Sydney invests $100 every month into an account that pays 0.8% annual interest, compounded monthly. Benny invests $80 every month into an account that pays 2.2% annual interest rate, compounded monthly.
a. Determine the amount in Sydney's account after 10 years.  
b. Determine the amount in Benny's account after 10 years.  
c. Who had more money in the account after 10 years? SelectSydneyBennyItem 3 
d. Determine the amount in Sydney's account after 30 years.  
e. Determine the amount in Benny's account after 30 years.  
f. Who had more money in the account after 30 years? SelectSydneyBennyItem 6 


We can solve this problem using the formula for the future value of a series of monthly investments with compound interest. The formula is:

\[
A = P \times \frac{{(1 + r/n)^{nt} - 1}}{{r/n}}
\]

Where:
- $ A $ is the future value of the investment.
- $ P $ is the monthly investment (payment).
- $ r $ is the annual interest rate (as a decimal).
- $ n $ is the number of times interest is compounded per year (for monthly compounding, $ n = 12 $).
- $ t $ is the number of years.

Let’s solve each part step by step:

### a. Determine the amount in Sydney's account after 10 years.
For Sydney:
- Monthly investment $ P = 100 $
- Annual interest rate $ r = 0.008 $ (0.8%)
- Compounded monthly $ n = 12 $
- Time $ t = 10 $ years

### b. Determine the amount in Benny's account after 10 years.
For Benny:
- Monthly investment $ P = 80 $
- Annual interest rate $ r = 0.022 $ (2.2%)
- Compounded monthly $ n = 12 $
- Time $ t = 10 $ years

I'll calculate these values now.### a. Sydney's account after 10 years:
Sydney will have approximately \$12,488.73.

### b. Benny's account after 10 years:
Benny will have approximately \$10,726.94.

### c. Who had more money after 10 years?
Sydney had more money after 10 years, with about \$12,488.73 compared to Benny's \$10,726.94.

Now, let's determine the amounts after 30 years.

### d. Determine the amount in Sydney's account after 30 years.
For Sydney:
- Monthly investment $ P = 100 $
- Annual interest rate $ r = 0.008 $
- Compounded monthly $ n = 12 $
- Time $ t = 30 $ years

### e. Determine the amount in Benny's account after 30 years.
For Benny:
- Monthly investment $ P = 80 $
- Annual interest rate $ r = 0.022 $
- Compounded monthly $ n = 12 $
- Time $ t = 30 $ years

I'll calculate these values now.### d. Sydney's account after 30 years:
Sydney will have approximately \$40,672.12.

### e. Benny's account after 30 years:
Benny will have approximately \$40,739.94.

### f. Who had more money after 30 years?
Benny had slightly more money after 30 years, with \$40,739.94 compared to Sydney's \$40,672.12.

Would you like to dive into more details or have any questions? Here are some related questions:

1. How does changing the interest rate affect the future value?
2. What would happen if the investment time was increased to 40 years?
3. How much more would Sydney need to invest monthly to surpass Benny after 30 years?
4. What is the impact of different compounding frequencies on the future value?
5. How much would both need to invest to reach \$100,000 in 30 years?

**Tip:** Compounding frequency and the interest rate have a significant impact on long-term investment growth.

Sydney invests $100 every month into an account that pays 0.8% annual interest, compounded monthly. Benny invests $80 every month into an account that pays 2.2% annual interest rate, compounded monthly.

a. Determine the amount in Sydney's account after 10 years.

b. Determine the amount in Benny's account after 10 years.

c. Who had more money in the account after 10 years? Select Sydney or Benny.

d. Determine the amount in Sydney's account after 30 years.

e. Determine the amount in Benny's account after 30 years.

f. Who had more money in the account after 30 years? Select Sydney or Benny.

Discover how Sydney's $100 monthly investment with 0.8% interest and Benny's $80 investment with 2.2% interest compare over 10 and 30 years. Detailed calculations and analysis of compound interest with monthly contributions, suitable for high school students in Grades 10-12.

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