Apothecaries (druggists) use the following set of measures in the English system: $$ \begin{aligned} 20 \text { grains ap } &=1 \text { scruple (exact) } \\ 3 \text { scruples } &=1 \text { dram ap (exact) } \\ 8 \text { dram ap } &=1 \mathrm{oz} \text { ap (exact) } \\ 1 \text { dram ap } &=3.888 \mathrm{~g} \end{aligned} $$ a. Is an apothecary grain the same as a troy grain? (See Exercise 45.) b. \(102 \mathrm{ap}=\)_____oz troy. c. An aspirin tablet contains \(5.00 \times 10^{2} \mathrm{mg}\) of active ingredient. What mass in grains ap of active ingredient does it contain? What mass in scruples? d. What is the mass of 1 scruple in grams?

We shall compare both metrics with a widely accepted unit of mass such as grams. From Exercise 45, we have that 1 grain troy = 0.0648 g. Therefore, to compare the apothecary grain to the troy grain, we need to convert 1 apothecary grain to grams: $$ 20 \text{ grains ap} = 1 \text{ scruple} $$ $$ 3 \text{ scruples} = 1 \text{ dram ap} $$ $$ 1 \text{ dram ap} = 3.888 \text{~g} $$ Multiplying these equations, we have: $$ (20 \times 3) \text{ grains ap} = 1 \text{ scruple} \times 1 \text{ dram ap} = 3.888 \text{~g} $$ $$ 60 \text{ grains ap} = 3.888 \text{~g} $$ Thus, 1 grain ap is equal to \( \frac{3.888}{60} \) g: $$ 1 \text{ grain ap} = 0.0648 \text{~g} $$ Therefore, 1 apothecary grain is the same as 1 troy grain.

We convert 102 grains ap to troy ounces using the given conversion relationships: $$ 102 \text{ grains ap} = \frac{102}{20} \text{ scruples} = 5.1 \text{ scruples} $$ $$ 5.1 \text{ scruples} = \frac{5.1}{3} \text{ drams ap} = 1.7 \text{ drams ap} $$ $$ 1.7 \text{ drams ap} = \frac{1.7}{8} \text{ oz ap} = 0.2125 \text{ oz ap} $$ Since 1 grain ap = 1 grain troy: $$ 102 \text{ grains ap} = 102 \text{ grains troy} = 0.2125 \text{ oz troy} $$

We first convert the given mass of the active ingredient from mg to g: $$ 5.00 \times 10^{2} \text{ mg} = 0.500 \text{~g} $$ Now we'll convert 0.500 g to grains ap. From our previous calculations, 1 grain ap = 0.0648 g. Thus: $$ 0.500 \text{~g} = \frac{0.500}{0.0648} \text{ grains ap} \approx 7.72 \text{ grains ap} $$ Finally, we convert grains ap to scruples: $$ 7.72 \text{ grains ap} = \frac{7.72}{20} \text{ scruples} \approx 0.386 \text{ scruples} $$ The aspirin tablet contains 7.72 grains ap and 0.386 scruples of the active ingredient.

From our initial calculations of conversion relationships, we have: $$ 1 \text{ scruple} = \frac{1}{3} \text{ dram ap} $$ and $$ 1 \text{ dram ap} = 3.888\text{~g} $$ To find the mass of 1 scruple in grams, we substitute the value of 1 dram ap in grams: $$ 1 \text{ scruple} = \frac{1}{3} \times 3.888 \text{~g} = 1.296 \text{~g} $$ Therefore, the mass of 1 scruple is 1.296 g.