Nella is purchasing a car for $30000, including taxes. She hopes to replace it in 4 years with a similar one. She estimates that in 4 years, the price ....

....will have increased by 25%, and the present car will have lost 60% of its value. GST of 7% is charged on the difference between the trade in value and the new car price. PST is charged on the price of the new car. She will start saving in 3 months, by making a payment every 3 months into an account paying 8% interest per annum, compounded quarterly. How much should each payment be so that she can pay the cash for the new car in 4 years

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This question is a combination of lump sums and annuities.

Since the new car is increasing by 25%, then the new price is

`N=30000(1+0.25)^1 = 37500`

The old car depreciates by 60%, so its sale price will be (also through a lump sum)

`O=30000(1-0.60)^1=12000`

The amount she has to pay is also going to be taxed on the difference, so we end up as the amount to be paid at:

`P=(37500-12000)(1+0.07)=27285`

Now to find the amount saved per quarter, we use the formula for future value annuities:

`P=R/i((1+i)^n-1)` which needs to be rearranged to find the rent:

`R={iP}/((1+i)^n-1)` now sub in values i=0.08/4=0.02, n=4x4=16 to get

`R={(0.02)(27285)}/{1.02^16-1}=1463.84`

**She needs to save $1463.84 every quarter.**

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