figures and $3.14159$ has 6 significant figures. The result must have 3 significant figures. Therefore, the final answer must be rounded off to the 3 significant figures. The last digit to be dropped is 5 thus the preceding digit is increased by 1.

Conclusion

The final answer is rounded off to 3 significant figures. The answer of the calculation to a correct number of significant figures is $565 cm3$ .

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Chapter 1, Problem 1.59P

(a)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$2.420 g+15.6 g4.8 g$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit. The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(a)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $3.8 g$ .

Explanation of Solution

The given expression is,

$2.420 g+15.6 g4.8 g$

The least significant digit in each number is underlined as follows:

$2.420_ g+15.6_ g4.8_ g$

Add $2.420_ g$ and $15.6_ g$ in the numerator as:

$2.420_ g+15.6_ g=18.0_2 g$

The result of the addition operation must carry the same number of decimal places as the measurement with fewest decimal places. Measurement $15.6_ g$ has one digit after the decimal point. Therefore, the least significant figure in the result is 0.

Divide $18.02 g$ by $4.8 g$ as:

$18.02 g4.8 g=3.7_54 g=3.8 g$

The result of the division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $18.02 g$ has 4 significant figures and $4.8 g$ has 2 significant figures. The result must have 2 significant figures. Therefore, the final answer is rounded off to the 2 significant figures.

The last digit to be dropped is 5 and the preceding digit is odd thus it is increased by 1.

Conclusion

The final answer is rounded off to 2 significant figures. The answer of the calculation to a correct number of significant figures is $3.8 g$ .

(b)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$7.87 mL16.1 mL−8.44 mL$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit.

The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(b)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $1.0$ .

Explanation of Solution

The given expression is,

$7.87 mL16.1 mL−8.44 mL$

The least significant digit in each number is underlined as follows:

$7.87_ mL16.1_ mL−8.44_ mL$

Subtract $8.44 mL$ from $16.1 mL$ in the denominator as,

$16.1_ mL−8.44_ mL=7.6_6 mL$

The result of the subtraction operation must carry the same number of decimal places as the measurement with fewest decimal places. Measurement $16.1 mL$ has one digit after the decimal point. Therefore, the result must also contain only one digit after the decimal point. The least significant figure in the result is 6.

Divide $7.87 mL$ by $7.66 mL$ .

$7.87_ mL7.6_6 mL=1.0_274=1.0$

The result of the division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $7.6_6 mL$ has 2 significant figures and $7.87_ mL$ has 3 significant figures. The result must have 2 significant figures. Therefore, the final answer must be rounded off to the 2 significant figures. The last digit to be dropped is 2. 2 is less than 5, therefore, the preceding digit remains the same.

Conclusion

The final answer is rounded off to 2 significant figures. The answer of the calculation to a correct number of significant figures is $1.0$ .

(c)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$V=πr2h$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit.

The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(c)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $565 cm3$ .

Explanation of Solution

The value of $r$ is $6.23 cm$ and the value of $h$ is $4.630 cm$ .

The formula to calculate the volume is,

$V=πr2h$

Substitute $6.23 cm$ for $r$ , $4.630 cm$ for $h$ and $3.14159$ for $π$ in the above equation.

$V=(3.14159_)(6.23_ cm)2(4.630_ cm)=564_.556 cm3=565 cm3$

The result of the multiplication and division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $6.23 cm$ has 3 significant figures, $4.630 cm$ has 4 significant figures and $3.14159$ has 6 significant figures. The result must have 3 significant figures. Therefore, the final answer must be rounded off to the 3 significant figures. The last digit to be dropped is 5 thus the preceding digit is increased by 1.

Conclusion

The final answer is rounded off to 3 significant figures. The answer of the calculation to a correct number of significant figures is $565 cm3$ .

Answer 6

Chapter 1, Problem 1.59P

(a)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$2.420 g+15.6 g4.8 g$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit. The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(a)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $3.8 g$ .

Explanation of Solution

The given expression is,

$2.420 g+15.6 g4.8 g$

The least significant digit in each number is underlined as follows:

$2.420_ g+15.6_ g4.8_ g$

Add $2.420_ g$ and $15.6_ g$ in the numerator as:

$2.420_ g+15.6_ g=18.0_2 g$

The result of the addition operation must carry the same number of decimal places as the measurement with fewest decimal places. Measurement $15.6_ g$ has one digit after the decimal point. Therefore, the least significant figure in the result is 0.

Divide $18.02 g$ by $4.8 g$ as:

$18.02 g4.8 g=3.7_54 g=3.8 g$

The result of the division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $18.02 g$ has 4 significant figures and $4.8 g$ has 2 significant figures. The result must have 2 significant figures. Therefore, the final answer is rounded off to the 2 significant figures.

The last digit to be dropped is 5 and the preceding digit is odd thus it is increased by 1.

Conclusion

The final answer is rounded off to 2 significant figures. The answer of the calculation to a correct number of significant figures is $3.8 g$ .

Answer 7

Chapter 1, Problem 1.59P

(a)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$2.420 g+15.6 g4.8 g$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit. The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

Answer 8

(a)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $3.8 g$ .

Answer 9

(a)

Expert Solution

Answer 10

(a)

Expert Solution

Answer 11

Expert Solution

Answer 12

Explanation of Solution

The given expression is,

$2.420 g+15.6 g4.8 g$

The least significant digit in each number is underlined as follows:

$2.420_ g+15.6_ g4.8_ g$

Add $2.420_ g$ and $15.6_ g$ in the numerator as:

$2.420_ g+15.6_ g=18.0_2 g$

The result of the addition operation must carry the same number of decimal places as the measurement with fewest decimal places. Measurement $15.6_ g$ has one digit after the decimal point. Therefore, the least significant figure in the result is 0.

Divide $18.02 g$ by $4.8 g$ as:

$18.02 g4.8 g=3.7_54 g=3.8 g$

The result of the division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $18.02 g$ has 4 significant figures and $4.8 g$ has 2 significant figures. The result must have 2 significant figures. Therefore, the final answer is rounded off to the 2 significant figures.

The last digit to be dropped is 5 and the preceding digit is odd thus it is increased by 1.

Answer 13

Conclusion

The final answer is rounded off to 2 significant figures. The answer of the calculation to a correct number of significant figures is $3.8 g$ .

Answer 14

(b)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$7.87 mL16.1 mL−8.44 mL$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit.

The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(b)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $1.0$ .

Explanation of Solution

The given expression is,

$7.87 mL16.1 mL−8.44 mL$

The least significant digit in each number is underlined as follows:

$7.87_ mL16.1_ mL−8.44_ mL$

Subtract $8.44 mL$ from $16.1 mL$ in the denominator as,

$16.1_ mL−8.44_ mL=7.6_6 mL$

The result of the subtraction operation must carry the same number of decimal places as the measurement with fewest decimal places. Measurement $16.1 mL$ has one digit after the decimal point. Therefore, the result must also contain only one digit after the decimal point. The least significant figure in the result is 6.

Divide $7.87 mL$ by $7.66 mL$ .

$7.87_ mL7.6_6 mL=1.0_274=1.0$

The result of the division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $7.6_6 mL$ has 2 significant figures and $7.87_ mL$ has 3 significant figures. The result must have 2 significant figures. Therefore, the final answer must be rounded off to the 2 significant figures. The last digit to be dropped is 2. 2 is less than 5, therefore, the preceding digit remains the same.

Conclusion

The final answer is rounded off to 2 significant figures. The answer of the calculation to a correct number of significant figures is $1.0$ .

Answer 15

(b)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$7.87 mL16.1 mL−8.44 mL$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit.

The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

Answer 16

(b)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $1.0$ .

Answer 17

(b)

Expert Solution

Answer 18

(b)

Expert Solution

Answer 19

Expert Solution

Answer 20

Explanation of Solution

The given expression is,

$7.87 mL16.1 mL−8.44 mL$

The least significant digit in each number is underlined as follows:

$7.87_ mL16.1_ mL−8.44_ mL$

Subtract $8.44 mL$ from $16.1 mL$ in the denominator as,

$16.1_ mL−8.44_ mL=7.6_6 mL$

The result of the subtraction operation must carry the same number of decimal places as the measurement with fewest decimal places. Measurement $16.1 mL$ has one digit after the decimal point. Therefore, the result must also contain only one digit after the decimal point. The least significant figure in the result is 6.

Divide $7.87 mL$ by $7.66 mL$ .

$7.87_ mL7.6_6 mL=1.0_274=1.0$

The result of the division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $7.6_6 mL$ has 2 significant figures and $7.87_ mL$ has 3 significant figures. The result must have 2 significant figures. Therefore, the final answer must be rounded off to the 2 significant figures. The last digit to be dropped is 2. 2 is less than 5, therefore, the preceding digit remains the same.

Answer 21

Conclusion

The final answer is rounded off to 2 significant figures. The answer of the calculation to a correct number of significant figures is $1.0$ .

Answer 22

(c)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$V=πr2h$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit.

The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

(c)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $565 cm3$ .

Explanation of Solution

The value of $r$ is $6.23 cm$ and the value of $h$ is $4.630 cm$ .

The formula to calculate the volume is,

$V=πr2h$

Substitute $6.23 cm$ for $r$ , $4.630 cm$ for $h$ and $3.14159$ for $π$ in the above equation.

$V=(3.14159_)(6.23_ cm)2(4.630_ cm)=564_.556 cm3=565 cm3$

The result of the multiplication and division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $6.23 cm$ has 3 significant figures, $4.630 cm$ has 4 significant figures and $3.14159$ has 6 significant figures. The result must have 3 significant figures. Therefore, the final answer must be rounded off to the 3 significant figures. The last digit to be dropped is 5 thus the preceding digit is increased by 1.

Conclusion

The final answer is rounded off to 3 significant figures. The answer of the calculation to a correct number of significant figures is $565 cm3$ .

Answer 23

(c)

Interpretation Introduction

Interpretation:

The following calculation is to be solved to the correct number of significant figures.

$V=πr2h$

Concept introduction:

Significant figures of a number are the digits which carry meaningful contribution to its measurement resolution. The rightmost digit of the quantity is the most uncertain digit.

The number of certain and uncertain digit in a quantity is considered as significant figures. The digit with a higher number of significant figures has a higher certainty of measurement.

To determine the number of significant figures in a quantity following steps is followed.

1. The quantity must has a decimal point.

2. Start counting from the left and proceed towards the right until the first nonzero digit is encountered. All nonzero digit and the zeroes between two nonzero digits are considered as significant figures. For example, 0.0000765 has three significant figures and 7009 has four significant figures.

3. Zeroes after a decimal point are significant figures. For example, 42.0 have three significant figures.

4. Trailing zeroes that do nothing but are used to set a decimal point are non-significant figures. However, exponential notation can be used to avoid confusion. For example, 4300 has 3 significant figures. It can be expressed in scientific notation as $4.30×103$ or

$4.300×103$ . The number of significant figures in $4.30×103$ and $4.300×103$ is 3 and 4 respectively.

5. Zeroes present before a trailing decimal point are significant figures. For example, 3200 has only two significant figures but 3200. has 4 significant figures.

Rules to determine significant figures in calculations are as follows:

(1) In multiplication and division operations the result carries the same number of significant figures as the operand or measurement with the fewest significant figures.

(2) In addition and subtraction operations, the result carries the same number of decimal places as the operand or measurement with fewest decimal places.

(3) Exact numbers do not affect the number of significant digits in the final answer.

In mathematical expression which involves mixed operations the result of each intermediate step with proper significant figures. Avoid rounding of the result at intermediate steps. Round off the final answer of the calculation. The rules to round off are as follows:

(1) If the last dropped digit is greater than 5 then increase the preceding digit by 1.

(2) If the last dropped digit is less than 5 then the preceding digit does not change.

(3) If the last digit dropped is 5, then the preceding digit is increased by 1 if it is odd and remains the same if it is even. Also, if 5 is followed by zeroes only then rule (3) is applicable and if it is followed by non-zero digit then rule (1) is applicable.

Answer 24

(c)

Expert Solution

Answer to Problem 1.59P

The answer of the calculation to a correct number of significant figures is $565 cm3$ .

Answer 25

(c)

Expert Solution

Answer 26

(c)

Expert Solution

Answer 27

Expert Solution

Answer 28

Explanation of Solution

The value of $r$ is $6.23 cm$ and the value of $h$ is $4.630 cm$ .

The formula to calculate the volume is,

$V=πr2h$

Substitute $6.23 cm$ for $r$ , $4.630 cm$ for $h$ and $3.14159$ for $π$ in the above equation.

$V=(3.14159_)(6.23_ cm)2(4.630_ cm)=564_.556 cm3=565 cm3$

The result of the multiplication and division operation must carry the same number of significant figures as the measurement with the fewest significant figures. $6.23 cm$ has 3 significant figures, $4.630 cm$ has 4 significant figures and $3.14159$ has 6 significant figures. The result must have 3 significant figures. Therefore, the final answer must be rounded off to the 3 significant figures. The last digit to be dropped is 5 thus the preceding digit is increased by 1.