A trader needs $ 300,000to improve her business, she deposits $ 110000 in her savings account at 5% per annum compound interest. She then adds $ 50,000 to her savings at the end of each year.
(a) Find her total savings after 3 years.
( b) By how much is her savings greater or less than $300000 at the end of the third year.
I interpret it that there are three payments of 50,000 aside from the current deposit of 110,000
amount after 3 years
= 110,000(1.05)^3 + 50,000( 1.05^3 - 1)/.05
= ....
Answer
Answer
To find the trader's total savings after 3 years, we need to calculate the amount of money accumulated in her savings account at the end of each year.
(a) To calculate the total savings after 3 years, we will use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount of savings
P = the principal amount (initial deposit)
r = annual interest rate (expressed as a decimal)
n = number of times that interest is compounded per year
t = number of years
Let's break down the problem step by step:
Step 1: Calculate the interest earned on the initial deposit of $110,000.
Interest = P(1 + r)^t - P
= $110,000(1 + 0.05)^1 - $110,000
= $110,000(1.05) - $110,000
= $110,000(0.05)
= $5,500
So, the interest earned on the initial deposit is $5,500.
Step 2: Calculate the interest earned on the additional deposits made at the end of each year.
For the first additional deposit of $50,000, the interest earned after 3 years would be:
Interest = P(1 + r)^t - P
= $50,000(1 + 0.05)^3 - $50,000
= $50,000(1.05)^3 - $50,000
= $50,000(1.157625) - $50,000
= $57,881.25
For the second additional deposit of $50,000, the interest earned after 2 years would be:
Interest = $50,000(1 + 0.05)^2 - $50,000
= $50,000(1.05)^2 - $50,000
= $50,000(1.1025) - $50,000
= $55,125
Total interest earned on the additional deposits = $57,881.25 + $55,125 = $113,006.25
Step 3: Calculate the total savings after 3 years.
Total savings = Initial deposit + Interest on initial deposit + Interest on additional deposits
Total savings = $110,000 + $5,500 + $113,006.25
= $228,506.25
Therefore, the trader's total savings after 3 years would be $228,506.25.
(b) To find out how much the trader's savings differ from $300,000 at the end of the third year, we need to subtract her total savings from $300,000.
Difference = $300,000 - $228,506.25
= $71,493.75
Therefore, the trader's savings are greater than $300,000 by $71,493.75 at the end of the third year.