- Evaluate each expression to its decimal form: \(A = 2.6\), \(B = -3.125\), \(C = 10.35\), \(D = -2.95\).
- Compare the numerical values: \(-3.125 < -2.95 < 2.6 < 10.35\).
- Map the decimals back to their original expressions.
- The order from least to greatest is \(\boxed{\frac{5}{-1.6}, -3 \frac{3}{10} - \left(-\frac{7}{20}\right), 5 \frac{6}{15} + \left(-2 \frac{4}{5}\right), -4.5 \times -2.3}\).
Explanation
- Analyzing the Expressions
We have four mathematical expressions involving fractions, mixed numbers, and decimals. To order them from least to greatest, we must evaluate each expression individually and then compare the results.
The expressions are:
- \(A = 5 \frac{6}{15} + \left(-2 \frac{4}{5}\right)\)
- \(B = \frac{5}{-1.6}\)
- \(C = -4.5 \times -2.3\)
- \(D = -3 \frac{3}{10} - \left(-\frac{7}{20}\right)\)
We will convert each expression to a standard decimal representation to facilitate direct comparison.
- Simplifying the Expressions
We simplify each expression step-by-step:
- For \(A = 5 \frac{6}{15} - 2 \frac{4}{5}\), simplify the fractions: \(5 \frac{6}{15} = 5 \frac{2}{5} = 5.4\), and \(2 \frac{4}{5} = 2.8\). So, \(A = 5.4 - 2.8 = 2.6\).
- For \(B = \frac{5}{-1.6}\), we perform the division: \(5 \div -1.6 = -3.125\).
- For \(C = -4.5 \times -2.3\), we multiply: Since both numbers are negative, the result is positive: \(4.5 \times 2.3 = 10.35\).
- For \(D = -3 \frac{3}{10} - \left(-\frac{7}{20}\right)\), simplify: \(-3.3 + \frac{7}{20} = -3.3 + 0.35 = -2.95\).
- Comparing and Ordering Values
We now have the numerical values:
\(A = 2.6\)
\(B = -3.125\)
\(C = 10.35\)
\(D = -2.95\)
Comparing these values:
- The smallest value is \(-3.125\) (Expression \(B\)).
- The next smallest value is \(-2.95\) (Expression \(D\)).
- The next value is \(2.6\) (Expression \(A\)).
- The largest value is \(10.35\) (Expression \(C\)).
Ordering them from least to greatest, we get: \(B < D < A < C\).
- Final Ordering
Substituting the original expressions back into our order, we find:
\(\frac{5}{-1.6} < -3 \frac{3}{10} - \left(-\frac{7}{20}\right) < 5 \frac{6}{15} + \left(-2 \frac{4}{5}\right) < -4.5 \times -2.3\).
Thus, the final ordered sequence is \(\frac{5}{-1.6}\), \(-3 \frac{3}{10} - \left(-\frac{7}{20}\right)\), \(5 \frac{6}{15} + \left(-2 \frac{4}{5}\right)\), \(-4.5 \times -2.3\). The final order is \(\boxed{\frac{5}{-1.6}, -3 \frac{3}{10} - \left(-\frac{7}{20}\right), 5 \frac{6}{15} + \left(-2 \frac{4}{5}\right), -4.5 \times -2.3}\).
Examples
Understanding how to order mixed operations with rational numbers is essential for financial planning and data analysis. For example, when managing multiple investment accounts, some might show a percentage loss (negative) while others show a gain (positive). By calculating the net value of each account using the correct order of operations, you can rank your assets from least profitable to most profitable, helping you make informed decisions about where to reallocate your funds for better overall performance. This logical ordering is a fundamental step in real-world financial management.