C++, Interest Calculator not displaying correct output - c++

Question

The question asks "Write a program that reads an initial investment balance and an interest rate, then prints the number of years it takes for the investment to reach one million dollars."
The inputs I've been putting in are 100 for amount, and 3 for interest rate. But when i compile and run, the output is 29 which is incorrect as the amount is only 187 which isn't at all close to a million.
/*
Question: Write a program that reads an initial
investment balance and an interest rate, then
prints the number of years it takes for the
investment to reach one million dollars.
*/
#include <iostream>
using namespace std;
int main()
{
//Obtain user amount
double amount;
cout << "Please enter an initial investment balance ($0.00): $";
cin >> amount;
//Obtain user interest rate
double interest_rate;
cout << "Please enter an interest rate: ";
cin >> interest_rate;
//Convert interest rate to decimal
interest_rate = interest_rate / 100;
int time = 1;
//Calculate how many years
while (amount < 1000000)
{
amount = amount * (1 + (interest_rate * time));
++time;
}
//Display years
cout << "Years to reach one million: " << time;
return 0;
}
The output i am expecting is:
"Years to reach one million: 333300"
since 333300 is exactly one million.

Answer 1

In one year, the amount will grow to
amount * (1 + interest_rate)
and in two years, the amount grows to
amount * (1 + interest_rate) * (1 + interest_rate)
assuming annual compounding of your interest rate. Your inclusion of time, and the continuous multiplication by amount are errors.
Note that there is a closed form solution. For rate r, initial amount I, final amount A, the number of years t is
t = ln(A / I) / ln(1 + r)
which you need to round up.