Mathematics: Applications & Interpretation SL
Financial mathematics — Topic 1
- 1.
Layla invests AED 4,000 in an account that pays a nominal annual interest rate of 6%, compounded monthly. Calculate the value of her investment after 3 years, giving your answer to the nearest AED.
[3 marks] - 2.
Marking analysis: A learner attempts the following task: “Layla invests AED 4,000 in an account that pays a nominal annual interest rate of 6%, compounded monthly. Calculate the value of her investment after 3 years, giving your answer to the nearest AED.” Their response addresses only this point: “Identifies n = 36 (monthly compounding over 3 years).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 3.
A delivery van bought for AED 90,000 depreciates at 15% per year using the reducing-balance method. Find its value after 4 complete years, and state the first complete year in which its value falls below AED 40,000.
[4 marks] - 4.
Marking analysis: A learner attempts the following task: “A delivery van bought for AED 90,000 depreciates at 15% per year using the reducing-balance method. Find its value after 4 complete years, and state the first complete year in which its value falls below AED 40,000.” Their response addresses only this point: “Uses the reducing-balance model V = 90000(0.85)^n.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 5.
A student borrows AED 12,000, to be repaid by equal monthly instalments over 2 years at a nominal annual interest rate of 9%, compounded monthly. (a) Calculate the monthly instalment. (b) Calculate the total amount repaid over the loan. (c) Calculate the total interest paid. (d) State how the total interest would change, without further calculation, if the loan term were extended to 3 years at the same rate.
[6 marks] - 6.
Marking analysis: A learner attempts the following task: “A student borrows AED 12,000, to be repaid by equal monthly instalments over 2 years at a nominal annual interest rate of 9%, compounded monthly. (a) Calculate the monthly instalment. (b) Calculate the total amount repaid over the loan. (c) Calculate the total interest paid. (d) State how the total interest would change, without further calculation, if the loan term were extended to 3 years at the same rate.” Their response addresses only this point: “Sets up the loan with PV = 12000, n = 24 monthly payments, and a monthly rate of 9%/12 = 0.75%.” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[6 marks] - 7.
Samir invests AED 5,000 for 4 years. Option A pays simple interest at 5% per year. Option B pays compound interest at 5% per year, compounded annually. Calculate the value of each option after 4 years, and state which option gives the greater return.
[3 marks] - 8.
Marking analysis: A learner attempts the following task: “Samir invests AED 5,000 for 4 years. Option A pays simple interest at 5% per year. Option B pays compound interest at 5% per year, compounded annually. Calculate the value of each option after 4 years, and state which option gives the greater return.” Their response addresses only this point: “Uses simple interest I = Prt to obtain Option A's value as AED 6,000.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 9.
A family borrows AED 8,000 to be repaid by equal monthly instalments over 2 years at a nominal annual interest rate of 9%, compounded monthly. (a) Calculate the monthly instalment. (b) Calculate the amount of the first instalment that is interest. (c) Calculate the amount of the first instalment that repays the principal.
[4 marks] - 10.
Marking analysis: A learner attempts the following task: “A family borrows AED 8,000 to be repaid by equal monthly instalments over 2 years at a nominal annual interest rate of 9%, compounded monthly. (a) Calculate the monthly instalment. (b) Calculate the amount of the first instalment that is interest. (c) Calculate the amount of the first instalment that repays the principal.” Their response addresses only this point: “Sets up PV = 8000, n = 24, monthly rate = 9%/12 = 0.75%.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 11.
Fatima deposits AED 200 at the end of each month into an account paying a nominal annual interest rate of 6%, compounded monthly. Calculate the value of her savings after 3 years.
[3 marks] - 12.
Marking analysis: A learner attempts the following task: “Fatima deposits AED 200 at the end of each month into an account paying a nominal annual interest rate of 6%, compounded monthly. Calculate the value of her savings after 3 years.” Their response addresses only this point: “Identifies this as the future value of a regular savings annuity with monthly payment AED 200 and a monthly rate of 0.5%.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 13.
A bank advertises a savings account with a nominal annual interest rate of 8%, compounded monthly. Calculate the effective annual interest rate of this account, giving your answer to 3 significant figures.
[2 marks] - 14.
Marking analysis: A learner attempts the following task: “A bank advertises a savings account with a nominal annual interest rate of 8%, compounded monthly. Calculate the effective annual interest rate of this account, giving your answer to 3 significant figures.” Their response addresses only this point: “Uses the effective rate formula (1 + nominal/m)^m − 1 with nominal = 0.08 and m = 12.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] - 15.
Inflation in a country is assumed to remain constant at 3% per year. A sum of AED 10,000 today is worth how much in real (today's) purchasing power in 5 years' time, assuming no other change in value? Calculate this real value, giving your answer to the nearest AED.
[2 marks] - 16.
Marking analysis: A learner attempts the following task: “Inflation in a country is assumed to remain constant at 3% per year. A sum of AED 10,000 today is worth how much in real (today's) purchasing power in 5 years' time, assuming no other change in value? Calculate this real value, giving your answer to the nearest AED.” Their response addresses only this point: “Recognises that inflation reduces purchasing power, so the real value must be discounted by the inflation rate.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] - 17.
A loan of AED 15,000 is to be repaid over 3 years by equal monthly instalments at a nominal annual interest rate of 7.2%, compounded monthly. (a) Calculate the monthly instalment. (b) Calculate the outstanding balance immediately after the 12th payment has been made.
[4 marks] - 18.
Marking analysis: A learner attempts the following task: “A loan of AED 15,000 is to be repaid over 3 years by equal monthly instalments at a nominal annual interest rate of 7.2%, compounded monthly. (a) Calculate the monthly instalment. (b) Calculate the outstanding balance immediately after the 12th payment has been made.” Their response addresses only this point: “Sets up PV = 15000, n = 36, monthly rate = 7.2%/12 = 0.6%.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 19.
An investment of AED 4,000 grows at a constant compound annual interest rate of 6%. (a) Use logarithms to find the exact number of years required for the investment to double in value. (b) Compare your answer to the estimate given by the 'rule of 72' (72 ÷ interest rate as a percentage).
[4 marks] - 20.
Marking analysis: A learner attempts the following task: “An investment of AED 4,000 grows at a constant compound annual interest rate of 6%. (a) Use logarithms to find the exact number of years required for the investment to double in value. (b) Compare your answer to the estimate given by the 'rule of 72' (72 ÷ interest rate as a percentage).” Their response addresses only this point: “Sets up the equation 2 = (1.06)^n.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 21.
AED 3,000 is invested for 5 years at a nominal annual rate of 5%. Calculate the final value of the investment if interest is compounded (a) annually and (b) monthly. (c) State which compounding frequency gives the greater return, and explain why.
[4 marks] - 22.
Marking analysis: A learner attempts the following task: “AED 3,000 is invested for 5 years at a nominal annual rate of 5%. Calculate the final value of the investment if interest is compounded (a) annually and (b) monthly. (c) State which compounding frequency gives the greater return, and explain why.” Their response addresses only this point: “Uses annual compounding to obtain a value of approximately AED 3,828.84.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 23.
A company buys machinery for AED 50,000, expected to have a salvage value of AED 5,000 after 10 years. (a) Using the straight-line method, calculate the machinery's value after 3 years. (b) Using the reducing-balance method with an annual depreciation rate of 15%, calculate the machinery's value after 3 years. (c) State which method gives a lower value after 3 years.
[5 marks] - 24.
Marking analysis: A learner attempts the following task: “A company buys machinery for AED 50,000, expected to have a salvage value of AED 5,000 after 10 years. (a) Using the straight-line method, calculate the machinery's value after 3 years. (b) Using the reducing-balance method with an annual depreciation rate of 15%, calculate the machinery's value after 3 years. (c) State which method gives a lower value after 3 years.” Their response addresses only this point: “Calculates the annual straight-line depreciation as (50000 − 5000) ÷ 10 = AED 4,500.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] - 25.
Noor wants to save AED 20,000 in 5 years by making equal monthly deposits into an account paying a nominal annual interest rate of 4%, compounded monthly. Calculate the monthly deposit required.
[3 marks] - 26.
Marking analysis: A learner attempts the following task: “Noor wants to save AED 20,000 in 5 years by making equal monthly deposits into an account paying a nominal annual interest rate of 4%, compounded monthly. Calculate the monthly deposit required.” Their response addresses only this point: “Identifies this as finding the regular payment of a sinking fund with target future value AED 20,000, n = 60 months, and a monthly rate of 4%/12.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 27.
A tourist exchanges USD 1,000 for UAE dirhams at a rate of 1 USD = 3.67 AED. The exchange office charges a 2% commission on the converted amount. (a) Calculate the amount in AED before commission. (b) Calculate the amount the tourist actually receives after commission.
[4 marks] - 28.
Marking analysis: A learner attempts the following task: “A tourist exchanges USD 1,000 for UAE dirhams at a rate of 1 USD = 3.67 AED. The exchange office charges a 2% commission on the converted amount. (a) Calculate the amount in AED before commission. (b) Calculate the amount the tourist actually receives after commission.” Their response addresses only this point: “Multiplies 1000 × 3.67 to obtain AED 3,670 before commission.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 29.
A credit card charges a nominal annual interest rate of 12%, compounded monthly, on any unpaid balance. Calculate the effective annual interest rate charged, to 3 significant figures, and comment on why this is higher than the advertised 12% rate.
[3 marks] - 30.
Marking analysis: A learner attempts the following task: “A credit card charges a nominal annual interest rate of 12%, compounded monthly, on any unpaid balance. Calculate the effective annual interest rate charged, to 3 significant figures, and comment on why this is higher than the advertised 12% rate.” Their response addresses only this point: “Uses the effective rate formula (1 + 0.12/12)^12 − 1.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 31.
Calculate the amount that must be invested today at a compound annual interest rate of 6.5% in order to have exactly AED 10,000 available in 6 years' time.
[3 marks] - 32.
Marking analysis: A learner attempts the following task: “Calculate the amount that must be invested today at a compound annual interest rate of 6.5% in order to have exactly AED 10,000 available in 6 years' time.” Their response addresses only this point: “Recognises this as a present-value problem: PV = FV ÷ (1 + r)^n.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks]