annuities - equations of value 
Chuck needs to purchase an item in $10$ years.  The item costs $ \$200$ today, but its price inflates at $4 \%$ per year.  To finance the purchase, Chuck deposits $ \$20$ into an account at the beginning of each year for $6$ years.  He deposits an additional $X$ at the beginning of years $4, 5,$ and $6$ to meet his goal.  The annual effective interest rate is $10 \%$.  Calculate $X$.

This is how i interpret the problem:  You have $5$ cash flows starting from $0$ to $5$ of $ \$ 20$.  You also have $3$ cash flows at $t=4,5,6$.  
I used annuity due formula to shift former cash flow to year $6$, and then accumulate it to year $10$ by the $4$ remaining years.  
I used the same approach for the latter:
$(20\ddot{s}_{\overline{5|}i=10\%})(1.1)^4 + (X \ddot{s}_{\overline{3|}i=10\%})(1.1)^3 = 200(1.04)^{10}\tag{1}$
But this does not give me the right answer.  Can someone please tell me what  I'm doing wrong?  Thanks in advance.
 A: It looks to me like your $\ddot{s}_{\overline{5|}i}$ should be $\ddot{s}_{\overline{6|}i}$. He makes a deposit at the beginning of every year for $6$ years. 
Also, the time for the deposits of $X$ aren't at time $t= 4, 5, 6$... they are at the beginning of year $4, 5, 6$, which is time $t = 3, 4, 5$, because we typically start at $t=0$ (which means that any time between time $t=0$ and $t=1$ is during the first year.)
I suggest drawing time diagrams any time you are not absolutely certain. They make it impossible to miscount.
A: Each time Chuck puts some money on the bank account, all you need to do is capitalize this amount until maturity.
First deposit of 20 will give him $20 * (1.1)^{10}$ at maturity. Second one $20 * (1.1)^{9}$.
So the 6 known deposits are treated easily. Then you do the same for the unknown deposits 'X'.
A X deposit at beginning of year 4 is capitalized into $x * (1.1)^{7}$ at maturity.
It is the same idea for deposits at year 5 and 6.
Then the total of the capitalized deposits must be equal to $200 * (1.04)^{10}$. Which should give you X easily.
