Account A and B each start out with $500. If Account A earns $50 each year and Account B earns 5% of its value each year, after how many years will Account B have more money than Account A? If necessary, round your answer to the nearest year.
Let x be the number of years after which A and B have the same amount of money.
500 + 50x = 500(1.05)^x
Work out x with a table of values or a graphing calculator to get x=0 (doesn't count) or x ~ 27 years.
To determine after how many years Account B will have more money than Account A, we need to compare their balances each year and find the year when Account B surpasses Account A.
Let's start by calculating the values for each account year by year.
For Account A, it earns a fixed $50 each year. So, after n years, the balance of Account A can be calculated using the formula:
Balance_A = 500 + 50n
For Account B, it earns 5% of its value each year. The value of Account B after the first year is:
Value_B1 = 500 + (0.05 * 500)
To calculate the value of Account B in subsequent years, we need to add the previous year's value and 5% of that value:
Value_Bn = Value_B(n-1) + (0.05 * Value_B(n-1))
Now, let's compare the balances of the two accounts year by year to find the year when Account B surpasses Account A.
We can set up the following equation:
Balance_A < Value_B
Substituting the balance formulas:
500 + 50n < Value_B(n-1) + (0.05 * Value_B(n-1))
This equation compares the balances after n years. By solving this equation, we will find the number of years when Account B surpasses Account A.
I'll solve this equation using a loop, calculating the balances year by year:
```
balance_A = 500
value_B = 500
years = 0
while balance_A <= value_B:
balance_A += 50
value_B += (0.05 * value_B)
years += 1
print(years)
```
So, after running the code, the output will give us the number of years it takes for Account B to have more money than Account A.