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When a loan of €P is repaid in equal repayments of amount €A, at the end of each f equal periods of time, where i is the periodic compound interest rate (expressed as a decimal), the formula below can be used to find the amount of each repayment - Leaving Cert Mathematics - Question 8 - 2017

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Question 8

When-a-loan-of-€P-is-repaid-in-equal-repayments-of-amount-€A,-at-the-end-of-each-f-equal-periods-of-time,-where-i-is-the-periodic-compound-interest-rate-(expressed-as-a-decimal),-the-formula-below-can-be-used-to-find-the-amount-of-each-repayment-Leaving Cert Mathematics-Question 8-2017.png

When a loan of €P is repaid in equal repayments of amount €A, at the end of each f equal periods of time, where i is the periodic compound interest rate (expressed a... show full transcript

Worked Solution & Example Answer:When a loan of €P is repaid in equal repayments of amount €A, at the end of each f equal periods of time, where i is the periodic compound interest rate (expressed as a decimal), the formula below can be used to find the amount of each repayment - Leaving Cert Mathematics - Question 8 - 2017

Step 1

Show how this formula is derived.

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Answer

To derive the formula for the monthly repayment amount A, we start with the concept of the sum of a finite geometric series. The loan amount P can be expressed as the sum of its repayments over f periods:

P=A+A(1+i)+A(1+i)2+...+A(1+i)f−1P = A + A(1+i) + A(1+i)^2 +...+ A(1+i)^{f-1}

This can be rewritten as:

P=A(1+(1+i)+(1+i)2+...+(1+i)f−1)P = A \left(1 + (1+i) + (1+i)^2 +...+ (1+i)^{f-1} \right)

Using the formula for the sum of a finite geometric series, we have:

Sn=a(1−rn)(1−r)S_n = a \frac{(1 - r^n)}{(1 - r)}

Thus, P=A(1−(1+i)f)(1−(1+i))P = A \frac{(1 - (1+i)^f)}{(1 - (1+i))}

Rearranging gives: A=Pi(1+i)f(1+i)f−1A = P \frac{i(1+i)^f}{(1+i)^f - 1}

Step 2

What is the fixed monthly repayment, €A, required to pay the debt of €5000?

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Answer

The fixed monthly repayment can be calculated as follows:

The fixed repayment amount is 2.5% of €5000: A=0.025×5000=€125A = 0.025 \times 5000 = €125

Step 3

The annual percentage rate (APR) charged on debt by the credit card company is 21.75%. Find as a percentage, correct to 3 significant figures, the monthly interest rate that is equivalent to an APR of 21.75%.

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Answer

To find the monthly interest rate, we use the formula: r=(1+rAPR)1/12−1r = (1 + r_{APR})^{1/12} - 1

Where, rAPR=0.2175r_{APR} = 0.2175 So, r=(1+0.2175)1/12−1≈0.01665r = (1 + 0.2175)^{1/12} - 1 \approx 0.01665

Thus, the monthly interest rate as a percentage is: r≈1.66%r \approx 1.66\%

Step 4

Complete the table below to show how the balance of the debt of €5000 is reducing each month for the first three months.

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Answer

Payment numberFixed monthly payment, €A€AInterestPrevious balance reduced by (€)New balance of debt (€)
012512582.5042.504957.50
112512581.0043.004914.30
212512580.0045.004869.30
312512579.0046.004823.30

Step 5

Find how long it would take to pay off a credit card debt of €5000.

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Answer

Using the derived formula, we set: 125=50000.01665(1+0.01665)f(1+0.01665)f−1125 = 5000 \frac{0.01665(1 + 0.01665)^f}{(1 + 0.01665)^f - 1} Solving this equation would require iterative methods or financial calculators to find f, which results in: t≈66 monthst \approx 66\text{ months}

Step 6

Find the amount of each weekly repayment.

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Answer

Using the formula: A=Pi(1+i)n(1+i)n−1A = P \frac{i(1 + i)^n}{(1 + i)^n - 1} Where, P=5000, i=0.058552, n=156P = 5000,\ i = \frac{0.0585}{52},\ n = 156 This computation yields: A≈€36.16A \approx €36.16

Step 7

How much will Alex save by paying off the credit card debt using the loan from the Credit Union?

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Answer

To calculate total payments: For the credit card: Total payment = 125 × 66 = €8250 For the Credit Union: Total payment = 36.16 × 156 = €5637.36

Thus, Alex saves: Savings = 8250 - 5637.36 = €2612.64

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