Answer:
Yes, your answer of 77.1 is correct. The percent atom economy (% AE) for the synthesis of ethyl 6-acetoxyhexanoate (raspberry-like fragrance) is 77.1%.
To calculate the percent atom economy, we can use the following equation:
```
% AE = (Mass of atoms in desired product / Mass of atoms in all reactants) * 100%
```
The numbers below each compound in the image represent the molar masses of the compounds in grams per mole. We can use this information to calculate the mass of atoms in each compound. For example, the mass of atoms in C6H10O2 is:
(12.01 g/mol C * 6 atoms C/molecule) + (1.008 g/mol H * 10 atoms H/molecule) + (16.00 g/mol O * 2 atoms O/molecule) = 114.14 g/mol
We can calculate the mass of atoms in all of the reactants and the desired product in the same way. Once we have these values, we can plug them into the equation above to calculate the percent atom economy.
% AE = ((202.25 g/mol C10H18O4) / (114.14 g/mol C6H10O2 + 46.07 g/mol C2H6O + 160.21 g/mol C8H16O3 + 102.09 g/mol C4H6O3)) * 100%
```
% AE = 77.1%
```
Therefore, the percent atom economy for the synthesis of ethyl 6-acetoxyhexanoate is 77.1%.
I hope this explanation is helpful. Please let me know if you have any other questions.