Sure! Let's break this down step-by-step:
### 1.2.1 Explain the term interest in this context.
Interest, in this context, is the amount of money earned on the initial investment or principal over a period of time, due to the annual interest rate being applied to the balance. In this case, the interest is compounded annually, which means that the interest earned each year is added to the principal, and the interest for the next year is calculated on the new balance (principal + previous interest).
### 1.2.2 Calculate the value of [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
To determine the values of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- [tex]\( A \)[/tex] represents the interest credited at the end of year 5.
- [tex]\( B \)[/tex] represents the balance at the end of year 5.
Let's complete the table through calculations:
End year 1:
- Opening Balance: R1,000.00
- Interest for the year: [tex]\( 1,000 \times 0.12 = R120.00 \)[/tex]
- Balance at the end of year 1: [tex]\( 1,000 + 120 = R1,120.00 \)[/tex]
End year 2:
- Opening Balance: R1,120.00
- Interest for the year: [tex]\( 1,120 \times 0.12 = R134.40 \)[/tex]
- Balance at the end of year 2: [tex]\( 1,120 + 134.40 = R1,254.40 \)[/tex]
End year 3:
- Opening Balance: R1,254.40
- Interest for the year: [tex]\( 1,254.40 \times 0.12 = R150.53 \)[/tex]
- Balance at the end of year 3: [tex]\( 1,254.40 + 150.53 = R1,404.93 \)[/tex]
End year 4:
- Opening Balance: R1,404.93
- Interest for the year: [tex]\( 1,404.93 \times 0.12 = R168.59 \)[/tex]
- Balance at the end of year 4: [tex]\( 1,404.93 + 168.59 = R1,573.52 \)[/tex]
End year 5 (Value of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]):
- Opening Balance: R1,573.52
- Interest for the year: [tex]\( 1,573.52 \times 0.12 = R188.82 \)[/tex]
- Balance at the end of year 5: [tex]\( 1,573.52 + 188.82 = R1,762.34 \)[/tex]
Therefore:
- [tex]\( A \)[/tex] (Interest at the end of year 5) = R188.82
- [tex]\( B \)[/tex] (Balance at the end of year 5) = R1,762.34
### 1.2.3 How much Interest did Michael earn over 6 years?
To calculate the total interest earned over the 6-year period, we sum the interest for each year:
- End year 1: R120.00
- End year 2: R134.40
- End year 3: R150.53
- End year 4: R168.59
- End year 5: R188.82
- End year 6: With a balance of R1,762.34 at the beginning of year 6, the interest for year 6 is: [tex]\( 1,762.34 \times 0.12 = R211.49 \)[/tex]
Adding all these interests:
[tex]\[ 120.00 + 134.40 + 150.53 + 168.59 + 188.82 + 211.49 = R973.82 \][/tex]
### Summary of Calculations
1. Define Interest: Interest is the amount earned on the investment over time, calculated annually here.
2. Calculated [tex]\( A \)[/tex] and [tex]\( B \)[/tex]: [tex]\( A \)[/tex] (Year 5 interest) = R188.82, [tex]\( B \)[/tex] (Year 5 balance) = R1,762.34.
3. Total Interest Earned: R973.82.
Thus,
- The interest credited at the end of year 5 is [tex]\( A = R188.82 \)[/tex].
- The balance at the end of year 5 is [tex]\( B = R1,762.34 \)[/tex].
- The total interest earned over the 6 years is R973.82.
