Question

The addition of two numbers $6.75 \times {10^3}\,cm$ and $4.52 \times {10^2}\,cm$ with regard to significant figures is:
A. $7.20 \times {10^3}\,cm$
B. $7.20 \times {10^3}\,mm$
C. $7.20 \times {10^4}\,cm$
D. None of these

Answer 1

In order to solve this question, we should know about significant figures and their laws. Significant figures are the number of digits in a numerical value which are needed to define the numeric number accurately. We will discuss $5$ basic laws of significant figures and then find the addition of two given numbers using these laws.

Complete step by step answer:
$5$ basic laws of significant figure:
(i) All the non-zero numbers in a numerical value are considered as significant figures.
(ii) If a numerical value starts with zeroes for example $0025$ then two zeroes do not count as significant figures.
(iii) Zeroes in between two significant digits are counted for example $2005$ two zeroes are counted as they lie between two and five which are non-zero significant figures.
(iv) Zeroes just after the decimal point are counted, for example $25.00$ two zeros after decimal are significant figures.
(v) Any number written in the form of $Y \times {10^n}$ where Y is a significant numerical value then a power of ten which is n are not significant digits.

Now, we have given two values as $6.75 \times {10^3}cm$ and $4.52 \times {10^2}cm$ while addition, we keep the power same of each value for evaluation so we can convert $4.52 \times {10^2}cm$ into $0.452 \times {10^3}cm$ Now, we have two values having same exponent of ten as $6.75 \times {10^3}cm$ and $0.452 \times {10^3}cm$ now, we will add significant digits of each number keeping ${10^3}$ as common which can be written as
$\text{Addition} = 6.75 \times {10^3} + 0.452 \times {10^3}$
$\Rightarrow \text{Addition} = (6.75 + 0.452){10^3}$
$\Rightarrow \text{Addition} = 7.202 \times {10^3}cm$

Since, least number of significant digits after decimal in two values $6.75 \times {10^3}\,cm$ and $0.452 \times {10^3}\,cm$ are two in $6.75 \times {10^3}\,cm$ so, addition of these two values will also contain only two digits after decimal.On rounding off up to two decimal values we get,
$\therefore \text{Addition} = 7.20 \times {10^3}\,cm$

Hence, the correct option is A.

Note: It should be remembered that, rounding off a decimal numerical value means to keep simple and close value of numerical values for example if a value is written as $52.026$ and we need to round off up to two decimals then we will look at the digit after two decimal which is $6$ in this example and if this digit is greater than $5$ then we add $1$ to last digit of rounding off which will became $52.03$ and if the rounding off digit is less than or equal to $5$ such as in example $52.024$ then we write simple $52.02$ ,these are the rules of rounding off a numerical value.