Question 4
Nolwazi wants to buy a new state of the art computer for R35 000 . She decides to save by depositing an
amount of R500 every month into an account earning interest per year, compounded monthly. How
many years (accurate to one decimal place) will it take Nolwazi to have R35 000 available?

Question

Question 4
Nolwazi wants to buy a new state of the art computer for R35 000 . She decides to save by depositing an
amount of R500 every month into an account earning

interest per year, compounded monthly. How
many years (accurate to one decimal place) will it take Nolwazi to have R35 000 available?

UpStudy AI · Accepted Answer

Let's denote:

- $$ R = 500 $$ (monthly deposit)
- $$ \text{FV} = 35000 $$ (future value)
- Annual interest rate $$ = 11.32\% $$
- Since interest is compounded monthly, the monthly interest rate is 
  $$
  i = \frac{0.1132}{12} \approx 0.0094333.
  $$
- Let $$ n $$ be the number of months required.

The future value of an annuity (multiple deposits) is given by:
$$
\text{FV} = R \times \frac{(1+i)^n - 1}{i}.
$$

Substitute the given values:
$$
35000 = 500 \times \frac{(1+0.0094333)^n - 1}{0.0094333}.
$$

Divide both sides by $$500$$:
$$
70 = \frac{(1+0.0094333)^n - 1}{0.0094333}.
$$

Multiply both sides by $$0.0094333$$:
$$
70 \times 0.0094333 = (1+0.0094333)^n - 1.
$$

Calculate $$70 \times 0.0094333$$:
$$
70 \times 0.0094333 \approx 0.66033.
$$

So,
$$
(1.0094333)^n = 1 + 0.66033 \approx 1.66033.
$$

Take the natural logarithm of both sides:
$$
\ln(1.0094333^n) = \ln(1.66033).
$$

Using the logarithm property $$\ln(a^n) = n \ln(a)$$:
$$
n \ln(1.0094333) = \ln(1.66033).
$$

Solve for $$n$$:
$$
n = \frac{\ln(1.66033)}{\ln(1.0094333)}.
$$

Calculating the logarithms:
$$
\ln(1.66033) \approx 0.508 \quad \text{and} \quad \ln(1.0094333) \approx 0.00939.
$$

Thus,
$$
n \approx \frac{0.508}{0.00939} \approx 54.1 \text{ months}.
$$

Convert months to years:
$$
\text{Years} = \frac{54.1}{12} \approx 4.5 \text{ years}.
$$

So, it will take Nolwazi approximately $$4.5$$ years to have $$\text{R}35\,000$$ in her account.
It will take Nolwazi approximately 4.5 years to save R35,000.

Question 4
Nolwazi wants to buy a new state of the art computer for R35 000 . She decides to save by depositing an
amount of R500 every month into an account earning interest per year, compounded monthly. How
many years (accurate to one decimal place) will it take Nolwazi to have R35 000 available?

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Question 4 Nolwazi wants to buy a new state of the art computer for R35 000 . She decides to save by depositing an amount of R500 every month into an account earning interest per year, compounded monthly. How many years (accurate to one decimal place) will it take Nolwazi to have R35 000 available?

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Question 4
Nolwazi wants to buy a new state of the art computer for R35 000 . She decides to save by depositing an
amount of R500 every month into an account earning interest per year, compounded monthly. How
many years (accurate to one decimal place) will it take Nolwazi to have R35 000 available?