Three $1000, 8% coupon rate bonds have ,10 and 20 years until maturity respectively. If interest rates move up to 10%, the following bond prices will occur: $828.36, $964.4, and $875.39?

The information is a bit sketchy. By interest rates moving up to 10% I assume you mean the market rate. And if it "moves up" was it lower before? Since there's no information indicating the market rate changed, or that this is anything other than the original bond issue, I'm going to assume that. And you're giving three different prices, but I only see two different maturities. Where are three numbers coming from?

And I'm not sure how you got these numbers since you haven't shown how you got them. And you also haven't said how often interest is paid, so I just made an assumption of annually. That could be incorrect. (But we can't know unless you provide this information.)

The price of a bond is the present value of the bond face value plus the present value of the series of interest payments. The interest payments in this case being $80 per year.

Present value of bond face value:

PV = \frac{FV}{(1+i)^n}

Present value of the series of interest payments:

PV=Pmt\,\left(\frac{1-(1+i)^{-n}}i\right)

Where i = interest per compounding period, at market rate, and n = total compounding periods.

This can also be done using charts, Excel or a financial calculator, but you didn't say how you solved this.