If $3 \frac{2}{3}$ is subtracted from $9 \frac{1}{9}$ and the difference is multiplied by 450, what is the final answer?

### Concept & Strategy This is a straight-forward application of the order of operations (BODMAS/PEMDAS) involving mixed fractions. Convert mixed numbers to improper fractions first to avoid borrowing errors during subtraction. ### Step-by-Step Solution * **Convert mixed fractions to improper fractions:** $$9 \frac{1}{9} = \frac{(9 \times 9) + 1}{9} = \frac{82}{9}$$ $$3 \frac{2}{3} = \frac{(3 \times 3) + 2}{3} = \frac{11}{3}$$ * **Perform the subtraction:** $$\text{Difference} = \frac{82}{9} - \frac{11}{3}$$ Make denominators equal (multiply numerator and denominator of the second fraction by 3): $$\text{Difference} = \frac{82}{9} - \frac{33}{9} = \frac{49}{9}$$ * **Perform the final multiplication:** The difference is multiplied by 450. $$\text{Final Answer} = \frac{49}{9} \times 450$$ Simplify by dividing 450 by 9: $\frac{450}{9} = 50$. $$\text{Final Answer} = 49 \times 50$$ $$49 \times 50 = 2450$$ * **Check against options:** The calculated answer is 2450. The given numerical options are 2045, 2250, and 2540. Since 2450 is missing, the answer must be "None of these". ### Exam Strategy & Shortcut Subtract the integer and fractional parts separately if you are comfortable with negative fractions. Integer difference: $9 - 3 = 6$ Fraction difference: $\frac{1}{9} - \frac{2}{3} = \frac{1}{9} - \frac{6}{9} = -\frac{5}{9}$ Total difference: $6 - \frac{5}{9} = \frac{54 - 5}{9} = \frac{49}{9}$. Multiply by 450: $\frac{49}{9} \times 450 = 49 \times 50 = 2450$. ### Common Pitfall A common error is misreading option (c) 2540 as 2450 under exam pressure. Test setters intentionally include transposed numbers (2450 vs 2540) to trap students who rush the final multiple-choice selection step. ### Final Answer **Therefore, the correct answer is None of these.**