1. Stating the problem: We are given several variables and equations involving $A$, and we want to find the value of $A$ given that $H = A + B + C + D + E + F = 8999.16$. 2. Given: - $B = \frac{A \times 3}{12} = \frac{3A}{12} = \frac{A}{4}$ - $C = 0$ - $D = 350$ - $E = 200$ - $F = \left(\frac{A}{30 \times 8} \times 0.05\right) + \left(\frac{A + B + C + D + E}{30 \times 8}\right) \times 52$ - $H = A + B + C + D + E + F = 8999.16$ 3. Substitute $B$, $C$, $D$, and $E$ into $H$: $$H = A + \frac{A}{4} + 0 + 350 + 200 + F = 8999.16$$ 4. Simplify the sum without $F$: $$A + \frac{A}{4} + 350 + 200 = \frac{4A}{4} + \frac{A}{4} + 550 = \frac{5A}{4} + 550$$ 5. Express $F$ in terms of $A$: First, calculate $\frac{A}{30 \times 8} = \frac{A}{240}$. Then, $$F = \left(\frac{A}{240} \times 0.05\right) + \left(\frac{A + \frac{A}{4} + 0 + 350 + 200}{240}\right) \times 52$$ Simplify numerator inside second term: $$A + \frac{A}{4} + 350 + 200 = \frac{5A}{4} + 550$$ So, $$F = \frac{0.05A}{240} + \frac{\frac{5A}{4} + 550}{240} \times 52 = \frac{0.05A}{240} + \frac{52}{240} \left(\frac{5A}{4} + 550\right)$$ 6. Simplify $F$: $$F = \frac{0.05A}{240} + \frac{52}{240} \times \frac{5A}{4} + \frac{52}{240} \times 550$$ Calculate each term: - $\frac{0.05A}{240} = \frac{0.05}{240} A = \frac{1}{4800} A$ - $\frac{52}{240} \times \frac{5A}{4} = \frac{52 \times 5}{240 \times 4} A = \frac{260}{960} A = \frac{13}{48} A$ - $\frac{52}{240} \times 550 = \frac{52 \times 550}{240} = \frac{28600}{240} = 119.166\overline{6}$ So, $$F = \frac{1}{4800} A + \frac{13}{48} A + 119.166\overline{6} = \left(\frac{1}{4800} + \frac{13}{48}\right) A + 119.166\overline{6}$$ 7. Find common denominator for coefficients of $A$: - $\frac{13}{48} = \frac{1300}{4800}$ - So sum is $\frac{1}{4800} + \frac{1300}{4800} = \frac{1301}{4800}$ Thus, $$F = \frac{1301}{4800} A + 119.166\overline{6}$$ 8. Now write $H$ in terms of $A$: $$H = \frac{5A}{4} + 550 + F = \frac{5A}{4} + 550 + \frac{1301}{4800} A + 119.166\overline{6}$$ Combine constants: $$550 + 119.166\overline{6} = 669.166\overline{6}$$ Combine $A$ terms: $$\frac{5A}{4} = \frac{6000}{4800} A$$ So total $A$ coefficient: $$\frac{6000}{4800} + \frac{1301}{4800} = \frac{7301}{4800}$$ Therefore, $$H = \frac{7301}{4800} A + 669.166\overline{6} = 8999.16$$ 9. Solve for $A$: $$\frac{7301}{4800} A = 8999.16 - 669.166\overline{6} = 8330$$ So, $$A = \frac{8330 \times 4800}{7301}$$ Calculate numerator: $$8330 \times 4800 = 39984000$$ Divide: $$A = \frac{39984000}{7301} \approx 5476.5$$ 10. Final answer: $$\boxed{A \approx 5476.5}$$