Answer:
annual interest rate = 9.336%, monthly 0.778%
Explanation:
Dealer A: down payment of $2,000 + 36 monthly payments of $564.05
Dealer B: down payment of $4,000 + 36 monthly payments of $500.14
we must find the interest rate at which both dealers' offers have the same present value:
$2,000 + PV monthly payments A = $4,000 + PV monthly payments B
PV monthly payments A = payment A x {1/r - 1 /[r x (1 + r)³⁶}
PV monthly payments B = payment B x {1/r - 1 /[r x (1 + r)³⁶}
we must use trial and error:
for r = 0.8% monthly, annually = 9.6%
PV monthly payments A = $564.05 x {1/0.008 - 1 /[r x (1 + r)³⁶} = $17,582.76
PV monthly payments B = $500.14 x {1/0.008 - 1 /[r x (1 + r)³⁶} = $15,590.54
the difference between them = $17,582.76 - $15,590.54 = $1,992.22 ≤ $2,000, so r must be a little lower
for r = 0.78% monthly, annually = 9.36%
PV monthly payments A = $564.05 x {1/0.008 - 1 /[r x (1 + r)³⁶} = $17,644.46
PV monthly payments B = $500.14 x {1/0.008 - 1 /[r x (1 + r)³⁶} = $15,645.24
the difference between them = $17,644.16 - $15,645.24 = $1,998.92 ≤ $2,000, so r must be a little lower
for r = 0.77% monthly, annually = 9.24%
PV monthly payments A = $564.05 x {1/0.008 - 1 /[r x (1 + r)³⁶} = $17,675.42
PV monthly payments B = $500.14 x {1/0.008 - 1 /[r x (1 + r)³⁶} = $15,672.70
the difference between them = $17,675.42 - $15,672.70 = $2,002.72 ≥ $2,000, so r must be a little higher
we continue until we find that r = 0.778% monthly, annually 9.336%
PV monthly payments A = $564.05 x {1/0.00778 - 1 /[r x (1 + r)³⁶} = $17,650.64
PV monthly payments B = $500.14 x {1/0.00778 - 1 /[r x (1 + r)³⁶} = $15,650.70
the difference between them = $17,650.64 - $15,650.70 = $2,000.06 ≈ $2,000, so that is our r
