Christina finances a piano for $3,070 by taking out an installment loan for 24 months. The payments were $153.50 per month and the total finance charge was $614. After 14 months, Christina decided to pay off the loan. After calculating the finance charge rebate, find her loan payoff.

I assume this is compound interest, since the time is more than 1 year.

We don't know the interest rate, let the monthly rate be i
then:
153.5(1 - (1+i)^-24)/i = 3070

a very tough equation to solve, I will use Wolfram:
http://www.wolframalpha.com/input/?i=153.5(1+-+(1%2Bx)%5E-24)%2Fx+%3D+3070
to get i = .0151308 (---> annual rate compounded monthly = 18.2%)

amount owing after 14 months = 3070(1.0151308)^14 =3788.32
value of 14 payments = 153.50(1.0151308^14 - 1)/.0151308 = 2373.70

amount owing = 3788.32 - 2373.70 = 1414.62

btw, the $614 total finance charge is bogus and has nothing to do with this.
If this is "simple interest" , then it would show an interest rate of 10%, totally misleading
since the true annual interest rate is over 18%

To find Christina's loan payoff after 14 months, we need to calculate the remaining balance of the loan at that point.

First, let's calculate the total amount paid over the 14 months. Since the monthly payment is $153.50 and there are 14 months, the total amount paid is $153.50 * 14 = $2,149.

Next, we need to calculate the remaining balance on the loan after 14 months. Subtracting the total amount paid ($2,149) from the original loan amount ($3,070), we get $3,070 - $2,149 = $921.

Now, we need to calculate the finance charge rebate. The finance charge rebate is the portion of the finance charge that Christina will get refunded since she paid off the loan earlier than expected. The total finance charge was $614, and since she paid off the loan after 14 months, she will receive a prorated rebate based on the remaining months of the loan.

To calculate the monthly finance charge, divide the total finance charge by the total number of months of the loan: $614 / 24 = $25.58 per month.

To find the finance charge rebate for 14 months, multiply the monthly finance charge by the remaining months: $25.58 * (24 - 14) = $255.8.

Now, subtract the finance charge rebate from the remaining balance on the loan: $921 - $255.8 = $665.2.

Therefore, Christina's loan payoff after 14 months is $665.2.