To solve the given algebraic expression \(\left(2a - \frac{1}{3}\right) \div \frac{b}{15}\) when \(a = -\frac{4}{5}\) and \(b = -2.25\), follow these steps:
- Substitute the given values of \(a\) and \(b\) into the expression.
- Simplify the expression inside the parentheses.
- Perform the division operation.
We start with the expression
\[
\left(2a - \frac{1}{3}\right) \div \frac{b}{15}
\]
Substituting \(a = -\frac{4}{5}\) and \(b = -2.25\) (which is equivalent to \(-\frac{9}{4}\)), we have:
\[
\left(2 \left(-\frac{4}{5}\right) - \frac{1}{3}\right) \div \frac{-\frac{9}{4}}{15}
\]
Calculating the numerator:
\[
2 \left(-\frac{4}{5}\right) = -\frac{8}{5}
\]
Now, we combine this with \(-\frac{1}{3}\):
\[
-\frac{8}{5} - \frac{1}{3}
\]
To combine these fractions, we find a common denominator, which is 15:
\[
-\frac{8}{5} = -\frac{24}{15}, \quad -\frac{1}{3} = -\frac{5}{15}
\]
Thus,
\[
-\frac{24}{15} - \frac{5}{15} = -\frac{29}{15}
\]
Next, we simplify the denominator:
\[
\frac{-\frac{9}{4}}{15} = -\frac{9}{4} \cdot \frac{1}{15} = -\frac{9}{60} = -\frac{3}{20}
\]
Now we perform the division:
\[
-\frac{29}{15} \div -\frac{3}{20} = -\frac{29}{15} \cdot -\frac{20}{3} = \frac{29 \cdot 20}{15 \cdot 3} = \frac{580}{45}
\]
We simplify \(\frac{580}{45}\):
\[
\frac{580 \div 5}{45 \div 5} = \frac{116}{9}
\]
To convert \(\frac{116}{9}\) to a mixed number:
\[
116 \div 9 = 12 \quad \text{(whole number)}
\]
\[
116 \mod 9 = 8 \quad \text{(remainder)}
\]
Thus,
\[
\frac{116}{9} = 12 \frac{8}{9}
\]
The final answer is
\[
\boxed{12 \frac{8}{9}}
\]
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