Answer
Tino can make 147 withdrawals of R20,000 from the fund.
Solution
To solve this problem, we need to determine how many monthly withdrawals Tino can make from his investment while accounting for the interest earned on the remaining balance in the fund.
### Known Conditions:
1. Initial investment (P): R1,600,000
2. Annual interest rate (r): 11.2% or 0.112
3. Monthly interest rate (i): \( \frac{0.112}{12} \)
4. Monthly withdrawal amount (W): R20,000
### Steps to Solve:
1. **Calculate the monthly interest rate:**
\[
i = \frac{0.112}{12}
\]
2. **Set up the formula for the future value of the investment after each withdrawal.** The balance after each withdrawal can be modeled using the formula for the future value of an annuity due to the monthly withdrawals.
3. **Use the formula for the present value of an annuity to find the number of withdrawals (n):**
The present value of the withdrawals can be expressed as:
\[
P = W \times \left( \frac{1 - (1 + i)^{-n}}{i} \right)
\]
Rearranging gives:
\[
n = -\frac{\log\left(1 - \frac{P \cdot i}{W}\right)}{\log(1 + i)}
\]
### Now, let's calculate the values step by step.
First, we will calculate the monthly interest rate \( i \) and then use it to find \( n \).
Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
\(\frac{1600000\times 0.112}{20000}\)
- step1: Multiply the numbers:
\(\frac{179200}{20000}\)
- step2: Reduce the fraction:
\(\frac{224}{25}\)
Calculate or simplify the expression \( 0.112/12 \).
Calculate the value by following steps:
- step0: Calculate:
\(\frac{0.112}{12}\)
- step1: Convert the expressions:
\(\frac{\frac{14}{125}}{12}\)
- step2: Multiply by the reciprocal:
\(\frac{14}{125}\times \frac{1}{12}\)
- step3: Reduce the numbers:
\(\frac{7}{125}\times \frac{1}{6}\)
- step4: Multiply the fractions:
\(\frac{7}{125\times 6}\)
- step5: Multiply:
\(\frac{7}{750}\)
Calculate or simplify the expression \( -\log(1 - 0.00933*1600000/20000)/\log(1 + 0.00933) \).
Calculate the value by following steps:
- step0: Calculate:
\(-\frac{\log_{10}{\left(1-\frac{0.00933\times 1600000}{20000}\right)}}{\log_{10}{\left(1+0.00933\right)}}\)
- step1: Reduce the fraction:
\(-\frac{\log_{10}{\left(1-\frac{933}{1250}\right)}}{\log_{10}{\left(1+0.00933\right)}}\)
- step2: Subtract the numbers:
\(-\frac{\log_{10}{\left(\frac{317}{1250}\right)}}{\log_{10}{\left(1+0.00933\right)}}\)
- step3: Add the numbers:
\(-\frac{\log_{10}{\left(\frac{317}{1250}\right)}}{\log_{10}{\left(1.00933\right)}}\)
- step4: Convert the expressions:
\(-\frac{\log_{10}{\left(\frac{317}{1250}\right)}}{\log_{10}{\left(\frac{100933}{100000}\right)}}\)
- step5: Use the logarithm base change rule:
\(-\log_{\frac{100933}{100000}}{\left(\frac{317}{1250}\right)}\)
- step6: Rewrite the expression:
\(-\log_{\frac{100933}{100000}}{\left(317\right)}+\log_{\frac{100933}{100000}}{\left(1250\right)}\)
### Calculations:
1. **Monthly Interest Rate (i):**
\[
i = \frac{0.112}{12} \approx 0.00933
\]
2. **Calculating \( \frac{P \cdot i}{W} \):**
\[
\frac{P \cdot i}{W} = \frac{1600000 \cdot 0.00933}{20000} \approx 8.96
\]
3. **Finding the number of withdrawals (n):**
Using the formula:
\[
n = -\frac{\log\left(1 - 8.96\right)}{\log(1 + 0.00933)}
\]
This simplifies to:
\[
n \approx 147.74
\]
Since Tino can only make whole withdrawals, we round down to the nearest whole number.
### Final Answer:
Tino will be able to make **147 withdrawals** of R20,000 from the fund.
Answered by UpStudy AI and reviewed by a Professional Tutor
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