1. Вычислите: a) \( \frac{1}{2} \cdot \frac{1}{19} - 9 \cdot \frac{2}{1} - 0.25 \cdot 31 \frac{1}{4} \)

Решение:

1. Первое действие: умножение.
\(\documentclass{article}\) \(\usepackage{amsmath}\) \(\begin{document}\) \(\begin{equation*}\) \(\frac{1}{2}\) \(\cdot\) \(\frac{1}{19}\) = \(\frac{1}{38}\) \(\end{equation*}\) \(\end{document}\)

Второе действие: умножение.

\(\documentclass{article}\) \(\usepackage{amsmath}\) \(\begin{document}\) \(\begin{equation*}\) 9 \(\cdot\) 2 = 18 \(\end{equation*}\) \(\end{document}\)

Третье действие: умножение.

\(\documentclass{article}\) \(\usepackage{amsmath}\) \(\begin{document}\) \(\begin{equation*}\) 0.25 \(\cdot\) 31 \(\frac{1}{4}\) = \(\frac{1}{4}\) \(\cdot\) \(\frac{125}{4}\) = \(\frac{125}{16}\) \(\end{equation*}\) \(\end{document}\)

Четвёртое действие: вычитание.

\(\documentclass{article}\) \(\usepackage{amsmath}\) \(\begin{document}\) \(\begin{equation*}\) \(\frac{1}{38}\) - 18 = \(\frac{1 - 18 \cdot 38}{38}\) = \(\frac{1 - 684}{38}\) = -\(\frac{683}{38}\) \(\end{equation*}\) \(\end{document}\)

Пятое действие: вычитание.

\(\documentclass{article}\) \(\usepackage{amsmath}\) \(\begin{document}\) \(\begin{equation*}\) -\(\frac{683}{38}\) - \(\frac{125}{16}\) = -\(\frac{683 \cdot 8 + 125 \cdot 19}{304}\) = -\(\frac{5464 + 2375}{304}\) = -\(\frac{7839}{304}\) \(\end{equation*}\) \(\end{document}\)

Ответ: $$ -\frac{7839}{304} $$.