Answer :
We start with the information that one grain has a mass of exactly [tex]$0.0648$[/tex] grams. This means that one gram is equal to
[tex]$$
\frac{1}{0.0648} = \frac{1250}{81} \text{ grains},
$$[/tex]
since [tex]$0.0648 = \frac{81}{1250}$[/tex] exactly.
Now, we want to convert [tex]$2.3$[/tex] grams into grains. First, express [tex]$2.3$[/tex] grams as a fraction:
[tex]$$
2.3 = \frac{23}{10} \text{ grams}.
$$[/tex]
Next, multiply this by the conversion factor [tex]$\frac{1250}{81}$[/tex] to get the number of grains:
[tex]$$
\frac{23}{10} \times \frac{1250}{81} = \frac{23 \times 1250}{10 \times 81} = \frac{28750}{810}.
$$[/tex]
We can simplify this fraction by dividing the numerator and the denominator by [tex]$10$[/tex]:
[tex]$$
\frac{28750}{810} = \frac{2875}{81}.
$$[/tex]
Thus, the exact number of grains in [tex]$2.3$[/tex] grams is
[tex]$$
\frac{2875}{81} \text{ grains}.
$$[/tex]
To express this as a mixed number, we divide [tex]$2875$[/tex] by [tex]$81$[/tex]. We note that
[tex]$$
35 \times 81 = 2835 \quad \text{and} \quad 2875 - 2835 = 40.
$$[/tex]
Therefore,
[tex]$$
\frac{2875}{81} = 35 \frac{40}{81}.
$$[/tex]
Thus, the conversion of [tex]$2.3$[/tex] grams to grains is
[tex]$$
\boxed{35 \frac{40}{81}}.
$$[/tex]
[tex]$$
\frac{1}{0.0648} = \frac{1250}{81} \text{ grains},
$$[/tex]
since [tex]$0.0648 = \frac{81}{1250}$[/tex] exactly.
Now, we want to convert [tex]$2.3$[/tex] grams into grains. First, express [tex]$2.3$[/tex] grams as a fraction:
[tex]$$
2.3 = \frac{23}{10} \text{ grams}.
$$[/tex]
Next, multiply this by the conversion factor [tex]$\frac{1250}{81}$[/tex] to get the number of grains:
[tex]$$
\frac{23}{10} \times \frac{1250}{81} = \frac{23 \times 1250}{10 \times 81} = \frac{28750}{810}.
$$[/tex]
We can simplify this fraction by dividing the numerator and the denominator by [tex]$10$[/tex]:
[tex]$$
\frac{28750}{810} = \frac{2875}{81}.
$$[/tex]
Thus, the exact number of grains in [tex]$2.3$[/tex] grams is
[tex]$$
\frac{2875}{81} \text{ grains}.
$$[/tex]
To express this as a mixed number, we divide [tex]$2875$[/tex] by [tex]$81$[/tex]. We note that
[tex]$$
35 \times 81 = 2835 \quad \text{and} \quad 2875 - 2835 = 40.
$$[/tex]
Therefore,
[tex]$$
\frac{2875}{81} = 35 \frac{40}{81}.
$$[/tex]
Thus, the conversion of [tex]$2.3$[/tex] grams to grains is
[tex]$$
\boxed{35 \frac{40}{81}}.
$$[/tex]
