Answer
The number of moles of $NH_4^+=0.272$ moles.
The number of moles of $SO_4^{2-}=0.136$ moles.
Work Step by Step
Moles of ammonium sulfate in 100.0 mL stock solution= $10.8 \text{ grams}*\frac{1 \text{ mol}}{132.14 \text{ grams}}=0.0817$ moles
Moles of ammonium sulfate in 10.00 mL solution = $0.0817\text{ moles}\div10=0.00817\text{ moles}$
Concentration of ammonium sulfate in 60.00 mL solution = $\frac{0.00817 \text{ moles}}{0.060 \text{ L}}=0.136$ M
Ammonium Sulfate's formula is $(NH_4)_2SO_4$ which separates into $2NH_4^{+}+SO_4^{2-}$
Therefore, the number of moles of $NH_4^+=2*0.136=0.272$ moles.
Therefore, the number of moles of $SO_4^{2-}=1*0.136=0.136$ moles.
