Statistics

Problem Statement for "AutoLoan"

Problem Statement

Auto dealerships frequently advertise tempting loan offers in order to make it easier for people to afford the "car of their dreams". A typical sales tactic is to show you various cars, and then talk in terms of what your monthly payment would be, to say nothing of how much you are actually paying for the car, how much interest you pay, or how long you have to make payments.

A typical auto loan is calculated using a fixed interest rate, and is set up so that you make the same monthly payment for a set period of time in order to fully pay off the balance. The balance of your loan starts out as the sticker price of the car. Each month, the monthly interest is added to your balance, and the amount of your payment is subtracted from your balance. (The payment is subtracted after the interest is added.) The monthly interest rate is 1/12 of the yearly interest rate. Thus, if your annual percentage rate is 12%, then 1% of the remaining balance would be charged as interest each month.

You have been checking out some of the cars at your local dealership, TopAuto. An excited salesman has just approached you, shouting about how you can have the car you are looking at for a payment of only monthlyPayment for only loanTerm months! You are to return a double indicating the annual percentage rate of the loan, assuming that the initial balance of the loan is price.

Definition

Class:
AutoLoan
Method:
interestRate
Parameters:
double, double, int
Returns:
double
Method signature:
double interestRate(double price, double monthlyPayment, int loanTerm)
(be sure your method is public)

Notes

  • Because of the way interest is compounded monthly, the actual interest accrued over the course of a year is not necessarily the same as (balance * yearly interest rate). In fact, it's usually more.
  • In a real situation, information like this would typically need to be disclosed, but since you aren't at a point of signing any paperwork, the salesman has no legal obligation to tell you anything.
  • The return value must be within 1e-9 absolute or relative error of the actual result.

Constraints

  • price will be between 1 and 1000000, inclusive.
  • monthlyPayment will be between 0 and price / 2, inclusive.
  • loanTerm will be between 1 and 600, inclusive.
  • The resulting interest rate will be between 0 and 100, inclusive.

Examples

  1. 6800

    100

    68

    Returns: 1.3322616182218813E-13

    Noting that 68 payments of 100 equals the total price of 6800, so there is no interest.

  2. 2000

    510

    4

    Returns: 9.56205462458368

    Here, we do pay a little interest. At 9.562% annual interest, that means each month we pay 0.7968% of the balance in interest. Our payment schedule looks like this: Month | + Interest | - Payment | = Balance ------------------------------------------ | | | 2000.00 1 | 15.94 | 510.00 | 1505.94 2 | 12.00 | 510.00 | 1007.94 3 | 8.03 | 510.00 | 505.97 4 | 4.03 | 510.00 | 0.00

  3. 15000

    364

    48

    Returns: 7.687856394581649

    This is similar to what purchasing a new car with no money down might look like, if you make payments for 4 years.

  4. 10000

    385

    30

    Returns: 11.473054204549669

  5. 200

    15

    600

    Returns: 89.99999999999982

  6. 1000000

    1667

    600

    Returns: 7.986158248041164E-4

  7. 724556

    50505

    575

    Returns: 83.64570854426711

  8. 332696

    10951

    136

    Returns: 38.98851536883803

  9. 23002

    4709

    6

    Returns: 74.54805516148613

  10. 492214

    2049

    471

    Returns: 3.9204806010895594

  11. 740192

    7185

    588

    Returns: 11.607783057395558

  12. 570542

    24374

    65

    Returns: 47.05796362542851

  13. 407050

    35078

    17

    Returns: 55.397079816319696

  14. 140192

    1491

    454

    Returns: 12.653412761497272

  15. 219252

    8934

    337

    Returns: 48.897089628821746

  16. 506973

    11900

    50

    Returns: 7.7628540286105885

  17. 281335

    16982

    54

    Returns: 68.87780115037442

  18. 286417

    9239

    145

    Returns: 38.30192971448798

  19. 140642

    1293

    561

    Returns: 10.965203755132924

  20. 389951

    12413

    163

    Returns: 37.9601927752288

  21. 973110

    337873

    3

    Returns: 24.80783488352443

  22. 230977

    67123

    4

    Returns: 75.65099005216278

  23. 61703

    16197

    4

    Returns: 23.76581938256987

  24. 44755

    8959

    5

    Returns: 0.3574309895975514

  25. 297007

    56775

    6

    Returns: 48.76330124289328

  26. 584755

    1613

    475

    Returns: 1.4303822503235695

  27. 5628

    2197

    3

    Returns: 100.00000000000003

  28. 6800.0

    100.0

    68

    Returns: 1.3322616182218813E-13

  29. 15000.0

    364.0

    48

    Returns: 7.687856394581649

  30. 2000.0

    510.0

    4

    Returns: 9.56205462458368

  31. 60000.0

    4999.0

    600

    Returns: 99.9799999999999

  32. 100001.0

    989.0

    599

    Returns: 11.834664337677792

  33. 1000.0

    500.0

    2

    Returns: 1.3322616182218813E-13

  34. 12.0

    1.0

    500

    Returns: 99.99999999999977

  35. 1000000.0

    2500.0

    600

    Returns: 1.7463073077668143

  36. 1.0

    0.142857142857143

    7

    Returns: 3.9968010105369827E-13

  37. 1000000.0

    79865.0

    599

    Returns: 95.83799999999995

  38. 999999.0

    3000.0

    599

    Returns: 2.6299652255705737

  39. 15000.0

    364.0

    600

    Returns: 29.11998354222205

  40. 9.999999991

    2.5

    4

    Returns: 4.320001689699897E-7

  41. 200.0

    100.0

    2

    Returns: 1.3322616182218813E-13

  42. 1000.0

    40.0

    34

    Returns: 22.418392731637795

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