Independent preparation for the SQA Higher Applications of Mathematics exam. Not affiliated with or endorsed by Qualifications Scotland (SQA).

Section 4: Interest and the time value of money

Match the effective rate to the period of the payments, then move every payment to the same date before you add or compare.

A lesson for this section

Quick checks: multiple choice

Question 1Calculator or spreadsheet

A savings account pays an effective rate of interest of 0.5% per month. Given that

1.005^{12}

= 1.061678

, what is the effective annual rate of interest, to 2 decimal places?

Question 2Calculator or spreadsheet

A loan has an effective annual rate of interest of 8%. Given that

1.08^{\frac{1}{12}}

= 1.006434

, what is the equivalent effective monthly rate of interest, to 3 significant figures?

Question 3Calculator or spreadsheet

Euan invests 2000 pounds. The effective annual rate of interest is 2% in the first year, 6% in the second year and 4% in the third year. What is his investment worth at the end of the third year, to the nearest penny?

Question 4Calculator or spreadsheet

Isla invests 5000 pounds. For the first year the account pays an effective rate of 1.5% per quarter. After that it pays an effective rate of 0.4% per month. She closes the account 3 years after opening it. Given that

1.015^{4}

= 1.061364

1.004^{12}

= 1.049070

and

1.004^{24}

= 1.100548

, how much is in the account when she closes it, to the nearest penny?

Question 5Calculator or spreadsheet

Three savings accounts. Account A: effective rate 4.2% per year. Account B: effective rate 0.35% per month. Account C: effective rate 2.1% per half year.
Account	Effective rate of interest
A	4.2% per year
B	0.35% per month
C	2.1% per half year

Three savings accounts pay interest as shown in the table. Given that

1.0035^{12}

= 1.042818

and

1.021^{2}

= 1.042441

, which account has the highest effective annual rate of interest?

Question 6Calculator or spreadsheet

Calum puts 4000 pounds in an account with an effective annual rate of interest of 5%. He makes no other payments or withdrawals. How much interest is added during the third year?

Question 7Calculator or spreadsheet

A fund earned effective annual rates of 3%, 5% and 7% in three successive years. Given that

\sqrt[3]{1.157205}

= 1.049873

, what constant effective annual rate would have given the same value at the end of the three years, to 2 decimal places?

Question 8Calculator or spreadsheet

Ross will receive 6000 pounds in 5 years' time. The effective annual rate of interest is 4%. Given that

1.04^{5}

= 1.216653

, what is the present value of the payment, to the nearest penny?

Question 9Calculator or spreadsheet

A payment of 10000 pounds is due in 3 years' time. The effective rate of interest is 0.5% per month. Given that

1.005^{36}

= 1.196681

, what is the present value of the payment, to the nearest penny?

Question 10Calculator or spreadsheet

Kirsty needs 8000 pounds in 4 years' time. The effective annual rate of interest will be 3% for the first 2 years and 5% for the last 2 years. Given that

1.03^{2} \times 1.05^{2}

= 1.169642

, how much must she invest now, to the nearest penny?

Question 11Calculator or spreadsheet

Spreadsheet output of 1.045 to the power n. n = 5: 1.2462. n = 6: 1.3023. n = 7: 1.3609. n = 8: 1.4221. n = 9: 1.4861.
$n$	$1.045^{n}$
5	1.2462
6	1.3023
7	1.3609
8	1.4221
9	1.4861

Mhairi invests 5000 pounds at an effective annual rate of interest of 4.5%, added at the end of each year. The spreadsheet shows

1.045^{n}

for some values of

n

. After how many complete years will her investment first be worth more than 7000 pounds?

Question 12Calculator or spreadsheet

An investment of 2500 pounds grows to 3000 pounds in 4 years with interest compounded each year. Given that

1.2^{\frac{1}{4}}

= 1.046635

, what is the effective annual rate of interest, to 2 decimal places?

Question 13Calculator or spreadsheet

Scott borrows 5000 pounds now. He repays 2000 pounds after 1 year and a final payment of

X

pounds after 3 years. The effective annual rate of interest is 6%. Given that

1.06^{2}

= 1.1236

and

1.06^{3}

= 1.191016

, what is

X

, to the nearest penny?

Question 14Calculator or spreadsheet

A payment of 11500 pounds is due 30 months from now. The effective rate of interest is 1.2% per quarter for the next 12 months, then 5% per year after that. Given that

1.012^{4}

= 1.048871

and

1.05^{1.5}

= 1.075930

, what is the present value of the payment, to the nearest pound?

Question 15Calculator or spreadsheet

A spreadsheet lists payments in column B and the number of years until each is paid in column A, starting in row 2. Cell E1 holds the effective annual rate of interest as a decimal. Which formula, entered in C2 and filled down, gives the present value of each payment?

Question 16Calculator or spreadsheet

Lewis pays 1500 pounds into a savings account at the end of each year for 5 years. The effective annual rate of interest is 4%. Given that

1.04^{5}

= 1.216653

, how much is in the account just after the fifth payment, to the nearest pound? The accumulated value of

n

payments of 1 at the end of each year is

\frac{(1 + i)^{n} - 1}{i}

Question 17Calculator or spreadsheet

Discount factors at 5% a year. 1 year: 0.952381. 2 years: 0.907029. 3 years: 0.863838. 4 years: 0.822702.
Years	Discount factor $1.05^{-t}$
1	0.952381
2	0.907029
3	0.863838
4	0.822702

A charity is promised 2000 pounds in 1 year, 3500 pounds in 2 years and 1000 pounds in 4 years. The effective annual rate of interest is 5%. The table shows the discount factors. What is the present value of the three payments, to the nearest penny?

Question 18Calculator or spreadsheet

In a spreadsheet, cell B2 holds the first of a series of yearly payments. Each payment is 3% more than the one before. Which formula, entered in B3 and filled down, gives the later payments?

Question 19Calculator or spreadsheet

A pension pays 10000 pounds at the end of the first year, and each later yearly payment is 2% more than the one before. Three payments are made. The effective annual rate of interest is 5%. Given that

1.05^{2}

= 1.1025

and

1.05^{3}

= 1.157625

, what is the present value of the three payments, to the nearest pound?

Question 20Calculator or spreadsheet

A trust will pay 3000 pounds at the end of each year for 5 years, with the first payment at the end of year 4. The effective annual rate of interest is 6%. Five payments of 1 at the end of each of the next 5 years have a present value of 4.212364. Given that

1.06^{2}

= 1.1236

1.06^{3}

= 1.191016

and

1.06^{4}

= 1.262477

, what is the present value of the payments now, to the nearest pound?

Question 21Calculator or spreadsheet

Skye pays 1000 pounds into an account at the end of each year for 3 years. The account pays an effective rate of 0.5% per month. Given that

1.005^{12}

= 1.061678

and

1.061678^{2}

= 1.127160

, how much is in the account just after the third payment, to the nearest penny?

Question 22Calculator or spreadsheet

Spreadsheet of a savings account. Month 1: opening balance 2000.00, interest 6.00, deposit 150.00, closing balance 2156.00. Month 2: opening 2156.00, interest 6.47, deposit 150.00, closing 2312.47. Month 3: opening 2312.47, interest 6.94, deposit 150.00, closing 2469.41. All amounts in pounds.
Month	Opening balance (pounds)	Interest (pounds)	Deposit (pounds)	Closing balance (pounds)
1	2000.00	6.00	150.00	2156.00
2	2156.00	6.47	150.00	2312.47
3	2312.47	6.94	150.00	2469.41

Ewan's savings account pays an effective rate of 0.3% per month. Each month, interest is added on the opening balance, rounded to the nearest penny, and then he pays in 150 pounds. The spreadsheet shows the first three months. What is the closing balance for month 4?

Question 23Calculator or spreadsheet

A business expects a machine to save it 5000 pounds at the end of the first year, with the saving falling by 1000 pounds each year after that, for 3 years in all. The effective annual rate of interest is 8%. The discount factors are

1.08^{-1}

= 0.925926

1.08^{-2}

= 0.857339

and

1.08^{-3}

= 0.793832

. What is the present value of the savings, to the nearest penny?

Question 24Calculator or spreadsheet

A loan of 15000 pounds is repaid by 5 equal payments at the end of each year. The effective annual rate of interest is 4.5%. The present value of 5 yearly payments of 1 is

\frac{1 - 1.045^{-5}}{0.045}

, and

1.045^{-5}

= 0.802451

. What is each payment, to the nearest penny?

Question 25Calculator or spreadsheet

Ailsa opens an account with 4000 pounds. She pays in 2500 pounds at the end of year 2 and takes out 1500 pounds at the end of year 3. The effective annual rate of interest is 3% for the first 2 years and 4% after that. Given that

1.03^{2}

= 1.0609

1.04^{2}

= 1.0816

and

1.04^{3}

= 1.124864

, how much is in the account at the end of year 5, to the nearest penny?

Question 26Calculator or spreadsheet

A credit union quotes an effective rate of interest of 2.4% per quarter. Given that

1.024^{\frac{1}{3}}

= 1.007937

, what is the equivalent effective monthly rate, to 3 significant figures?

Question 27Calculator or spreadsheet

Callum pays 1200 pounds into an account at the end of each year for 4 years. The effective annual rate of interest is 3% for the first 2 years and 5% for the last 2 years. Given that

1.05^{2}

= 1.1025

, how much is in the account just after the fourth payment, to the nearest penny?

Question 28Calculator or spreadsheet

Discount factors at 5% a year. 1 year: 0.952381. 2 years: 0.907029. 3 years: 0.863838. 4 years: 0.822702.
Years	Discount factor $1.05^{-t}$
1	0.952381
2	0.907029
3	0.863838
4	0.822702

Ewan owes 4000 pounds due in 1 year and 6000 pounds due in 4 years. The lender agrees to replace these with two equal payments of

X

pounds, at the ends of years 2 and 3, of the same value at an effective annual rate of 5%. The table gives the discount factors. What is

X

, to the nearest pound?

Question 29Calculator or spreadsheet

A loan is repaid by 10 payments of 2400 pounds at the end of each year, at an effective annual rate of 6%. At 6%, level yearly payments of 1 have a present value of 3.465106 over 4 years, 4.212364 over 5 years and 7.360087 over 10 years. How much is still owed just after the sixth payment, to the nearest penny?

Question 30Calculator or spreadsheet

A business can be paid for a contract in one of three ways. Plan A: 10300 pounds now. Plan B: 3000 pounds at the end of each year for 4 years. Plan C: 1500, 2500, 3500 and 4800 pounds at the ends of years 1, 2, 3 and 4. At 6% a year the discount factors are

1.06^{-1}

= 0.943396

1.06^{-2}

= 0.889996

1.06^{-3}

= 0.839619

and

1.06^{-4}

= 0.792094

. Which plan has the highest present value?

Question 31Calculator or spreadsheet

Isla must repay 12000 pounds in one sum in 3 years' time. She will save the same amount at the end of each month for 3 years in an account with an effective annual rate of 8%. Given that

1.08^{\frac{1}{12}}

= 1.006434

and

1.08^{3}

= 1.259712

, how much must she save each month, to the nearest penny? Monthly payments of 1 for

n

months at a monthly rate

j

accumulate to

\frac{(1 + j)^{n} - 1}{j}

Question 32Calculator or spreadsheet

A grant pays 5000 pounds at the end of year 1 and 5000 pounds at the end of year 3. The effective annual rate of interest is 4% for the first 2 years and 6% in year 3. Given that

1.04^{2}

= 1.0816

, what is the present value of the grant, to the nearest penny?

Paper-style questions

Choosing a savings accountQuestion paper style · 8 marks

Gemma has 8000 pounds to save for 3 years. She makes no other payments or withdrawals. The table shows three accounts she is considering.

Three savings accounts. Account X: effective rate 3.9% per year, fixed for 3 years. Account Y: effective rate 0.32% per month, fixed for 3 years. Account Z: variable rate, expected to be 3.5% in year 1, 4.0% in year 2 and 4.5% in year 3.
Account	Effective rate of interest	Fixed or variable
X	3.9% per year	Fixed for 3 years
Y	0.32% per month	Fixed for 3 years
Z	Expected: 3.5% in year 1, 4.0% in year 2, 4.5% in year 3	Variable

(a) Calculate the effective annual rate of interest for Account Y, to 2 decimal places. You may use

1.0032^{12}

= 1.039083

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Calculate the value of Gemma's savings after 3 years in Account Y, to the nearest penny. You may use

1.0032^{36}

= 1.121891

. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

1.039^{3}

= 1.121622

. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) Calculate the value of her savings after 3 years in Account Z if the rates are as expected, to the nearest penny. [2]

Show your working on paper; type the final answer here. Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) Gemma says Account Z is the best choice. Comment on her statement, using your answers. [2]

0 words · a sentence or two

Taking a prizeQuestion paper style · 10 marks

Ava has won a prize. She can take it in one of four ways, shown in the table. All payments in Options B, C and D are made at the end of the year named. She values each option using an effective annual rate of interest of 4%. Ten payments of 1 at the end of each of the next 10 years have a present value of 8.110896 at 4%.

Four ways to take the prize. Option A: a lump sum of 50000 pounds now. Option B: 6000 pounds a year for 10 years, the first payment at the end of year 1. Option C: 4000 pounds at the end of year 1, then each payment 5% more than the one before, 12 payments in all. Option D: 7000 pounds a year for 10 years, the first payment at the end of year 3.
Option	Payments
A	50000 pounds now
B	6000 pounds a year for 10 years, first payment at the end of year 1
C	4000 pounds at the end of year 1, rising by 5% each year, 12 payments
D	7000 pounds a year for 10 years, first payment at the end of year 3

(a) Calculate the present value of Option B, to the nearest pound. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Ava sets up a spreadsheet to value Option C. Complete the payment and present value for years 2 and 3, to the nearest penny. [2]

Year	Payment (pounds)	Discount factor $1.04^{-t}$	Present value (pounds)
1	4000.00	0.961538	3846.15
2		0.924556
3		0.888996

Type one value in each empty cell.

(c) The payments for Option C are in column B, with year 1 in row 2. Which formula, entered in B3 and filled down, gives the later payments? [1]

(d) The spreadsheet gives the present value of Option C as 48675 pounds. Calculate the present value of Option D, to the nearest pound. [3]

Show your working on paper; type the final answer here. Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) Ava decides to choose Option D. Give one reason for this choice and one risk of it. [2]

0 words · a sentence or two