The table shows the balances (in dollars) of two accounts earning simple interest for x years. After how many years (total) will the accounts have the same balance?
x
Account A
Account B
0
250
295
1
255
297
2
260
299
3
265
301
4
270
303
AAAaannndd the bot gets it wrong yet again!
250+5x = 295+2x
3x = 45
x = 15
To determine after how many years the accounts will have the same balance, we need to find the value of x when Account A's balance is equal to Account B's balance.
Looking at the table, we can see that the balances of Account A are increasing by $5 each year, while the balances of Account B are increasing by $2 each year.
Account A's balance is initially $250, and Account B's balance is initially $295.
Let's set up an equation to represent this information:
Account A's balance at any year x = 250 + 5x
Account B's balance at any year x = 295 + 2x
To find the value of x when the balances are equal, we set the two expressions equal to each other:
250 + 5x = 295 + 2x
Simplifying the equation:
3x = 295 - 250
3x = 45
Dividing both sides by 3:
x = 45/3
x = 15
Therefore, after 15 years, Account A and Account B will have the same balance.
To find out after how many years the accounts will have the same balance, we need to look for the point where the balance of Account A is equal to the balance of Account B.
Looking at the table, we can see that the difference in the balances of Account A and Account B is decreasing by $2 each year. Initially, the difference is $45 (295 - 250), and after 1 year, it becomes $42 (297 - 255).
Since the difference decreases by $2 each year, we can determine that it will take 45 / 2 = 22.5 years for the difference to become $0.
However, we cannot have fractional years, so we need to determine the exact point at which the balance becomes equal.
To do that, we need to find the year when the difference is less than $1. Looking at the table, after 22 years, the difference is still $7 (303 - 270). But in the 23rd year, the difference will become $5 (303 - 270 - 2), which is less than $7.
Therefore, it will take a total of 23 years for Account A and Account B to have the same balance.