For future value annuities, we regularly save the same amount of money into an account, which earns a
certain rate of compound interest, so that we have money for the future.
Deriving the formula (EMCG2)
Note: for this section is it important to be familiar with the formulae for the sum of a
geometric series (Chapter 1):
\begin{align*}
S_{n} = \frac{a\left({r}^{n}1\right)}{r1} & \qquad \text{for } r >1 \\
S_{n} = \frac{a\left(1{r}^{n}\right)}{1r} & \qquad \text{for } r < 1
\end{align*}
In the worked example above, the total value of Kobus' investment at the end of the four year
period is calculated by summing the accumulated amount for each deposit:
\(\begin{array}{[email protected]{\;}[email protected]{\;}[email protected]{\;}[email protected]{\;}[email protected]{\;}}
\text{R}\,\text{2 320,50} &= \text{R}\,\text{500,00} \quad + & \text{R}\,\text{550,00}
\quad + & \text{R}\,\text{605,00} \quad + & \text{R}\,\text{665,50} \\
&= \text{500}(1 + \text{0,1})^{0} \enspace + & \text{500}(1 + \text{0,1})^{1} \enspace +
& \text{500}(1 + \text{0,1})^{2} \enspace + & \text{500}(1 + \text{0,1})^{3}
\end{array}\)
We notice that this is a geometric series with a constant ratio \(r = 1 + \text{0,1}\).
Using the formula for the sum of a geometric series:
\begin{align*}
a &= \text{500} \\
r &= \text{1,1} \\
n &= 4 \\
& \\
S_{n} &= \frac{a\left({r}^{n}1\right)}{r1} \\
&= \frac{\text{500}\left({\text{1,1}}^{4}1\right)}{\text{1,1}  1} \\
&= \text{2 320,50}
\end{align*}
We can therefore use the formula for the sum of a geometric series to derive a formula for
the future value (\(F\)) of a series of (\(n\)) regular payments of an amount (\(x\))
which are subject to an interest rate (\(i\)):
\begin{align*}
a &= x \\
r &= 1 + i \\
& \\
S_{n} &= \frac{a\left({r}^{n}1\right)}{r1} \\
\therefore F &= \frac{x\left[(1 + i)^{n}1\right]}{(1 + i)1} \\
&= \frac{x\left[(1 + i)^{n}1\right]}{i}
\end{align*}
Future value of payments:
\[F = \frac{x\left[(1 + i)^{n}1\right]}{i}\]
If we are given the future value of a series of payments, then we can calculate the value of
the payments by making \(x\) the subject of the above formula.
Payment amount:
\[x = \frac{F \times i}{\left[(1 + i)^{n}1\right]}\]
Worked example 4: Future value annuities
Ciza decides to start saving money for the future. At the end of each month
she deposits \(\text{R}\,\text{900}\) into an account at Harringstone
Mutual Bank, which earns \(\text{8,25}\%\) interest p.a. compounded
monthly.
 Determine the balance of Ciza's account after \(\text{29}\) years.
 How much money did Ciza deposit into her account over the
\(\text{29}\) year period?
 Calculate how much interest she earned over the \(\text{29}\) year
period.
Write down the given information and the future value formula
\[F = \frac{x\left[(1 + i)^{n}1\right]}{i}\]
\begin{align*}
x &= \text{900} \\
i &= \frac{\text{0,0825}}{12} \\
n &= 29 \times 12 = \text{348}
\end{align*}
Substitute the known values and use a calculator to determine
\(F\)
\begin{align*}
F &= \dfrac{\text{900}\left[(1 +
\frac{\text{0,0825}}{12})^{\text{348}}1\right]}{\frac{\text{0,0825}}{12}} \\
&= \text{R}\,\text{1 289 665,06}
\end{align*}
Remember: do not round off at any of the interim steps of a calculation as
this will affect the accuracy of the final answer.
Calculate the total value of deposits into the account
Ciza deposited \(\text{R}\,\text{900}\) each month for \(\text{29}\) years:
\begin{align*}
\text{Total deposits: } &= \text{R}\,\text{900} \times 12 \times 29 \\
&= \text{R}\,\text{313 200}
\end{align*}
Calculate the total interest earned
\begin{align*}
\text{Total interest } &= \text{final account balance }  \text{total value
of all deposits} \\
&= \text{R}\,\text{1 289 665,06}  \text{R}\,\text{313 200} \\
&= \text{R}\,\text{976 465,06}
\end{align*}
Useful tips for solving problems:
 Timelines are very useful for summarising the given information in a visual way.

When payments are made more than once per annum, we determine the total
number of payments (\(n\)) by multiplying the number of years by \(p\):
Term 
\(p\) 
yearly / annually 
\(\text{1}\) 
halfyearly / biannually 
\(\text{2}\) 
quarterly 
\(\text{4}\) 
monthly 
\(\text{12}\) 
weekly 
\(\text{52}\) 
daily 
\(\text{365}\) 

If a nominal interest rate \(\left( i^{(m)} \right)\) is given, then use the
following formula to convert it to an effective interest rate:
\[1 + i = \left( 1 + \frac{i^{(m)}}{m} \right)^{m}\]
Worked example 5: Calculating the monthly payments
Kosma is planning a trip to Canada to visit her friend in two years' time.
She makes an itinerary for her holiday and she expects that the trip
will cost \(\text{R}\,\text{25 000}\). How much must she save at the end
of every month if her savings account earns an interest rate of
\(\text{10,7}\%\) per annum compounded monthly?
Write down the given information and the future value formula
\[F = \frac{x\left[(1 + i)^{n}1\right]}{i}\]
To determine the monthly payment amount, we make \(x\) the subject of the
formula:
\[x = \frac{F \times i}{\left[(1 + i)^{n}1\right]}\]
\begin{align*}
F &= \text{25 000} \\
i &= \frac{\text{0,107}}{12} \\
n &= 2 \times 12 = \text{24}
\end{align*}
Substitute the known values and calculate \(x\)
\begin{align*}
x &= \dfrac{\text{25 000} \times \frac{\text{0,107}}{12}}{\left[(1 +
\frac{\text{0,107}}{12})^{24}1\right]} \\
&= \text{R}\,\text{938,80}
\end{align*}
Write the final answer
Kosma must save \(\text{R}\,\text{938,80}\) each month so that she can afford
her holiday.
Worked example 6: Determining the value of an investment
Simon starts to save for his retirement. He opens an investment account and
immediately deposits \(\text{R}\,\text{800}\) into the account, which
earns \(\text{12,5}\%\) per annum compounded monthly. Thereafter, he
deposits \(\text{R}\,\text{800}\) at the end of each month for
\(\text{20}\) years. What is the value of his retirement savings at the
end of the \(\text{20}\) year period?
Write down the given information and the future value formula
\[F = \frac{x\left[(1 + i)^{n}1\right]}{i}\]
\begin{align*}
x &= \text{800} \\
i &= \frac{\text{0,125}}{12} \\
n &= 1 + (20 \times 12) = \text{241}
\end{align*}
Note that we added one extra month to the \(\text{20}\) years because Simon
deposited \(\text{R}\,\text{800}\) immediately.
Substitute the known values and calculate \(F\)
\begin{align*}
F &= \frac{\text{800}\left[(1 +
\frac{\text{0,125}}{12})^{\text{241}}1\right]}{ \frac{\text{0,125}}{12}} \\
&= \text{R}\,\text{856 415,66}
\end{align*}
Write the final answer
Simon will have saved \(\text{R}\,\text{856 415,66}\) for his retirement.
Future value annuities
Textbook
Exercise 3.2
Determine the balance of Shelly's account after
\(\text{35}\) years.
Write down the given information and the future value
formula:
\[F = \frac{x\left[(1 + i)^{n}1\right]}{i}\]
\begin{align*}
x &= \text{500} \\
i &= \frac{\text{0,0596}}{4} \\
n &= 35 \times 4 = \text{140}
\end{align*}
Substitute the known values and use a calculator to
determine \(F\):
\begin{align*}
F &= \dfrac{\text{500}\left[(1 +
\frac{\text{0,0596}}{4})^{\text{140}}1\right]}{\frac{\text{0,0596}}{4}}
\\
&= \text{R}\,\text{232 539,41}
\end{align*}
How much money did Shelly deposit into her account
over the \(\text{35}\) year period?
Calculate the total value of deposits into the
account:
Shelly deposited \(\text{R}\,\text{500}\) each
quarter
for \(\text{35}\) years:
\begin{align*}
\text{Total deposits: } &= \text{R}\,\text{500}
\times 4 \times 35 \\
&= \text{R}\,\text{70 000}
\end{align*}
Calculate how much interest she earned over the
\(\text{35}\) year period.
Calculate the total interest earned:
\begin{align*}
\text{Total interest } &= \text{final account
balance }  \text{total value of all deposits} \\
&= \text{R}\,\text{232 539,41}  \text{R}\,\text{70
000} \\
&= \text{R}\,\text{162539,41}
\end{align*}
Gerald wants to buy a new guitar worth \(\text{R}\,\text{7
400}\) in a year's time. How much must he deposit at the
end of each month into his savings account, which earns
a interest rate of \(\text{9,5}\%\) p.a. compounded
monthly?
Write down the given information and the future value
formula:
\[F = \frac{x\left[(1 + i)^{n}1\right]}{i}\]
To determine the monthly payment amount, we make \(x\) the
subject of the formula:
\[x = \frac{F \times i}{\left[(1 + i)^{n}1\right]}\]
\begin{align*}
F &= \text{7 400} \\
i &= \frac{\text{0,095}}{12} \\
n &= 1 \times 12 = \text{12}
\end{align*}
Substitute the known values and calculate \(x\):
\begin{align*}
x &= \dfrac{\text{7 400} \times
\frac{\text{0,095}}{12}}{\left[(1 +
\frac{\text{0,095}}{12})^{12}1\right]} \\
&= \text{R}\,\text{590,27}
\end{align*}
Write the final answer:
Gerald must deposit \(\text{R}\,\text{590,27}\) each month so
that he can afford his guitar.
How much money will Grace have in her account after
\(\text{29}\) years?
\begin{align*}
F & = \frac{x\left[(1+i)^n  1\right]}{i} \\
\text{Where: } \quad x & = \text{R}\,\text{1 100} \\
i &= \text{0,089} \\
n & = \text{29}
\end{align*}
\begin{align*}
F & = \frac{(\text{1 100}) \left[ \left(1 +
\frac{\text{0,089}}{\text{12}}\right) ^{(\text{29}
\times \text{12})}  1 \right]}
{\left(\frac{\text{0,089}}{\text{12}} \right)} \\
& = \text{R}\,\text{1 792 400,11}
\end{align*}
After \(\text{29}\) years, Grace will have
\(\text{R}\,\text{1 792 400,11}\) in her
account.
How much money did Grace deposit into her account by
the end of the \(\text{29}\) year period?
The total amount of money Grace saves each
year is \(\text{1 100}
\times \text{12} = \text{R}\,\text{13 200}\).
From that we can determine the total amount she
saves by multiplying by the number of years:
\(\text{13 200} \times \text{29} =
\text{R}\,\text{382 800}\).
After \(\text{29}\) years, Grace deposited a total of
\(\text{R}\,\text{382 800}\) into her account.
Ruth decides to save for her retirement so she opens a
savings account and immediately deposits
\(\text{R}\,\text{450}\) into the account. Her savings
account earns \(\text{12}\%\) per annum compounded
monthly. She then deposits \(\text{R}\,\text{450}\) at
the end of each month for \(\text{35}\) years. What is
the value of her retirement savings at the end of the
\(\text{35}\) year period?
Write down the given information and the future value
formula:
\[F = \frac{x\left[(1 + i)^{n}1\right]}{i}\]
\begin{align*}
x &= \text{450} \\
i &= \frac{\text{0,12}}{12} \\
n &= 1 + (35 \times 12) = \text{421}
\end{align*}
Substitute the known values and calculate \(F\):
\begin{align*}
F &= \frac{\text{450}\left[(1 +
\frac{\text{0,12}}{12})^{\text{421}}1\right]}{
\frac{\text{0,12}}{12}} \\
&= \text{R}\,\text{2 923 321,08}
\end{align*}
Write the final answer:
Ruth will have saved \(\text{R}\,\text{2 923 321,08}\) for
her retirement.
How much must Monique deposit into her account each
month in order to reach her goal?
\begin{align*}
F & = \frac{x\left[(1+i)^n  1\right]}{i} \\
F & = \text{R}\,\text{750 000} \\
i & = \text{0,0613} \\
n & = \text{35}
\end{align*}
\begin{align*}
\text{750 000} & = \frac{x \left[ \left(1 +
\frac{\text{0,0613}}{\text{12}}\right) ^{(\text{35}
\times \text{12})} \right]}
{\left(\frac{\text{0,0613}}{\text{12}} \right)} \\
\therefore x &= \frac{ \text{750 000} \times
\left(\frac{\text{0,0613}}{\text{12}} \right) }{\left[
\left(1 + \frac{\text{0,0613}}{\text{12}}\right)
^{(\text{35} \times \text{12})} \right]} \\
&= \text{510,84927} \ldots
\end{align*}
In order to save \(\text{R}\,\text{750 000}\) in
\(\text{35}\) years, Monique will need to save
\(\text{R}\,\text{510,85}\) in her account every
month.
How much money, to the nearest rand, did Monique
deposit into her account by the end of the
\(\text{35}\) year period?
The final amount calculated in the question above
includes the money Monique deposited into the
account plus the interest paid by the bank. The
total amount of money Monique put into her
account during the \(\text{35}\) year is the
product of \(\text{12}\) payments per year,
\(\text{35}\) years, and the payment amount
itself:
\[\text{R}\,\text{510,85} \times 12 \times 35 =
\text{R}\,\text{214 557,00}\]
After \(\text{35}\) years, Monique deposited a total
of \(\text{R}\,\text{214 557}\) into her
account.
Lerato plans to buy a car in five and a half years' time. She
has saved \(\text{R}\,\text{30 000}\) in a separate
investment account which earns \(\text{13}\%\) per annum
compound interest. If she doesn't want to spend more
than \(\text{R}\,\text{160 000}\) on a vehicle and her
savings account earns an interest rate of
\(\text{11}\%\) p.a. compounded monthly, how much must
she deposit into her savings account each month?
First calculate the accumulated amount for the
\(\text{R}\,\text{30 000}\) in Lerato's investment
account:
\[A = P(1 + i)^{n}\]
\begin{align*}
P &= \text{30 000} \\
i &= \text{0,13} \\
n &= \text{5,5}
\end{align*}
\begin{align*}
A &= \text{30 000}(1 + \text{0,13})^{\text{5,5}}\\
&= \text{R}\,\text{58 756,06}
\end{align*}
In five and a half years' time, Lerato needs to have saved
\(\text{R}\,\text{160 000}  \text{R}\,\text{58 756,06}
= \text{R}\,\text{101 243,94}\).
\[x = \frac{F \times i}{\left[(1 + i)^{n}1\right]}\]
\begin{align*}
F &= \text{101 243,94} \\
i &= \frac{\text{0,11}}{12} \\
n &= \text{5,5} \times 12 = \text{66}
\end{align*}
Substitute the known values and calculate \(x\):
\begin{align*}
x &= \dfrac{\text{101 243,94} \times
\frac{\text{0,11}}{12}}{\left[(1 +
\frac{\text{0,11}}{12})^{66}1\right]} \\
&= \text{R}\,\text{1 123,28}
\end{align*}
Write the final answer:
Lerato must deposit \(\text{R}\,\text{1 123,28}\) each month
into her savings account.
Every Monday Harold puts \(\text{R}\,\text{30}\) into
a savings account at the King Bank, which
accrues interest of \(\text{6,92}\%\) p.a.
compounded weekly. How long will it take
Harold's account to reach a balance of
\(\text{R}\,\text{4 397,53}\). Give the answer
as a number of years and days to the nearest
integer.
\begin{align*}
F & = \frac{x\left[(1+i)^n  1\right]}{i} \\
F & = \text{R}\,\text{4 397,53} \\
x & = \text{R}\,\text{30} \\
i &= \text{0,0692}
\end{align*}
\begin{align*}
\text{4 397,53} & = \frac{(\text{30}) \left[
\left(1 + \frac{\text{0,0692}}{\text{52}}\right) ^{(n
\times \text{52})}  1 \right]}
{\left(\frac{\text{0,0692}}{\text{52}} \right)} \\
\text{4 397,53} & = \frac{(\text{30}) \left[
\left( \text{1,00133} \right) ^{\text{52}n}  1 \right]}
{\text{0,00133} \ldots}
\end{align*}
\begin{align*}
(\text{0,00133} \ldots)(\text{4 397,53}) & =
(\text{30}) \left[ \left( \text{1,00133} \ldots \right)
^{\text{52}n}  1 \right] \\
\frac{\text{5,85209} \ldots}{\text{30}} & = \left[
\left( \text{1,00133} \ldots \right) ^{\text{52}n}  1
\right]
\end{align*}
\begin{align*}
\text{0,19506} \ldots + 1 & = \left( \text{1,00133}
\ldots \right) ^{\text{52}n} \\
\text{1,19506} \ldots & = \left( \text{1,00133}
\ldots \right) ^{\text{52}n} \\
\text{Change to logarithmic form: } \quad \text{52}n
& = \log_{\text{1,00133} \ldots} (\text{1,19506}
\ldots) \\
\text{52}n & = \text{134} \\
n & = \frac{\text{134}}{\text{52}} \\
n & = \text{2,57692} \ldots
\end{align*}
To get to the final answer for this question, convert
\(\text{2,57692} \ldots\) years into years and
days.
\(\qquad (\text{0,57692} \ldots) \times
\frac{\text{365}}{\text{year}} =
\text{210,577}\) days
Harold's investment takes \(\text{2}\) years and
\(\text{211}\) days to reach the final value of
\(\text{R}\,\text{4 397,53}\).
How much interest will Harold receive from the bank
during the period of his investment?
The total amount Harold invests is as follows:
\(\quad \text{30} \times \text{52} \times
\text{2,57692} \ldots = \text{R}\,\text{4
020,00}\)
Therefore, the total amount of interest paid by the
bank: \(\text{R}\,\text{4 397,53} 
\text{R}\,\text{4 020,00} =
\text{R}\,\text{377,53}\).
Sinking funds (EMCG3)
Vehicles, equipment, machinery and other similar assets, all depreciate in value as a result
of usage and age. Businesses often set aside money for replacing outdated equipment or
old vehicles in accounts called sinking funds. Regular deposits, and sometimes lump sum
deposits, are made into these accounts so that enough money will have accumulated by the
time a new machine or vehicle needs to be purchased.
Worked example 7: Sinking funds
Wellington Courier Company buys a delivery truck for \(\text{R}\,\text{296
000}\). The value of the truck depreciates on a reducingbalance basis
at \(\text{18}\%\) per annum. The company plans to replace this truck in
seven years' time and they expect the price of a new truck to increase
annually by \(\text{9}\%\).
 Calculate the book value of the delivery truck in seven years' time.
 Determine the minimum balance of the sinking fund in order for the
company to afford a new truck in seven years' time.
 Calculate the required monthly deposits if the sinking fund earns an
interest rate of \(\text{13}\%\) per annum compounded monthly.
Determine the book value of the truck in seven years' time
\begin{align*}
P &= \text{296 000} \\
i &= \text{0,18} \\
n &= \text{7} \\
& \\
A &= P(1  i)^{n} \\
&= \text{296 000}(1  \text{0,18})^{\text{7}} \\
&= \text{R}\,\text{73 788,50}
\end{align*}
Determine the minimum balance of the sinking fund
Calculate the price of a new truck in seven years' time:
\begin{align*}
P &= \text{296 000} \\
i &= \text{0,09} \\
n &= \text{7} \\
& \\
A &= P(1 + i)^{n} \\
&= \text{296 000}(1 + \text{0,09})^{\text{7}} \\
&= \text{R}\,\text{541 099,58}
\end{align*}
Therefore, the balance of the sinking fund (\(F\)) must be greater than the
cost of a new truck in seven years' time minus the money from the sale
of the old truck:
\begin{align*}
F &= \text{R}\,\text{541 099,58}  \text{R}\,\text{73 788,50} \\
&= \text{R}\,\text{467 311,08}
\end{align*}
Calculate the required monthly payment into the sinking fund
\[x = \frac{F \times i}{\left[(1 + i)^{n}1\right]}\]
\begin{align*}
F &= \text{467 311,08} \\
i &= \frac{\text{0,13}}{12} \\
n &= \text{7} \times 12 = \text{84}
\end{align*}
Substitute the values and calculate \(x\):
\begin{align*}
x &= \dfrac{\text{467 311,08} \times \frac{\text{0,13}}{12}}{\left[(1 +
\frac{\text{0,13}}{12})^{84}1\right]} \\
&= \text{R}\,\text{3 438,77}
\end{align*}
Therefore, the company must deposit \(\text{R}\,\text{3 438,77}\) each month.
Sinking funds
Textbook Exercise 3.3
How much money will be in the fund in \(\text{6}\)
years' time, when Mfethu wants to buy the new
truck?
\begin{align*}
F & = \frac{x\left[(1+i)^n  1\right]}{i} \\
\text {Where: } \quad & \\
x & = \text{3 100} \\
i & = \text{0,053} \\
n & = \text{6}
\end{align*}
Interest is compounded monthly: \(\; i = \text{0,053}
\rightarrow \frac{\text{0,053}}{12} \;\) and
\(\; n = \text{6} \rightarrow \text{6} \times
(12) \;\).
\begin{align*}
F & = \frac{(\text{3 100}) \left[ \left(1 +
\frac{\text{0,053}}{12}\right) ^{(\text{6} \times 12)} 
1 \right]} {\left(\frac{\text{0,053}}{12} \right)} \\
& = \text{R}\,\text{262 094,55}
\end{align*}
After \(\text{6}\) years, Mfethu will have
\(\text{R}\,\text{262 094,55}\) in his sinking
fund.
If a new truck costs \(\text{R}\,\text{285 000}\) in
\(\text{6}\) years' time, will Mfethu have
enough money to buy it?
No, Mfethu does not have enough money in his account:
\[\text{R}\,\text{285 000}  \text{R}\,\text{262 094,55}
= \text{R}\,\text{22 905,45}\]
Calculate the book value of the van in five
years' time.
\begin{align*}
P &= \text{265 000} \\
i &= \text{0,17} \\
n &= \text{5} \\
& \\
A &= P(1  i)^{n} \\
&= \text{265 000}(1  \text{0,17})^{\text{5}}
\\
&= \text{R}\,\text{104 384,58}
\end{align*}
Determine the amount of money needed in the
sinking fund for the company to be able to afford a new
van in five years' time.
\begin{align*}
P &= \text{265 000} \\
i &= \text{0,12} \\
n &= \text{5} \\
& \\
A &= P(1 + i)^{n} \\
&= \text{265 000}(1 + \text{0,12})^{\text{5}}
\\
&= \text{R}\,\text{467 020,55}
\end{align*}
Therefore, the balance of the sinking fund (\(F\))
must be greater than the cost of a new van in
five years' time less the money from the sale of
the old van:
\begin{align*}
F &= \text{R}\,\text{467 020,55} 
\text{R}\,\text{104 384,58} \\
&= \text{R}\,\text{362 635,97}
\end{align*}
Calculate the required monthly deposits if
the sinking fund earns an interest rate of
\(\text{11}\%\) per annum compounded monthly.
Calculate the required monthly payment into the
sinking fund:
\[x = \frac{F \times i}{\left[(1 + i)^{n}1\right]}\]
\begin{align*}
F &= \text{362 635,97} \\
i &= \frac{\text{0,11}}{12} \\
n &= \text{5} \times \text{12} = \text{60}
\end{align*}
Substitute the values and calculate \(x\):
\begin{align*}
x &= \dfrac{\text{362 635,97} \times
\frac{\text{0,11}}{12}}{\left[(1 +
\frac{\text{0,11}}{12})^{60}1\right]} \\
&= \text{R}\,\text{4 560,42}
\end{align*}
Therefore, the company must deposit
\(\text{R}\,\text{4 560,42}\) each month.
How much money must Tonya save quarterly so that
there will be enough money in the account to buy
the new computer?
\begin{align*}
F & = \frac{x\left[(1+i)^n  1\right]}{i} \\
\text{Where: } \quad & \\
F & = \text{8 450} \\
i & = \text{0,0767} \\
n & = \text{7}
\end{align*}
Interest is compounded per quarter, therefore \(\; i
= \text{0,0767} \rightarrow
\frac{\text{0,0767}}{\text{4}}\) and \(n =
\text{7} \rightarrow \text{7} \times \text{4}\)
\begin{align*}
\text{8 450} & = \frac{x \left[ \left(1 +
\frac{\text{0,0767}}{\text{4}}\right) ^{(\text{7} \times
\text{4})}  1 \right]}
{\left(\frac{\text{0,0767}}{\text{4}} \right)} \\
\therefore x & = \frac{\left( \text{8 450}
\times \frac{\text{0,0767}}{\text{4}} \right)}{ \left[
\left(1 + \frac{\text{0,0767}}{\text{4}}\right)
^{(\text{7} \times \text{4})}  1 \right]} \\
&= \text{230,80273} \ldots
\end{align*}
Tonya must deposit \(\text{R}\,\text{230,80}\) into
the sinking fund quarterly.
How much interest (to the nearest rand) does the bank
pay into the account by the end of the
\(\text{7}\) year period?
Total savings:
\[\text{R}\,\text{230,80} \times \text{4} \times
\text{7} = \text{R}\,\text{6 462,40}\]
Interest earned:
\[\quad \text{R}\,\text{8 450}  \text{R}\,\text{6
462,40} = \text{R}\,\text{1 987,60}\]
To the nearest rand, the bank paid
\(\text{R}\,\text{1 988}\) into the account.