Question

For a precious stone, carat is used for specifying its mass. If $1{\text{carat = 3}}{\text{.168grains}}$ (a unit of mass) and \[1{\text{g = 15}}{\text{.4grains}}\], find the total mass in kilogram of the ring that contains \[0.5{\text{carat}}\] diamond and \[7{\text{g}}\] gold.
A. \[7.1 \times {10^4}{\text{kg}}\]
B. \[7.1 \times {10^{ - 4}}{\text{kg}}\]
C. \[7.1 \times {10^3}{\text{kg}}\]
D. \[7.1 \times {10^{ - 3}}{\text{kg}}\]

Answer 1

Hint: Carat is a metric unit of weight of the gemstone in industry. It is equal to one-fifth of a gram or five carats in a gram. Grains are generally used for measuring cereals. \[1{\text{g}}\] is equal to \[15.4{\text{grains}}\].From this the total mass can be calculated.
Given data:
$1{\text{carat = 3}}{\text{.168grains}}$
\[1{\text{g = 15}}{\text{.4grains}}\]

Complete step by step answer:
Weight of gold is commonly measured in troy ounces. But now, a metric system has been introduced. Therefore it is measured in grams. While purity of gold is measured in carats. \[24{\text{carats}}\] of gold indicates that it is \[100\% \] pure. Generally gold is mixed with harder metals. Different countries use carats for different measurements.
It is given that $1{\text{carat = 3}}{\text{.168grains}}$
And \[15.4{\text{grains}} = 1{\text{g}}\]
Therefore \[1{\text{grain}} = \dfrac{1}{{15.4}}{\text{g}}\]
So \[1{\text{carat}}\] can be written with respect to gram.
 $ 3.168{\text{grains}} = \dfrac{{3.168}}{{15.4}}{\text{g}} \\
  {\text{1carat = 3}}{\text{.168grains = }}\dfrac{{3.168}}{{15.4}}{\text{g}} = 0.206{\text{g}} \\ $
Now the ring contains \[0.5{\text{carat}}\] diamond and \[7{\text{g}}\] gold.
We have to calculate the total mass in a kilogram of ring.
\[1{\text{carat}} = 0.206{\text{g}}\]
\[0.5{\text{carat}} = 0.206{\text{g}} \times 0.5 = 0.103{\text{g}}\]
So the ring contains \[0.103{\text{g}}\] diamond and \[7{\text{g}}\] gold.
Therefore it can be said that the ring contains total of \[0.103{\text{g}} + 7{\text{g = 7}}{\text{.103g}}\]
This total mass of \[7.103{\text{g}}\] can be converted to kilograms.
\[1000{\text{g}} = 1{\text{kg}}\]
 $ 1{\text{g}} = \dfrac{1}{{1000}}{\text{kg}} = 0.001{\text{kg}} \\
  \therefore {\text{7}}{\text{.103g = 0}}{\text{.001}} \times {\text{7}}{\text{.103 = 7}}{\text{.103}} \times {\text{1}}{{\text{0}}^{ - 3}}{\text{kg}} \\ $
Therefore, the total mass in kilogram in ring is \[7.103 \times {10^{ - 3}}{\text{kg}}\].

Additional information- Diamonds are generally weighed using carats. Carat is divided into 100 points. For example, a diamond weighing \[0.5{\text{carats}}\] is referred to as a fifty pointer.


Note:
Bullets and propellants are commonly measured using grain. It is a standard unit for measuring weight of arrows. 1 gram (g) = 15.4323584 grains or "troy grains" (gr) = 1000000 microgram.