Question

How many micrograms make $1kg$?

Answer 1

Hint:
We know that one kilogram is equal to a thousand grams and one microgram is equal to ${10^{ - 6}}$ grams. Start with making two separate equations for kilogram/gram and microgram/gram. Now combine these two equations eliminating the “gram” and form a single equation between kilogram and microgram.

Complete step by step answer:
Here in this problem, we have to change the weight of one kilogram into the unit of micrograms. And answer the question of how many micrograms are combined to make one kilogram.
Before starting with the solution, we must understand a few concepts of a gram. Gram (g) is a unit of mass or weight that is used especially in the centimeter-gram-second system of measurement. One gram is equal to $0.001kg$. The gram is very nearly equal to the mass of one cubic centimeter of pure water at $4^\circ C\left( {39.2^\circ F} \right)$, the temperature at which water reaches its maximum density under normal terrestrial pressures.
A microgram is a unit of mass that is equal to one-millionth of a gram. A microgram is a millionth of a gram. The abbreviation mcg stands for microgram. A microgram is a unit of mass that is equal to one-millionth of a gram.
Therefore, according to the definition:
$ \Rightarrow 1{\text{ }}microgram = {10^{ - 6}}gram{\text{ or }}1\mu g = 1 \times {10^{ - 6}}g$ ………(A)
The prefix used in kilogram, i.e. “kilo”, means thousand times of one gram.
$ \Rightarrow 1{\text{ kilogram}} = 1 \times {10^3}{\text{ }}gram{\text{ or }}1kg = 1000g$ ……….(B)
So for finding the micrograms in kilogram, we just need to express the gram in the RHS of (B) in the form of micrograms.
Now transposing ${10^{ - 6}}$ from RHS to LHS in (A), we get:
$ \Rightarrow 1\mu g = 1 \times {10^{ - 6}}g \Rightarrow \dfrac{{{{10}^{ - 6}}}}{{{{10}^{ - 6}}}}g = \dfrac{1}{{{{10}^{ - 6}}}}\mu g$
Therefore, the relation between one gram and microgram will be
$ \Rightarrow \dfrac{{{{10}^{ - 6}}}}{{{{10}^{ - 6}}}}g = \dfrac{1}{{{{10}^{ - 6}}}}\mu g \Rightarrow 1g = {10^6}\mu g$
Let’s use this relation in the equation (B)
$ \Rightarrow 1kg = 1000g \Rightarrow 1kg = 1000 \times \left( {{{10}^6}g} \right)$
On further simplifying it, we get:
$ \Rightarrow 1kg = 1000 \times \left( {{{10}^6}\mu g} \right) = {10^9}\mu g$
Thus, there are ${10^{^9}}$ micrograms in one kilogram of mass.

Note:
Remember that the prefix “kilo” (symbol: ‘k’) or prefix “micro” (symbol: $'\mu '$) will represent the same factor if it is attached to quantities of length or mass. As we know one micrometer is equal to the length of ${10^{ - 6}}$ meters and one kilometer is equal to $1000$ meters. An alternate approach to this problem can be to transform equation (A) to express the gram in RHS in form of a kilometer. This will give you a relation between kilometer and micrometer.