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Question: If \(1gc{m^{ - 3}} = {10^3}kg{m^{ - 3}}\) then \({10^6}kg{m^{ - 3}} = \) (A) \({10^{ - 3}}kgc{m^...

If 1gcm−3=103kgm−31gc{m^{ - 3}} = {10^3}kg{m^{ - 3}} then 106kgm−3={10^6}kg{m^{ - 3}} =
(A) 10−3kgcm−3{10^{ - 3}}kgc{m^{ - 3}}
(B) 106gcm−3{10^6}gc{m^{ - 3}}
(C) 103gcm−3{10^3}gc{m^{ - 3}}
(D) 10−6gcm−3{10^{ - 6}}gc{m^{ - 3}}

Explanation

Solution

Since the conversion factor is already given in the question you don’t have to change everything in terms of one unit. We need to use this information to write 10 kg/m in terms of the required quantity.

Complete step-by-step solution
In the question it is given that 1gcm−3=103kgm−31gc{m^{ - 3}} = {10^3}kg{m^{ - 3}} so 106kgm−3={10^6}kg{m^{ - 3}} = x value in terms of gcm−3gc{m^{ - 3}}
1gcm−3=103kgm−3 xgcm−3=106kgm−3 x=106×1103 x=106−3 x=103gcm−3  1 g c{m^{ - 3}} = {10^3}kg{m^{ - 3}} \\\ x gc{m^{ - 3}} = {10^6}kg{m^{ - 3}} \\\ x = \dfrac{{{{10}^6} \times 1}}{{{{10}^3}}} \\\ x = {10^{6 - 3}} \\\ x = {10^3}gc{m^{ - 3}} \\\

Hence, the value of 106kgm−3{10^6}kg{m^{ - 3}} is equivalent to 103gcm−3{10^3}gc{m^{ - 3}} and the correct option among them is C.

Note: This can also be solved by converting all the units that are kilogram (kg) into grams (g) and meter (m) into centimeter (cm). We know the following conversion factor
1 g = 10−3{10^{ - 3}} kg which implies that 1 kg = 1000 g = 10−3{10^{ - 3}} g
1 cm = 10−2{10^{ - 2}} m which implies that 1 m = 100 cm = 10−2{10^{ - 2}} cm
Here are some prefix values which can be used for conversions

| S I Prefixes|
---|---|---
Multiple| Prefix| Symbol
1015{10^{15}} | peta| P
1012{10^{12}} | tera| T
109{10^9} | giga| G
106{10^6} | mega| M
103{10^3} | kilo| k
102{10^2} | hecto| h
10| deka| da
10−1{10^{ - 1}} | deci| d
10−2{10^{ - 2}} | centi| c
10−3{10^{ - 3}} | milli| m
10−6{10^{ - 6}} | micro| μ\mu
10−9{10^{ - 9}} | nano| n
10−12{10^{ - 12}} | pico| p
10−15{10^{ - 15}} | femto| f