Leaving Cert Mathematics Financial Maths: Donagh is arranging a loan and i

Donagh is arranging a loan and is examining two different repayment options - Leaving Cert Mathematics - Question 6 - 2015

Question 6

Donagh is arranging a loan and is examining two different repayment options. (i) Bank A will charge him a monthly interest rate of 0.35%. Find, correct to three sig... show full transcript

Worked Solution & Example Answer:Donagh is arranging a loan and is examining two different repayment options - Leaving Cert Mathematics - Question 6 - 2015

Step 1

Bank A will charge him a monthly interest rate of 0.35%. Find, correct to three significant figures, the annual percentage rate (APR) that is equivalent to a monthly interest rate of 0.35%.

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Answer

To compute the APR from the monthly interest rate, we can use the formula:

F = P(1 + i)^{12}

Where:

$F$ is the APR
$P$ is the principal amount (which can be any non-zero value)
$i$ is the monthly interest rate (0.35% = 0.0035)

Substituting the values:

F = P(1 + 0.0035)^{12}

Choosing $P = 1$ for simplicity, we have:

F = (1 + 0.0035)^{12} \ F \approx 1.042818 \ \Rightarrow APR \approx (1.042818 - 1) \times 100 \approx 4.28\%

Thus, the annual percentage rate (APR) is approximately 4.28%.

Step 2

Bank B will charge him a rate that is equivalent to an APR of 4.5%. Find, correct to three significant figures, the monthly interest rate that is equivalent to an APR of 4.5%.

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Answer

For Bank B, we can express the APR in terms of the monthly interest rate using the formula:

F = P(1 + i)^{12}

Given that $F = 1.045$ , we need to solve for $i$ :

1.045 = (1 + i)^{12}

Taking the 12th root:

1 + i = 1.045^{1/12}

Now calculate:

1 + i \approx 1.00367 \\ \Rightarrow i \approx 0.00367\%

Thus, the monthly interest rate is approximately 0.367%.

Step 3

Donagh borrowed €80,000 at a monthly interest rate of 0.35%, fixed for the term of the loan, from Bank A. The loan is to be repaid in equal monthly repayments over ten years. The first repayment is due one month after the loan is issued. Calculate, correct to the nearest euro, the amount of each monthly repayment.

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Answer

To find the monthly repayment amount, we will use the formula for the annuity:

A = P \left( \frac{i(1+i)^{n}}{(1+i)^{n} - 1} \right)

Where:

$A$ is the amount of the monthly repayment
$P = 80000€$
$i = 0.0035$ (monthly interest rate)
$n = 10 \times 12 = 120$ months

Substituting the values:

A = 80000 \left( \frac{0.0035(1 + 0.0035)^{120}}{(1 + 0.0035)^{120} - 1} \right)

Calculating:

A \approx 80000 \left( 0.005322846 \right) \approx 818€

Thus, the amount of each monthly repayment is approximately €818.

Donagh is arranging a loan and is examining two different repayment options - Leaving Cert Mathematics - Question 6 - 2015

Worked Solution & Example Answer:Donagh is arranging a loan and is examining two different repayment options - Leaving Cert Mathematics - Question 6 - 2015

Bank A will charge him a monthly interest rate of 0.35%. Find, correct to three significant figures, the annual percentage rate (APR) that is equivalent to a monthly interest rate of 0.35%.

Bank B will charge him a rate that is equivalent to an APR of 4.5%. Find, correct to three significant figures, the monthly interest rate that is equivalent to an APR of 4.5%.

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