Chapter 1, Problem 1.36QP

Carry out the following operations as if they were calculations of experimental results, and express each answer in the correct units with the correct number of significant figures:

1. (a) 7.310 km ÷ 5.70 km
2. (b) (3.26 × 10⁻³ mg) × (7.88 × 10⁻⁵ mg)
3. (c) (4.02 × 10⁶ dm) + (7.74 × 10⁷ dm)
4. (d) (7.8 m − 0.34 m)/(1.15 s + 0.82 s)

Interpretation Introduction

Interpretation:

The given mathematical operation has to be done in right way and the answer should be written in correct unit with the correct number of significant figures.

Concept introduction:

Significant figures are all the digits in a measurement that are known with certainty.

Rules for significant digits

• Digits from 1 to 9 are always significant
• Zeros between two other significant digits are always significant.
• One or more additional zeroes to the right of both the decimal place and other significant digits are significant.
• Zeroes used solely for spacing the decimal point are not significant.

Rules for rounding off numbers

If the digits to the immediate right of the last significant figure are less than five do not change.

Example: $2.532\to 2.53$

If the digit to the immediate right of the last significant figures is greater than five, round up the last significant figures.

Example: $2.536\to 2.54$

Explanation of Solution

Given,

$7.310km÷5.70km=?$

All the given terms are in same unit so conversions of units are not required. This division operation can be done as follows,

$7.310km÷5.70km=1.283$

The three in $1.283$ is a non-significant digit because the original number $5.70$ only has three significant digits. Therefore should have only three significant digits.

The answer with correct unit with the correct number of significant figures is $1.28$

Interpretation Introduction

Interpretation:

The given mathematical operation has to be done in right way and the answer should be written in correct unit with the correct number of significant figures.

Concept introduction:

Significant figures are all the digits in a measurement that are known with certainty.

Rules for significant digits

• Digits from 1 to 9 are always significant
• Zeros between two other significant digits are always significant
• One or more additional zeroes to the right of both the decimal place and other significant digits are significant.
• Zeroes used solely for spacing the decimal point are not significant.

Rules for rounding off numbers

If the digits to the immediate right of the last significant figure are less than five do not change.

Example: $2.532\to 2.53$

If the digit to the immediate right of the last significant figures is greater than five, round up the last significant figures.

Example: $2.536\to 2.54$

Explanation of Solution

Given,

$\left(3.26\times {10}^{-3}mg\right)-\left(7.88\times {10}^{-5}mg\right)=?$

All the given terms are in same unit so conversions of units are not required. This subtraction operation can be done as follows,

Writing both numbers in decimal notation,

$\begin{array}{l}0.00326mg-\\ \underset{\_}{0.0000788mg}\\ 0.0031812mg\end{array}$

The answer with correct unit with the correct number of significant figures is $0.00318mg$

The bolded digits in $0.0031812mg$ are not significant; because the number $0.00326$ has five digits to the right of the decimal point.

Interpretation Introduction

Interpretation:

The given mathematical operation has to be done in right way and the answer should be written in correct unit with the correct number of significant figures.

Concept introduction:

Significant figures are all the digits in a measurement that are known with certainty.

Rules for significant digits

• Digits from 1 to 9 are always significant
• Zeros between two other significant digits are always significant.
• One or more additional zeroes to the right of both the decimal place and other significant digits are significant.
• Zeroes used solely for spacing the decimal point are not significant.

Rules for rounding off numbers

If the digits to the immediate right of the last significant figure are less than five do not change.

Example: $2.532\to 2.53$

If the digit to the immediate right of the last significant figures is greater than five, round up the last significant figures.

Example: $2.536\to 2.54$

Explanation of Solution

Given,

$\left(4.02\times {10}^{6}dm\right)+\left(7.74\times {10}^{7}dm\right)=?$

All the given terms are in same unit so conversions of units are not required. This addition operation can be done as follows,

Writing both number with exponents which is equal to $+7$

$\left(0.402\times {10}^{7}dm\right)+\left(7.74\times {10}^{7}dm\right)=8.14\times 10^{7}dm$

The answer with correct unit with the correct number of significant figures is $8.14\times 10^{7}dm$

Interpretation Introduction

Interpretation:

The given mathematical operation has to be done in right way and the answer should be written in correct unit with the correct number of significant figures.

Concept introduction:

Significant figures are all the digits in a measurement that are known with certainty.

Rules for significant digits

• Digits from 1 to 9 are always significant
• Zeros between two other significant digits are always significant.
• One or more additional zeroes to the right of both the decimal place and other significant digits are significant.
• Zeroes used solely for spacing the decimal point are not significant.

Rules for rounding off numbers

If the digits to the immediate right of the last significant figure are less than five do not change.

Example: $2.532\to 2.53$

If the digit to the immediate right of the last significant figures is greater than five, round up the last significant figures.

Example: $2.536\to 2.54$

Explanation of Solution

Given,

$\left(7.8m-0.34m\right)/\left(1.15s+0.82s\right)=?$

This subtraction, addition and division operation can be done as follows,

$\frac{\left(7.8m-0.34m\right)}{\left(1.15s+0.82s\right)}=\frac{7.5m}{1.97s}=3.8m/s$

The answer with correct unit with the correct number of significant figures is $3.8m/s$