What is 45% of 25% of $\frac{4}{5}$th of 850?

### Concept & Strategy This is a sequential percentage and fraction problem. The most effective strategy is to translate the English statement directly into a single continuous mathematical equation, replacing "of" with multiplication, and converting standard percentages into their simplest fraction forms before calculating. ### Step-by-Step Solution **Given:** * Equation: $45\%$ of $25\%$ of $\frac{4}{5}$ of $850$ **Calculation / Deduction:** * Convert the percentages to fractions for easier cancellation: $25\% = \frac{1}{4}$ $45\% = \frac{45}{100} = \frac{9}{20}$ * Rewrite the entire expression using these fractions: $$\frac{9}{20} \times \frac{1}{4} \times \frac{4}{5} \times 850$$ * Cancel out the $4$ in the numerator and denominator: $$\frac{9}{20} \times \frac{1}{5} \times 850$$ * Combine the denominators ($20 \times 5 = 100$): $$\frac{9}{100} \times 850$$ * Multiply $9 \times 850$: $7650$ * Divide by $100$: $$\frac{7650}{100} = 76.5$$ ### Exam Strategy & Shortcut Memorizing basic percentage-to-fraction conversions is crucial. Instantly seeing $25\%$ as $\frac{1}{4}$ allows you to immediately cancel it with the numerator in $\frac{4}{5}$, leaving just $\frac{1}{5}$. Then you simply find $\frac{1}{5}$ of $850$ ($170$) and take $45\%$ of $170$. $45\% = 50\% - 5\%$. $50\%$ of $170 = 85$. $5\%$ is $8.5$. $85 - 8.5 = 76.5$. This mental math completely bypasses heavy multiplication. ### Common Pitfall Students often try to calculate the values step-by-step from left to right (e.g., finding $45\%$ of $25\%$ first). This leads to ugly decimals. Always expand the entire expression and cancel terms vertically and diagonally before doing any actual multiplication. ### Final Answer Therefore, the correct answer is **76.5**.