Three Years Ago You Earned 2000 Working As A Lifeguard

Three Years Ago You Earned 2000 Working As a Lifeguard Youdeposits

Three years ago, you earned $2000 working as a lifeguard. You deposited the money in a bank that paid simple interest of 4.75%. One of the other lifeguards with whom you work deposited the $2000 she earned in a different bank. Her bank only paid 4.70%, but it compounded the interest annually. How less/much money was in your account after the end of the first year? How much was in your friend's account? How much did each of you have in the bank after the end of the second year? Why did your friend earn?

Paper For Above instruction

The scenario involves comparing the growth of two bank deposits over a period of three years, one with simple interest and the other with compounded interest. The core goal is to analyze the differences in accumulated amounts after specified periods, particularly after the first and second years.

Introduction

Understanding the differences between simple interest and compound interest is fundamental in personal finance. Simple interest accrues on the original principal only, while compound interest accumulates on the principal and previously earned interest. This paper examines a real-world scenario involving two individuals who deposited the same amount of money into different banks with differing interest schemes. Analyzing their account balances over time illustrates the significant impact of interest compounding versus simple interest over multiple years.

Initial Deposit and Interest Rate Context

Three years ago, both individuals deposited $2000 into their respective bank accounts. The first individual’s bank paid simple interest at a rate of 4.75% annually, while the second individual’s bank paid 4.70% annual interest compounded yearly. The core difference in interest calculation methods affects the total accumulated wealth over time.

Year 1: Calculation of Balances after One Year

Simple Interest Calculation

Simple interest (SI) is calculated using the formula:

SI = P × r × t

where P = principal ($2000), r = annual interest rate (4.75%, or 0.0475), t = time in years (1).

Interest earned in the first year:

SI = 2000 × 0.0475 × 1 = $95

Thus, total amount after one year:

Amount = principal + interest = 2000 + 95 = $2095

Compound Interest Calculation

Compound interest (CI), with compounded annually, uses the formula:

A = P(1 + r)^t

where P = $2000, r = 0.047 (4.70%), t = 1 year.

Amount after one year:

A = 2000 × (1 + 0.047)^1 ≈ 2000 × 1.047 = $2094

Comparison After First Year

After the first year, the simple interest account has $2095, whereas the compound interest account has approximately $2094. The difference is:

$2095 - $2094 = $1

Therefore, your account with simple interest had $1 more than your friend’s account after the first year. The reason for this is that at exactly one year, simple interest effectively results in slightly more due to the specific rounding or interest calculation method, but usually, the difference is negligible for a single year with close interest rates. Over longer periods, the effects become more pronounced.

Year 2: Calculations for the End of the Second Year

Simple Interest

The simple interest grows linearly:

Interest for second year = 2000 × 0.0475 × 1 = $95

Total at end of second year:

Principal after first year = $2095

Interest for second year on this amount:

Interest = 2095 × 0.0475 ≈ 2095 × 0.0475 ≈ $99.46

Total amount after two years:

2095 + 99.46 ≈ $2194.46

Compound Interest

The amount at the end of the first year was approximately $2094. The second year involves compounding on this new amount:

A after 2 years = 2000 × (1 + 0.047)^2 ≈ 2000 × 1.047^2 ≈ 2000 × 1.096209 ≈ $2192.42

Final Comparison After Two Years

At the end of the second year, the simple interest account holds approximately $2194.46, and the compounded interest account holds approximately $2192.42. The simple interest account has about $2.04 more than the compounded account. Over two years, simple interest accrues a slightly higher total due to linear accumulation, but the difference diminishes compared to potential long-term gains with compounding.

Why Did Your Friend Earn Less?

The key reason your friend earned less after each year is that her bank paid interest compounded annually at a slightly lower rate (4.70%) compared to your bank's simple interest rate (4.75%). While simple interest offers straightforward growth based solely on the principal, compounding interest, especially at lower rates, can sometimes result in slightly lower accumulation over short periods. However, over extended periods, the power of compounding generally surpasses simple interest, particularly if the rates are comparable or higher, making it more advantageous over the long term.

Conclusion

This comparison underscores the importance of understanding different interest calculations on savings accounts. Simple interest can sometimes appear more beneficial over short durations or specific interest rates, but altogether, compound interest generally leads to greater wealth accumulation over time, especially with higher rates and longer periods. Awareness of these concepts enables better financial decisions and optimized savings strategies.

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