Question

The distance between city $A$ and $B$ is $100$ times the distance between city $B$ and $C.C$ is $38\times {10}^a$ miles away from city $B$ and city $A$ is $38\times {10}^b$ miles away from city $B$. Choose the expression representing the relation between the distances.
[Hint: Convert both the distance in scientific notation and compare]

Answer 1

Detailed step-by-step solution:


Let the distance between $\text{city A and city B}$ be $S_{AB}$ and distance between $\text{city B and city C be}{S}_{BC}$

$S_{BC}=38\times {10}^a\text{miles}=3.8\times {10}^{a+1}(38=3.8\times {10}^1)$

$S_{AB}=3.8\times {10}^b\text{miles}$
$S_{AB}=100S_{BC}$

$\Rightarrow 3.8\times {10}^b=100\times 3.8\times {10}^{a+1}$

$\Rightarrow {10}^b={10}^2\times {10}^{a+1}$

$\Rightarrow {10}^b={10}^{a+1+2}$ (by using product of powers property, i.e., $\frac{a^m}{a^n}=a^{m+n}$ )
$\Rightarrow {10}^b={10}^{a+3}$

As the bases are equal, we can equate the exponents.
$b=a+3$
$\Rightarrow b-a=3$

➡ Option D is correct.