7.1 How many years will it take for an investment to double in value, if it earns interest at a rate of 8,5% p.a., compounded quarterly? 7.2 A company purchased machinery for R500 000. After 5 years, the machinery was sold for R180 000 and new machinery was bought. 7.2.1 Calculate the rate of depreciation of the old machinery over the 5 years, using the reducing-balance method. 7.2.2 The rate of inflation for the cost of the new machinery is 6,3% p.a. over the 5 years. What will the new machinery cost at the end of 5 years? 7.2.3 The company set up a sinking fund and made the first payment into this fund on the day the old machinery was bought. The last payment was made three months before the new machinery was purchased at the end of the 5 years. The interest earned on the sinking fund was 10,25% p.a., compounded monthly. The money from the sinking fund and the R180 000 from the sale of the old machinery was used to pay for the new machinery. Calculate the monthly payment into the sinking fund.

NSC Mathematics Finance growth and decay: 7.1 How many years will it take

7.1 How many years will it take for an investment to double in value, if it earns interest at a rate of 8,5% p.a., compounded quarterly? 7.2 A company purchased machinery for R500 000 - NSC Mathematics - Question 7 - 2022 - Paper 1

Question 7

7.1 How many years will it take for an investment to double in value, if it earns interest at a rate of 8,5% p.a., compounded quarterly? 7.2 A company purchased mac... show full transcript

Worked Solution & Example Answer:7.1 How many years will it take for an investment to double in value, if it earns interest at a rate of 8,5% p.a., compounded quarterly? 7.2 A company purchased machinery for R500 000 - NSC Mathematics - Question 7 - 2022 - Paper 1

Step 1

How many years will it take for an investment to double in value, if it earns interest at a rate of 8,5% p.a., compounded quarterly?

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Answer

To determine the number of years required for an investment to double in value, we use the compound interest formula:

$A = P(1 + i)^n$

Where:

$A$ is the amount of money accumulated after n years, including interest.
$P$ is the principal amount (initial investment).
$i$ is the interest rate per period.
$n$ is the number of periods.

Setting $A = 2P$ , we have:

$2P = P(1 + 0.085/4)^{4n}$

This simplifies to:

$2 = (1 + 0.02125)^{4n}$

Taking log on both sides:

$4n = rac{ ext{log}(2)}{ ext{log}(1 + 0.02125)}$

From calculations, we find:

$n = 8.24 ext{ years}$

Step 2

Calculate the rate of depreciation of the old machinery over the 5 years, using the reducing-balance method.

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Answer

Using the reducing-balance method, we start with:

$R180,000 = R500,000(1 - r)^5$

Where:

$R180,000$ is the salvage value after 5 years.
$R500,000$ is the original value.
$r$ is the depreciation rate.

This gives us:

$(1 - r)^5 = rac{R180,000}{R500,000}$

Taking the fifth root:

$1 - r = rac{R180,000^{1/5}}{R500,000^{1/5}}$

Calculating this, we find the depreciation rate $r ext{ as approximately } 18.48\%$ .

Step 3

The rate of inflation for the cost of the new machinery is 6,3% p.a. over the 5 years. What will the new machinery cost at the end of 5 years?

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Answer

We can calculate the future value of the new machinery using the compound interest formula again:

$A = P(1 + i)^n$

Given:

$P = R500,000 + 0.063/R500,000$ (adding the inflation component)
$i = 0.063$ (6.3% inflation rate)
$n = 5$ (number of years)
Finally, we get $A \approx R678,635.11$ .

Step 4

The company set up a sinking fund and made the first payment into this fund on the day the old machinery was bought. The last payment was made three months before the new machinery was purchased at the end of the 5 years. Calculate the monthly payment into the sinking fund.

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Answer

To find the monthly payment into the sinking fund, we use:

$S = R498,635.11$

Where $S$ is the value of the sinking fund. To calculate the monthly payment:

Interest rate $i = rac{10.25\%}{12} = 0.00854167$
Time $t = 58$ months

The formula for the sinking fund is:

$R = rac{S imes i}{(1+i)^t - 1}$

Solving gives:

$R = R6,510.36$

7.1 How many years will it take for an investment to double in value, if it earns interest at a rate of 8,5% p.a., compounded quarterly? 7.2 A company purchased machinery for R500 000 - NSC Mathematics - Question 7 - 2022 - Paper 1

Worked Solution & Example Answer:7.1 How many years will it take for an investment to double in value, if it earns interest at a rate of 8,5% p.a., compounded quarterly? 7.2 A company purchased machinery for R500 000 - NSC Mathematics - Question 7 - 2022 - Paper 1

How many years will it take for an investment to double in value, if it earns interest at a rate of 8,5% p.a., compounded quarterly?

Calculate the rate of depreciation of the old machinery over the 5 years, using the reducing-balance method.

The rate of inflation for the cost of the new machinery is 6,3% p.a. over the 5 years. What will the new machinery cost at the end of 5 years?

The company set up a sinking fund and made the first payment into this fund on the day the old machinery was bought. The last payment was made three months before the new machinery was purchased at the end of the 5 years. Calculate the monthly payment into the sinking fund.

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