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

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.