📘 2.

4 Apply Mathematical Techniques for Calculating

Interest
Understanding Interest

Interest is the cost of borrowing money or the return on invested funds. It's paid by borrowers to
lenders or earned by depositors from financial institutions. There are two main types of interest:

1. Simple Interest – calculated only on the original principal.

2. Compound Interest – calculated on both the principal and previously earned interest.

2.4.1 Simple Interest

Definition:

Simple interest is calculated on the original principal only, over a period of time.

Formula:

I=P×r×t\boxed{I = P \times r \times t}I=P×r×t

Where:

 III = Simple Interest

 PPP = Principal (initial amount of money)
 rrr = Annual interest rate (decimal form)
 ttt = Time in years

Other Related Formulas:

 Amount (Future Value):

A=P+I=P(1+rt)A = P + I = P(1 + rt)A=P+I=P(1+rt)

 To find Principal:

P=IrtP = \frac{I}{rt}P=rtI

 To find Rate:

r=IPtr = \frac{I}{Pt}r=PtI

 To find Time:
 t=IPrt = \frac{I}{Pr}t=PrI

✅ Simple Interest - Example 1

Problem:
Ato Kassahun borrows Br.10,000 for 9 months at 6% simple annual interest. What is the interest
and total amount to repay?

Solution:

 P=10,000P = 10,000P=10,000, r=6%=0.06r = 6\% = 0.06r=6%=0.06, t=9/12=0.75t =

9/12 = 0.75t=9/12=0.75 years
 I=P×r×t=10,000×0.06×0.75=Br.450I = P \times r \times t = 10,000 \times 0.06 \times
0.75 = Br.450I=P×r×t=10,000×0.06×0.75=Br.450
 A=P+I=10,000+450=Br.10,450A = P + I = 10,000 + 450 =
Br.10,450A=P+I=10,000+450=Br.10,450

✅ Simple Interest - Example 2

Problem:
How long will it take for Br.10,000 to double at 10% annual simple interest?

Solution:

 P=10,000P = 10,000P=10,000, A=20,000A = 20,000A=20,000, I=10,000I =

10,000I=10,000, r=0.10r = 0.10r=0.10
 Use t=IPr=10,00010,000×0.10=10t = \frac{I}{Pr} = \frac{10,000}{10,000 \times 0.10} =
10t=PrI=10,000×0.1010,000=10 years

2.4.2 Compound Interest

📌 Definition:

Compound interest is interest calculated on the initial principal, which also includes all
accumulated interest from previous periods.

📐 Formula:
A=P(1+i)n\boxed{A = P (1 + i)^n}A=P(1+i)n

Where:

 AAA = Future Value (Compound Amount)

 PPP = Principal
 rrr = Annual interest rate
 mmm = Number of compounding periods per year
 i=rmi = \frac{r}{m}i=mr = Rate per period
 n=m×tn = m \times tn=m×t = Total number of periods

Compound Interest (CI):

CI=A−PCI = A - PCI=A−P

🔁 Compounding Frequencies and Rate per Period:

Compounding Frequency Periods per Year (m) i = r/m

Annually 1 r/1
Semi-annually 2 r/2
Quarterly 4 r/4
Monthly 12 r/12

Compound Interest - Example 1

Problem:
Invest Br.10,000 at 12% annual interest compounded quarterly for 1 year. What is the compound
amount and interest?

Solution:

 P=10,000P = 10,000P=10,000, r=0.12r = 0.12r=0.12, m=4m = 4m=4, t=1t = 1t=1

 i=0.124=0.03i = \frac{0.12}{4} = 0.03i=40.12=0.03, n=4×1=4n = 4 \times 1 = 4n=4×1=4
 A=10,000(1+0.03)4=10,000(1.1255)=Br.11,255.09A = 10,000 (1 + 0.03)^4 = 10,000
(1.1255) = Br.11,255.09A=10,000(1+0.03)4=10,000(1.1255)=Br.11,255.09
 CI=11,255.09−10,000=Br.1,255.09CI = 11,255.09 - 10,000 =
Br.1,255.09CI=11,255.09−10,000=Br.1,255.09

✅ Compound Interest - Example 2

Problem:
How much will Br.5,000 become in 3 years if invested at 8% interest compounded semi-
annually?

Solution:

 P=5,000P = 5,000P=5,000, r=0.08r = 0.08r=0.08, m=2m = 2m=2, t=3t = 3t=3

 i=0.08/2=0.04i = 0.08/2 = 0.04i=0.08/2=0.04, n=2×3=6n = 2 \times 3 = 6n=2×3=6
 A=5,000(1+0.04)6=5,000(1.2653)=Br.6,326.50A = 5,000 (1 + 0.04)^6 = 5,000 (1.2653)
= Br.6,326.50A=5,000(1+0.04)6=5,000(1.2653)=Br.6,326.50
 CI=6,326.50−5,000=Br.1,326.50CI = 6,326.50 - 5,000 =
Br.1,326.50CI=6,326.50−5,000=Br.1,326.50

✅ Key Differences Between Simple and Compound Interest:

Feature Simple Interest Compound Interest
Interest Base Only on original principal On principal + accumulated interest
Formula I=PrtI = PrtI=Prt A=P(1+i)nA = P(1 + i)^nA=P(1+i)n
Suitable For Short-term loans Long-term investments/loans
Growth Pattern Linear Exponential

📌 Important Notes:
 Always match time (t) and rate (r) units (e.g., if time is in months, divide by 12).
 Use 360-day year for ordinary interest, or 365-day year for exact time.
 Compound interest grows faster due to "interest on interest".

2.5 Applying Mathematical Techniques for

Calculating Break-Even Point
What is the Break-Even Point?
The break-even point (BEP) is the point at which total revenue equals total cost, resulting in
zero profit and zero loss. It represents the minimum amount of sales a business needs to avoid
losing money.

 At break-even:

Total Revenue=Fixed Costs+Variable Costs\text{Total Revenue} = \text{Fixed Costs}

+ \text{Variable Costs}Total Revenue=Fixed Costs+Variable Costs
Why Break-Even Analysis Matters
 Determines minimum sales needed to stay in business.
 Helps in pricing strategy.
 Assists in cost control and planning expansion.
 Useful in assessing new products or investments.

Key Components of Break-Even Analysis

1. Fixed Costs (FC):

Costs that do not change with the number of units produced or sold.
Examples: Rent, salaries, insurance, equipment depreciation.

2. Variable Costs (VC):

Costs that vary with production volume.

Examples: Raw materials, direct labor, packaging, commissions.

3. Sales Price (SP):

Selling price per unit of product or service.

4. Contribution Margin (CM):

The amount that contributes to covering fixed costs after paying for variable costs.

Contribution Margin per Unit=Sales Price per Unit−Variable Cost per Unit\text{Contribution
Margin per Unit} = \text{Sales Price per Unit} - \text{Variable Cost per
Unit}Contribution Margin per Unit=Sales Price per Unit−Variable Cost per Unit
Contribution Margin Ratio (CMR)=Contribution MarginSales Price×100\text{Contribution
Margin Ratio (CMR)} = \frac{\text{Contribution Margin}}{\text{Sales Price}} \times
100Contribution Margin Ratio (CMR)=Sales PriceContribution Margin×100

Break-Even Point Formulas

A. In Units:

Break-Even Point (Units)=Fixed CostsSales Price per Unit−Variable Cost per Unit\boxed{\
text{Break-Even Point (Units)} = \frac{\text{Fixed Costs}}{\text{Sales Price per Unit} - \
text{Variable Cost per Unit}}}Break-
Even Point (Units)=Sales Price per Unit−Variable Cost per UnitFixed Costs
B. In Sales Revenue (Currency):

Break-Even Point (Sales Revenue)=Fixed CostsContribution Margin Ratio\boxed{\text{Break-

Even Point (Sales Revenue)} = \frac{\text{Fixed Costs}}{\text{Contribution Margin
Ratio}}}Break-Even Point (Sales Revenue)=Contribution Margin RatioFixed Costs

✅ Examples for Better Understanding

✳️Example 1: Break-Even Point in Units

Scenario:
Jane runs a soda shop. Her costs are:

 Fixed Costs = Br. 20,000

 Sales Price per Unit (bottle) = Br. 2.00
 Variable Cost per Unit = Br. 1.50

Formula:

Break-Even Units=20,0002.00−1.50=20,0000.50=40,000 bottles\text{Break-Even Units} = \

frac{20,000}{2.00 - 1.50} = \frac{20,000}{0.50} = \boxed{40,000 \text{ bottles}}Break-
Even Units=2.00−1.5020,000=0.5020,000=40,000 bottles

👉 She must sell 40,000 bottles to cover all costs and break even.

xample 2: Break-Even Point in Sales Revenue

Scenario:
A company has:

 Fixed Costs = Br. 1,000,000

 Gross Margin (same as Contribution Margin Ratio here) = 37%

Formula:

Break-Even Sales=1,000,0000.37=Br.2,702,703\text{Break-Even Sales} = \frac{1,000,000}

{0.37} = \boxed{Br. 2,702,703}Break-Even Sales=0.371,000,000=Br.2,702,703

👉 The company must generate Br. 2.7 million in revenue to break even.

✳️Example 3: Canned Product Business

Scenario:
  Fixed Costs = Br. 2,000
 Variable Cost = Br. 0.40 per can
 Sales Price = Br. 1.50 per can

Step 1:

Contribution Margin=1.50−0.40=Br.1.10 per can\text{Contribution Margin} = 1.50 - 0.40 = Br.

1.10 \text{ per can}Contribution Margin=1.50−0.40=Br.1.10 per can

Step 2:

Break-Even (Units)=2,0001.10=1,819 cans (rounded)\text{Break-Even (Units)} = \frac{2,000}

{1.10} = \boxed{1,819 \text{ cans (rounded)}}Break-Even (Units)=1.102,000
=1,819 cans (rounded)

👉 The business must sell 1,819 cans to break even.

Break-Even Chart (Optional)

A break-even chart is a graphical representation showing:

 X-axis: Number of units sold

 Y-axis: Sales and costs in currency
 Shows three lines:
o Fixed cost line (horizontal)
o Total cost line (starts from fixed cost)
o Revenue line (starts from 0)

👉 The intersection point of total cost and revenue is the break-even point.

🧠 Key Takeaways from Break-Even Analysis

Term Description
Fixed Costs Costs that remain constant regardless of output
Variable Costs Costs that vary directly with production volume
Contribution Margin Profit left after variable cost, used to cover fixed costs
Break-Even Sales Minimum revenue required to cover all costs
Break-Even Units Minimum units to sell to avoid loss
Contribution Ratio Contribution margin as a % of sales price

When to Use Break-Even Analysis

 Launching a new product or business
 Evaluating price changes
  Budget planning and goal setting
 Deciding whether to expand operations

factors That Can Affect Break-Even Point

Factor Impact on BEP
↑ Fixed Costs BEP increases (need more sales to cover costs)
↓ Variable Costs BEP decreases (more contribution per unit)
↑ Selling Price BEP decreases
↓ Selling Price BEP increases
Improved efficiency Lowers variable costs and reduces BEP

Break-Even Sensitivity Analysis

You can change one variable (price, costs, etc.) at a time to study how it affects the BEP. This
helps in:

 Making strategic pricing decisions

 Planning for different sales volumes
 Managing risk and uncertainty

🧮 Practice Problem for You

Your bakery has:

 Fixed costs = Br. 12,000

 Variable cost per loaf = Br. 5
 Selling price per loaf = Br. 8

Questions:

1. What is the break-even point in units?

2. What is the break-even point in Br.?

Let me know if you’d like me to solve this one with you step-by-step or if you want a PDF
worksheet or Excel template.

Would you like me to generate:

 An Excel calculator for break-even point?

 A PowerPoint slide to present this?
  A fill-in-the-blank worksheet for student practice?

📘 2.6 Applying Mathematical Techniques for

Calculating Annuity

🔷 2.6.1 What is an Annuity?

An annuity is a financial product or arrangement where a series of equal payments are made or
received at regular intervals (e.g., monthly, quarterly, annually) over a period of time.

🟦 Examples:

 Monthly rent or loan repayments

 Insurance premiums
 Retirement income distributions

📑 Types of Annuities
There are two major types:

✅ 1. Ordinary Annuity

 Payments are made at the end of each period.

 Most common type in financial accounting and personal finance.

✅ 2. Annuity Due

 Payments are made at the beginning of each period.

 Example: Rent is usually paid at the beginning of the month.

🔧 Mathematical Techniques Used to Calculate Annuities

Annuities are calculated based on the Time Value of Money (TVM) — meaning money now is
worth more than the same amount in the future due to earning potential.
🟩 I. Ordinary Annuity
📌 A. Present Value of an Ordinary Annuity (PVOA)

Used to determine how much a series of future payments is worth today, if the payments are
made at the end of each period.

✅ Formula:

PV=PMT×(1−(1+r)−nr)PV = PMT \times \left(\frac{1 - (1 + r)^{-n}}{r}\

right)PV=PMT×(r1−(1+r)−n)

Where:

 PMT = Payment per period

 r = Interest rate per period
 n = Total number of periods
 PV = Present Value

🔷 Example:

You receive $50,000 per year for 5 years, at an interest rate of 7%.

PV=50,000×(1−(1+0.07)−50.07)PV = 50,000 \times \left(\frac{1 - (1 + 0.07)^{-5}}{0.07}\

right)PV=50,000×(0.071−(1+0.07)−5) PV=50,000×(4.1002)=$205,010PV = 50,000 \times
(4.1002) = \boxed{\$205,010}PV=50,000×(4.1002)=$205,010

📌 B. Future Value of an Ordinary Annuity (FVOA)

Used to calculate the total amount accumulated in the future, if equal payments are made at the
end of each period.

✅ Formula:

FV=PMT×((1+r)n−1r)FV = PMT \times \left(\frac{(1 + r)^n - 1}{r} \right)FV=PMT×(r(1+r)n−1

)

🔷 Example:

Invest $100,000 annually for 5 years at 7% annual interest.

FV=100,000×((1+0.07)5−10.07)FV = 100,000 \times \left(\frac{(1 + 0.07)^5 - 1}{0.07}\

right)FV=100,000×(0.07(1+0.07)5−1) FV=100,000×(5.7507)=$575,074FV = 100,000 \times
(5.7507) = \boxed{\$575,074}FV=100,000×(5.7507)=$575,074
🧮 Monthly Compounding Example:

Invest $8,000/month for 5 years, with monthly interest rate of 0.5833% (7% annual / 12):

FV=8,000×((1+0.005833)60−10.005833)FV = 8,000 \times \left(\frac{(1 + 0.005833)^{60} - 1}

{0.005833}\right)FV=8,000×(0.005833(1+0.005833)60−1) FV=8,000×(71.5921)=$572,737FV
= 8,000 \times (71.5921) = \boxed{\$572,737}FV=8,000×(71.5921)=$572,737
II. Annuity Due
Here, payments are made at the beginning of each period.

A. Present Value of an Annuity Due (PVAD)

Used to determine the present value of a set of payments made at the start of each period.

✅ Formula:

PV=PMT×(1−(1+r)−nr)×(1+r)PV = PMT \times \left(\frac{1 - (1 + r)^{-n}}{r} \right) \times (1

+ r)PV=PMT×(r1−(1+r)−n)×(1+r)

🔷 Example:

You invest 2,000,000 for 4 years at 10% interest.

PV=2,000,000×(1−(1+0.10)−40.10)×(1.10)PV = 2,000,000 \times \left(\frac{1 - (1 + 0.10)^{-

4}}{0.10} \right) \times (1.10)PV=2,000,000×(0.101−(1+0.10)−4)×(1.10)
PV=2,000,000×(3.1699)×1.10=Br.6,973,780PV = 2,000,000 \times (3.1699) \times 1.10 = \
boxed{Br. 6,973,780}PV=2,000,000×(3.1699)×1.10=Br.6,973,780

B. Future Value of an Annuity Due (FVAD)

Calculates the amount accumulated in the future when payments are made at the start of each
period.

✅ Formula:

FV=PMT×((1+r)n−1r)×(1+r)FV = PMT \times \left(\frac{(1 + r)^n - 1}{r} \right) \times (1 +

r)FV=PMT×(r(1+r)n−1)×(1+r)

🔷 Example:

John deposits $5,000 at the beginning of each year for 7 years at 5% interest:

FV=5,000×((1+0.05)7−10.05)×(1.05)FV = 5,000 \times \left(\frac{(1 + 0.05)^7 - 1}{0.05} \

right) \times (1.05)FV=5,000×(0.05(1+0.05)7−1)×(1.05)
FV=5,000×(8.142)×1.05=$42,745.50FV = 5,000 \times (8.142) \times 1.05 = \boxed{\
$42,745.50}FV=5,000×(8.142)×1.05=$42,745.50
Summary Table
Payment
Type Formula Used Use Case Example
Timing
How much future
PMT×1−(1+r)−nr\text{PMT} \times \frac{1 - (1 +
PVOA End of period payments are worth
r)^{-n}}{r}PMT×r1−(1+r)−n
today
PMT×(1+r)n−1r\text{PMT} \times \frac{(1 + r)^n How much will be
FVOA End of period
- 1}{r}PMT×r(1+r)n−1 saved in future
Beginning of Present value of
PVAD PVOA × (1 + r)
period annuity paid in advance
Beginning of Future value of
FVAD FVOA × (1 + r)
period advance-paid annuity

🧠 Real-Life Applications of Annuities

Application Type of Annuity Example
Retirement planning Annuity due / Ordinary Monthly income after retirement
Loan repayment Ordinary annuity Monthly car or mortgage loan payments
Lease agreements Annuity due Rent paid at the beginning of the month
Saving for education Ordinary / Annuity due College fund through annual contributions
Lottery payout options Ordinary annuity Choosing annual payouts vs. lump sum

Tips for Solving Annuity Problems

 Use annuity tables or financial calculators when manual calculation is too complex.
 Remember:
o "End" = Ordinary Annuity
o "Beginning" = Annuity Due
 Ensure interest rates and time periods match:
o Annual rates → yearly periods
o Monthly rates → monthly periods