Mathematical Applications in Business: Detailed Notes & Exam Questions | Grade 12 Mathematics Unit 5

Grade 12

Mathematical Applications in Business : Detailed Notes & Exam Questions | Grade 12 Mathematics Unit 5

1. Introduction to Business Mathematics

Dear student, welcome to Unit 5! In this unit, we learn how mathematics is used in real business situations. Whether you are borrowing money from a bank, paying tax, calculating profit, or buying something on installments — mathematics is involved everywhere.

The main topics we will cover are:

  • Simple Interest — interest calculated on the original amount only
  • Compound Interest — interest calculated on the amount that keeps growing
  • Hire Purchase — buying goods by paying in installments
  • Taxation — income tax and value-added tax (VAT)
  • Profit, Loss, and Discount — basic business calculations

Let’s begin with one of the most important topics: interest!

2. Simple Interest

When you borrow money from a bank, you pay extra money called interest. When you deposit money in a bank, the bank pays YOU interest. Simple interest is the easiest type — it is calculated only on the original amount (called the principal).

2.1 Key Terms

  • Principal (P): The original amount of money borrowed or invested
  • Rate (R): The percentage charged or earned per year (always express as a percentage)
  • Time (T): The duration in years
  • Simple Interest (I): The extra money paid or received
  • Amount (A): Principal + Interest = Total money at the end

2.2 Simple Interest Formula

\[ I = \frac{P \times R \times T}{100} \]
\[ A = P + I = P\left(1 + \frac{RT}{100}\right) \]

Think about it: if you borrow 1000 Birr at 10% for 2 years, the interest each year is 100 Birr. For 2 years, it’s 200 Birr. The interest never changes because it’s always calculated on the original 1000 Birr. That’s why it’s called “simple”!

Worked Example 1: Abebe borrows 5,000 Birr from a bank at 8% simple interest per year for 3 years. Find: (a) the interest, (b) the total amount to repay.

Solution:

\( P = 5000 \), \( R = 8\% \), \( T = 3 \) years

(a) \( I = \frac{5000 \times 8 \times 3}{100} = \frac{120000}{100} = 1200 \) Birr

(b) \( A = P + I = 5000 + 1200 = 6200 \) Birr

Worked Example 2: Tigist deposits 12,000 Birr in a savings account that pays 6% simple interest per year. How much will she have after 2.5 years?

Solution:

\( P = 12000 \), \( R = 6\% \), \( T = 2.5 \) years

\[ I = \frac{12000 \times 6 \times 2.5}{100} = \frac{180000}{100} = 1800 \text{ Birr} \] \[ A = 12000 + 1800 = 13800 \text{ Birr} \]

Worked Example 3: A sum of money earns 3,600 Birr as simple interest in 4 years at 9% per year. Find the principal.

Solution:

\( I = 3600 \), \( R = 9\% \), \( T = 4 \), \( P = ? \)

\[ P = \frac{I \times 100}{R \times T} = \frac{3600 \times 100}{9 \times 4} = \frac{360000}{36} = 10000 \text{ Birr} \]

Worked Example 4: At what rate of simple interest will 8,000 Birr grow to 10,400 Birr in 3 years?

Solution:

\( P = 8000 \), \( A = 10400 \), \( T = 3 \), \( R = ? \)

\( I = A – P = 10400 – 8000 = 2400 \) Birr

\[ R = \frac{I \times 100}{P \times T} = \frac{2400 \times 100}{8000 \times 3} = \frac{240000}{24000} = 10\% \]
Key Exam Notes — Simple Interest:
  • Always check the units: Time must be in YEARS. If given in months, divide by 12.
  • Rate is always a percentage. If given as a decimal (e.g., 0.08), multiply by 100 first.
  • To find P: rearrange as \( P = \frac{I \times 100}{R \times T} \)
  • To find R: rearrange as \( R = \frac{I \times 100}{P \times T} \)
  • To find T: rearrange as \( T = \frac{I \times 100}{P \times R} \)
Practice Question 1

Find the simple interest on 15,000 Birr at 7.5% per year for 4 years. Also find the total amount.

\[ I = \frac{15000 \times 7.5 \times 4}{100} = \frac{450000}{100} = 4500 \text{ Birr} \] \[ A = 15000 + 4500 = 19500 \text{ Birr} \]
Practice Question 2

In how many years will 6,000 Birr earn 1,800 Birr as simple interest at 10% per year?

\[ T = \frac{I \times 100}{P \times R} = \frac{1800 \times 100}{6000 \times 10} = \frac{180000}{60000} = 3 \text{ years} \]

3. Compound Interest

Compound interest is different from simple interest — and much more powerful! In compound interest, the interest earned each period is added to the principal, and the NEXT period’s interest is calculated on this NEW, larger amount.

Imagine you deposit 1000 Birr at 10%. After year 1, you have 1100 Birr. In year 2, the 10% is calculated on 1100 Birr (not 1000), so you earn 110 Birr, giving you 1210 Birr. Your money is “earning interest on interest” — that’s the power of compounding!

3.1 Compound Interest Formula

\[ A = P\left(1 + \frac{R}{100}\right)^n \]
\[ CI = A – P = P\left(1 + \frac{R}{100}\right)^n – P \]
Where:
  • \( A \) = Amount after \( n \) periods
  • \( P \) = Principal
  • \( R \) = Rate per period (as percentage)
  • \( n \) = Number of compounding periods
  • \( CI \) = Compound Interest earned

3.2 Important: Matching Rate and Time

The rate \( R \) and the number of periods \( n \) must match! Here are common cases:

CompoundingIf rate is per yearNumber of periods in \( t \) years
Annually (yearly)\( R\% \) per year\( n = t \)
Semi-annually (half-yearly)\( \frac{R}{2}\% \) per half-year\( n = 2t \)
Quarterly\( \frac{R}{4}\% \) per quarter\( n = 4t \)
Monthly\( \frac{R}{12}\% \) per month\( n = 12t \)

Worked Example 5: Find the compound interest on 10,000 Birr at 8% per year compounded annually for 3 years.

Solution:

\( P = 10000 \), \( R = 8\% \), \( n = 3 \)

\[ A = 10000\left(1 + \frac{8}{100}\right)^3 = 10000(1.08)^3 \] \[ = 10000 \times 1.259712 = 12597.12 \text{ Birr} \] \[ CI = 12597.12 – 10000 = 2597.12 \text{ Birr} \]

Worked Example 6: Compare: Find both the simple interest and compound interest on 20,000 Birr at 10% for 2 years.

Solution:

Simple Interest:

\[ SI = \frac{20000 \times 10 \times 2}{100} = 4000 \text{ Birr} \]

Compound Interest:

\[ A = 20000(1.10)^2 = 20000 \times 1.21 = 24200 \text{ Birr} \] \[ CI = 24200 – 20000 = 4200 \text{ Birr} \]

Difference: \( CI – SI = 4200 – 4000 = 200 \) Birr

Compound interest is always greater than simple interest (for rate > 0 and time > 1 year). The difference grows as time increases!

Worked Example 7: Find the compound interest on 5,000 Birr at 12% per year compounded semi-annually for 2 years.

Solution:

Compounded semi-annually: Rate per half-year = \( \frac{12}{2} = 6\% \), Number of periods = \( 2 \times 2 = 4 \)

\[ A = 5000\left(1 + \frac{6}{100}\right)^4 = 5000(1.06)^4 \] \[ = 5000 \times 1.262477 = 6312.39 \text{ Birr} \] \[ CI = 6312.39 – 5000 = 1312.39 \text{ Birr} \]

Worked Example 8: How long will it take for 8,000 Birr to double itself at 10% compound interest per year?

Solution:

We want \( A = 2P = 16000 \)

\[ 16000 = 8000(1.10)^n \] \[ 2 = (1.10)^n \] \[ n = \frac{\log 2}{\log 1.10} = \frac{0.3010}{0.0414} \approx 7.27 \text{ years} \]

It takes approximately 7.27 years (about 7 years and 3 months).

Exam Note — Compound Interest:
  • Always match the rate to the compounding period. Semi-annually → halve the rate, double the periods.
  • For “compounded annually,” use the formula directly with yearly rate and years.
  • To find the difference between CI and SI for 2 years: \( CI – SI = P \times \frac{R^2}{100^2} \)
  • To find the difference between CI and SI for 3 years: \( CI – SI = P \times \frac{R^2(300+R)}{100^3} \)
  • These shortcut formulas save time in exams!
Exam-Style Question 1

(a) Find the compound interest on 25,000 Birr at 8% per year for 3 years, compounded annually.

(b) What would be the difference between compound interest and simple interest for the same data?

(a)

\[ A = 25000(1.08)^3 = 25000 \times 1.259712 = 31492.80 \text{ Birr} \] \[ CI = 31492.80 – 25000 = 6492.80 \text{ Birr} \]

(b)

\[ SI = \frac{25000 \times 8 \times 3}{100} = 6000 \text{ Birr} \] \[ \text{Difference} = 6492.80 – 6000 = 492.80 \text{ Birr} \]

Or using the shortcut for 3 years:

\[ CI – SI = P \times \frac{R^2(300+R)}{100^3} = 25000 \times \frac{64 \times 308}{1000000} = 25000 \times \frac{19712}{1000000} = 492.80 \text{ Birr} \]

4. Hire Purchase

Hire Purchase (HP) is a way of buying goods by paying in installments instead of paying the full price at once. You usually pay a deposit first, then pay the rest in equal monthly installments over a period of time.

4.1 Key Terms

  • Cash Price: The actual price of the item if you pay in full immediately
  • Deposit: The initial amount paid upfront
  • Installment: Equal periodic payments (usually monthly)
  • Hire Purchase Price: Deposit + Total of all installments
  • Interest (Extra Pay): Hire Purchase Price − Cash Price
\[ \text{Hire Purchase Price} = \text{Deposit} + (\text{Number of Installments} \times \text{Installment Amount}) \]
\[ \text{Interest (Extra Paid)} = \text{Hire Purchase Price} – \text{Cash Price} \]

Worked Example 9: A television has a cash price of 12,000 Birr. Under hire purchase, a deposit of 3,000 Birr is required, followed by 12 monthly installments of 850 Birr. Find: (a) the hire purchase price, (b) the extra amount paid, (c) the flat rate of interest charged on the balance.

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Solution:

(a) Hire Purchase Price = Deposit + Total installments

\[ HP = 3000 + (12 \times 850) = 3000 + 10200 = 13200 \text{ Birr} \]

(b) Extra amount paid = HP Price − Cash Price

\[ \text{Extra} = 13200 – 12000 = 1200 \text{ Birr} \]

(c) The balance after deposit = \( 12000 – 3000 = 9000 \) Birr. This is the amount on which interest is charged for 1 year (12 months).

\[ \text{Flat Rate} = \frac{\text{Extra} \times 100}{\text{Balance} \times \text{Time (years)}} = \frac{1200 \times 100}{9000 \times 1} = \frac{120000}{9000} = 13.33\% \]

Worked Example 10: A refrigerator costs 18,000 Birr cash. On hire purchase, a deposit of 20% of the cash price is required, and the balance is paid in 18 monthly installments of 1,000 Birr each. Find the extra amount paid and the rate of interest.

Solution:

Deposit = 20% of 18000 = \( 0.20 \times 18000 = 3600 \) Birr

Balance = \( 18000 – 3600 = 14400 \) Birr

Total installments = \( 18 \times 1000 = 18000 \) Birr

HP Price = \( 3600 + 18000 = 21600 \) Birr

Extra paid = \( 21600 – 18000 = 3600 \) Birr

Time = 18 months = 1.5 years

\[ \text{Rate} = \frac{3600 \times 100}{14400 \times 1.5} = \frac{360000}{21600} = 16.67\% \]
Exam Note — Hire Purchase:
  • The deposit is usually a percentage of the CASH PRICE (not the HP price).
  • The balance = Cash Price − Deposit. Interest is charged on this balance.
  • Time in years = Number of monthly installments ÷ 12.
  • The “flat rate” treats the entire balance as if it were owed for the full period (unlike reducing balance method).
  • Hire purchase is ALWAYS more expensive than paying cash — the extra amount is the cost of borrowing.
Practice Question 3

A laptop has a cash price of 15,000 Birr. Under hire purchase terms, a deposit of 25% is paid and the remainder is paid in 12 equal monthly installments of 1,100 Birr. Find: (a) the deposit, (b) the hire purchase price, (c) the extra amount paid.

(a) Deposit = 25% of 15000 = \( 0.25 \times 15000 = 3750 \) Birr

(b) HP Price = 3750 + (12 × 1100) = 3750 + 13200 = 16950 Birr

(c) Extra = 16950 − 15000 = 1950 Birr

5. Taxation

Tax is money collected by the government from individuals and businesses to fund public services like roads, schools, and hospitals. In Grade 12, we focus on two main types: Income Tax and Value Added Tax (VAT).

5.1 Income Tax

Income tax is a tax on the money people earn. In Ethiopia (and many countries), income tax uses a progressive tax system — higher income is taxed at a higher rate.

\[ \text{Income Tax} = \text{Taxable Income} \times \text{Tax Rate} \]
\[ \text{Taxable Income} = \text{Gross Income} – \text{Allowances (Deductions)} \]
\[ \text{Net Income (Take-Home Pay)} = \text{Gross Income} – \text{Income Tax} \]

Ethiopian Income Tax Brackets (Monthly Taxable Income):

Taxable Income (Birr/month)RateTax (Birr)
0 – 6000%0
601 – 1,65010%10% of amount exceeding 600
1,651 – 3,20015%105 + 15% of amount exceeding 1,650
3,201 – 5,25020%337.50 + 20% of amount exceeding 3,200
5,251 – 7,80025%747.50 + 25% of amount exceeding 5,250
7,801 – 10,90030%1,385 + 30% of amount exceeding 7,800
Above 10,90035%2,312 + 35% of amount exceeding 10,900

Worked Example 11: A teacher earns a gross monthly salary of 6,000 Birr. The tax-free allowance is 600 Birr. Calculate: (a) taxable income, (b) income tax, (c) net income.

Solution:

(a) Taxable Income = 6000 − 600 = 5400 Birr

(b) This falls in the bracket 5,251 – 7,800 at 25%:

\[ \text{Tax} = 747.50 + 25\% \times (5400 – 5250) \] \[ = 747.50 + 0.25 \times 150 \] \[ = 747.50 + 37.50 = 785 \text{ Birr} \]

(c) Net Income = 6000 − 785 = 5215 Birr

Worked Example 12: An engineer earns 12,000 Birr per month gross. The tax-free allowance is 600 Birr. Calculate the income tax and net pay.

Solution:

Taxable Income = 12000 − 600 = 11400 Birr

This falls in the bracket “Above 10,900” at 35%:

\[ \text{Tax} = 2312 + 35\% \times (11400 – 10900) \] \[ = 2312 + 0.35 \times 500 \] \[ = 2312 + 175 = 2487 \text{ Birr} \]

Net Pay = 12000 − 2487 = 9513 Birr

Practice Question 4

A shopkeeper has a monthly gross income of 4,500 Birr with a tax-free allowance of 600 Birr. Calculate the income tax and net income.

Taxable Income = 4500 − 600 = 3900 Birr

This falls in the bracket 3,201 – 5,250 at 20%:

\[ \text{Tax} = 337.50 + 20\% \times (3900 – 3200) \] \[ = 337.50 + 0.20 \times 700 \] \[ = 337.50 + 140 = 477.50 \text{ Birr} \]

Net Income = 4500 − 477.50 = 4022.50 Birr

5.2 Value Added Tax (VAT)

VAT is a tax added to the selling price of goods and services. In Ethiopia, the standard VAT rate is 15%.

\[ \text{VAT Amount} = \text{Selling Price (before VAT)} \times \frac{\text{VAT Rate}}{100} \]
\[ \text{Price including VAT} = \text{Selling Price} + \text{VAT} = \text{Selling Price} \times \left(1 + \frac{\text{VAT Rate}}{100}\right) \]
\[ \text{If price includes VAT:} \quad \text{Selling Price} = \frac{\text{Price including VAT}}{1 + \frac{\text{VAT Rate}}{100}} \]

Worked Example 13: A shop sells a product for 2,000 Birr before VAT. If the VAT rate is 15%, find: (a) the VAT amount, (b) the price the customer pays.

Solution:

\[ \text{VAT} = 2000 \times \frac{15}{100} = 300 \text{ Birr} \] \[ \text{Price including VAT} = 2000 + 300 = 2300 \text{ Birr} \]

Worked Example 14: A customer pays 4,600 Birr for an item including 15% VAT. What is the price before VAT, and what is the VAT amount?

Solution:

\[ \text{Price before VAT} = \frac{4600}{1.15} = \frac{4600}{1.15} = 4000 \text{ Birr} \] \[ \text{VAT} = 4600 – 4000 = 600 \text{ Birr} \]

Check: 15% of 4000 = 600. 4000 + 600 = 4600 ✓

Exam Note — VAT:
  • To ADD VAT: multiply by \( 1.15 \) (for 15% VAT)
  • To REMOVE VAT (find original price): divide by \( 1.15 \)
  • A common mistake is to find 15% of the INCLUDING-VAT price and subtract. This gives the WRONG answer! Always divide by 1.15.
Exam-Style Question 2

A restaurant bill is 1,150 Birr including 15% VAT. A service charge of 10% (on the pre-VAT price) is also included. What is the actual food cost before any charges?

Let the food cost = \( x \)

Service charge = 10% of \( x \) = \( 0.1x \)

Price before VAT = \( x + 0.1x = 1.1x \)

Price including VAT = \( 1.1x \times 1.15 = 1.265x \)

\[ 1.265x = 1150 \] \[ x = \frac{1150}{1.265} = 909.09 \text{ Birr} \]

The actual food cost is approximately 909.09 Birr.

6. Profit, Loss, and Discount

6.1 Profit and Loss

When a business buys and sells goods:

  • Profit = Selling Price − Cost Price (when SP > CP)
  • Loss = Cost Price − Selling Price (when CP > SP)
\[ \text{Profit \%} = \frac{\text{Profit}}{\text{CP}} \times 100\% \]
\[ \text{Loss \%} = \frac{\text{Loss}}{\text{CP}} \times 100\% \]
\[ \text{SP} = \text{CP} \times \left(1 + \frac{\text{Profit \%}}{100}\right) \quad \text{(when profit)} \]
\[ \text{SP} = \text{CP} \times \left(1 – \frac{\text{Loss \%}}{100}\right) \quad \text{(when loss)} \]

Worked Example 15: A trader buys goods for 5,000 Birr and sells them for 6,200 Birr. Find the profit and profit percentage.

Solution:

\[ \text{Profit} = 6200 – 5000 = 1200 \text{ Birr} \] \[ \text{Profit \%} = \frac{1200}{5000} \times 100\% = 24\% \]

Worked Example 16: A shopkeeper sells an article at a loss of 15%. If the selling price is 3,400 Birr, find the cost price.

Solution:

\[ \text{SP} = \text{CP} \times \left(1 – \frac{15}{100}\right) = \text{CP} \times 0.85 \] \[ 3400 = 0.85 \times \text{CP} \] \[ \text{CP} = \frac{3400}{0.85} = 4000 \text{ Birr} \]

6.2 Discount

A discount is a reduction in the marked price (list price) of an item.

\[ \text{Discount} = \text{Marked Price} – \text{Selling Price} \]
\[ \text{Discount \%} = \frac{\text{Discount}}{\text{Marked Price}} \times 100\% \]
\[ \text{SP} = \text{MP} \times \left(1 – \frac{\text{Discount \%}}{100}\right) \]

Worked Example 17: A shirt is marked at 800 Birr. A 12.5% discount is offered. Find: (a) the discount amount, (b) the selling price.

Solution:

\[ \text{Discount} = 800 \times \frac{12.5}{100} = 100 \text{ Birr} \] \[ \text{SP} = 800 – 100 = 700 \text{ Birr} \]

Worked Example 18: A shopkeeper marks an article at 25% above the cost price and then gives a discount of 10%. If the cost price is 2,000 Birr, find: (a) the marked price, (b) the selling price, (c) the actual profit percentage.

Solution:

(a) Marked Price = \( 2000 \times 1.25 = 2500 \) Birr

(b) Selling Price = \( 2500 \times (1 – 0.10) = 2500 \times 0.90 = 2250 \) Birr

(c) Profit = 2250 − 2000 = 250 Birr

\[ \text{Profit \%} = \frac{250}{2000} \times 100\% = 12.5\% \]

Interesting! The shopkeeper marked up by 25% but gave 10% discount, and ended up with only 12.5% profit. This shows that markup and profit are NOT the same thing!

Quick Formula — Successive Discount: If two discounts of \( d_1\% \) and \( d_2\% \) are given:
\[ \text{SP} = \text{MP} \times \left(1 – \frac{d_1}{100}\right) \times \left(1 – \frac{d_2}{100}\right) \]
\[ \text{Single Equivalent Discount} = d_1 + d_2 – \frac{d_1 \times d_2}{100} \]
Practice Question 5

An article is marked at 1,500 Birr. Two successive discounts of 20% and 10% are offered. Find: (a) the selling price, (b) the single equivalent discount.

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(a)

\[ \text{SP} = 1500 \times 0.80 \times 0.90 = 1500 \times 0.72 = 1080 \text{ Birr} \]

(b)

\[ \text{Equivalent Discount} = 20 + 10 – \frac{20 \times 10}{100} = 30 – 2 = 28\% \]

Check: \( 1500 \times 0.28 = 420 \) discount. \( 1500 – 420 = 1080 \) ✓

7. Summary of Key Exam Notes

  • Simple Interest: \( I = \frac{PRT}{100} \), always calculated on original P
  • Compound Interest: \( A = P(1 + R/100)^n \), always match R and n to the compounding period
  • CI > SI always (for positive rate and time > 1 year)
  • Hire Purchase: Extra paid = HP Price − Cash Price; Interest is on the balance after deposit
  • Income Tax: Use the tax bracket table; taxable income = gross − allowance
  • VAT: To add: multiply by 1.15; to remove: divide by 1.15 (for 15% VAT)
  • Profit/Loss %: Always calculated on Cost Price, never on Selling Price
  • Discount %: Always calculated on Marked Price, never on Cost Price
  • Successive discounts: Multiply the factors; equivalent discount = \( d_1 + d_2 – \frac{d_1 d_2}{100} \)
Exam-Style Question 3

A merchant buys goods for 50,000 Birr. He marks them up by 30% and offers a discount of 15% on the marked price. He sells the goods and deposits the entire selling price in a bank at 10% compound interest per year for 2 years. Find: (a) the marked price, (b) the selling price, (c) the profit percentage on cost, (d) the total amount in the bank after 2 years.

(a) Marked Price = \( 50000 \times 1.30 = 65000 \) Birr

(b) Selling Price = \( 65000 \times 0.85 = 55250 \) Birr

(c) Profit = 55250 − 50000 = 5250 Birr

\[ \text{Profit \%} = \frac{5250}{50000} \times 100\% = 10.5\% \]

(d) Amount after 2 years at 10% CI:

\[ A = 55250 \times (1.10)^2 = 55250 \times 1.21 = 66852.50 \text{ Birr} \]

Quick Revision Notes — Business Mathematics

1. All Key Formulas

SIMPLE INTEREST:

\[ I = \frac{P \times R \times T}{100}, \quad A = P + I \] \[ P = \frac{100I}{RT}, \quad R = \frac{100I}{PT}, \quad T = \frac{100I}{PR} \]

COMPOUND INTEREST:

\[ A = P\left(1 + \frac{R}{100}\right)^n, \quad CI = A – P \]

Semi-annually: use \( R/2 \) and \( 2n \)

Quarterly: use \( R/4 \) and \( 4n \)

CI − SI SHORTCUTS:

\[ \text{For 2 years: } CI – SI = P \times \frac{R^2}{100^2} \] \[ \text{For 3 years: } CI – SI = P \times \frac{R^2(300 + R)}{100^3} \]

HIRE PURCHASE:

\[ \text{HP Price} = \text{Deposit} + n \times \text{Installment} \] \[ \text{Extra Paid} = \text{HP Price} – \text{Cash Price} \] \[ \text{Flat Rate} = \frac{\text{Extra} \times 100}{\text{Balance} \times \text{Time (years)}} \]

INCOME TAX:

\[ \text{Taxable Income} = \text{Gross Income} – \text{Allowances} \] \[ \text{Net Income} = \text{Gross Income} – \text{Tax} \]

VAT (15%):

\[ \text{Price with VAT} = \text{Price} \times 1.15 \] \[ \text{Price without VAT} = \frac{\text{Price with VAT}}{1.15} \]

PROFIT & LOSS:

\[ \text{Profit} = SP – CP, \quad \text{Loss} = CP – SP \] \[ \text{Profit \%} = \frac{\text{Profit}}{CP} \times 100, \quad \text{Loss \%} = \frac{\text{Loss}}{CP} \times 100 \] \[ SP = CP(1 + p/100), \quad SP = CP(1 – l/100) \]

DISCOUNT:

\[ \text{Discount} = MP – SP, \quad \text{Discount \%} = \frac{\text{Discount}}{MP} \times 100 \] \[ SP = MP(1 – d/100) \] \[ \text{Successive: } SP = MP(1-d_1/100)(1-d_2/100) \] \[ \text{Equivalent: } d = d_1 + d_2 – \frac{d_1 d_2}{100} \]

2. Important Definitions

  • Principal: Original amount borrowed or invested
  • Amount: Principal + Interest
  • Simple Interest: Interest calculated only on the original principal
  • Compound Interest: Interest calculated on principal plus accumulated interest
  • Cash Price: Full price paid immediately
  • Hire Purchase Price: Total paid via deposit + installments
  • Taxable Income: Income after removing tax-free allowances
  • VAT: Tax added to the selling price of goods/services
  • Cost Price (CP): Price at which goods are bought
  • Selling Price (SP): Price at which goods are sold
  • Marked Price (MP): Listed/label price before any discount
  • Markup: The amount added to CP to get MP

3. Common Mistakes to Avoid

  1. Wrong units for time: If time is in months, convert to years by dividing by 12 before using in the simple interest formula.
  2. Not matching R and n in CI: For semi-annual compounding, halve R and double n. Don’t forget!
  3. Using SI formula for CI: These are different! SI uses \( \frac{PRT}{100} \); CI uses \( P(1+R/100)^n \).
  4. VAT removal error: To find price before VAT, DIVIDE by 1.15, don’t subtract 15%. The results are different!
  5. Profit % on SP instead of CP: Profit percentage is ALWAYS on cost price. A 20% profit on CP of 100 gives SP = 120. But 20% of 120 = 24, which is wrong.
  6. Discount % on CP: Discount is calculated on the marked price, NOT the cost price.
  7. Confusing markup with profit: Markup is on CP to get MP. Profit is on CP from actual SP (after discount).
  8. Hire Purchase deposit: Deposit is usually a percentage of the CASH price, not the HP price.
  9. Income Tax bracket error: The tax in each bracket is calculated ONLY on the amount EXCEEDING the lower limit, not the entire amount.
  10. Successive discount vs single discount: Two discounts of 20% and 10% give 28% equivalent, NOT 30%. Don’t just add them!

4. Quick Examples

Q: SI on 3000 at 5% for 4 years?

A: \( I = \frac{3000 \times 5 \times 4}{100} = 600 \) Birr

Q: CI on 5000 at 10% for 2 years?

A: \( A = 5000(1.1)^2 = 6050 \). CI = 1050 Birr

Q: Price before VAT if price with 15% VAT is 3450 Birr?

A: \( \frac{3450}{1.15} = 3000 \) Birr

Q: CP = 400, SP = 500. Profit %?

A: Profit = 100. \( \frac{100}{400} \times 100 = 25\% \)

Q: MP = 1000, discount 20%. SP?

A: \( 1000 \times 0.80 = 800 \) Birr

Challenge Exam Questions — Business Mathematics

These questions test your deep understanding. Try each one fully before checking the answer!

Section A: Multiple Choice Questions

Question 1
MCQ

The compound interest on 10,000 Birr at 20% per year for 2 years is:

A) 2,000 Birr    B) 4,000 Birr    C) 4,400 Birr    D) 2,400 Birr

Answer: C) 4,400 Birr

\[ A = 10000(1.20)^2 = 10000 \times 1.44 = 14400 \] \[ CI = 14400 – 10000 = 4400 \text{ Birr} \]
Question 2
MCQ

If an item is sold at 2,250 Birr including 15% VAT, the price before VAT is:

A) 1,912.50 Birr    B) 2,587.50 Birr    C) 1,956.52 Birr    D) 1,875 Birr

Answer: C) 1,956.52 Birr

\[ \text{Price before VAT} = \frac{2250}{1.15} = 1956.52 \text{ Birr} \]

Note: Option A is wrong — that would be 2250 − 15% of 2250 = 1912.50, which is the common mistake of subtracting VAT from the wrong base!

Question 3
MCQ

The single equivalent discount of two successive discounts of 25% and 20% is:

A) 45%    B) 40%    C) 5%    D) 42.5%

Answer: B) 40%

\[ \text{Equivalent} = 25 + 20 – \frac{25 \times 20}{100} = 45 – 5 = 40\% \]
Question 4
MCQ

A sum of money triples itself in 20 years at simple interest. The rate of interest is:

A) 10%    B) 15%    C) 20%    D) 5%

Answer: A) 10%

If it triples, \( A = 3P \), so \( I = 2P \).

\[ 2P = \frac{P \times R \times 20}{100} \] \[ R = \frac{200}{20} = 10\% \]
Question 5
MCQ

The difference between compound interest and simple interest on 8,000 Birr for 2 years at 5% per year is:

A) 20 Birr    B) 40 Birr    C) 10 Birr    D) 80 Birr

Answer: A) 20 Birr

\[ CI – SI = P \times \frac{R^2}{100^2} = 8000 \times \frac{25}{10000} = 8000 \times 0.0025 = 20 \text{ Birr} \]

Section B: Fill in the Blanks

Question 6
Fill in the Blank

If the cost price is 500 Birr and the selling price is 450 Birr, the loss percentage is ________

Answer: 10%

\[ \text{Loss} = 500 – 450 = 50, \quad \text{Loss \%} = \frac{50}{500} \times 100\% = 10\% \]
Question 7
Fill in the Blank

The compound interest on 1,000 Birr at 10% per year compounded semi-annually for 1 year is ________

Answer: 102.50 Birr

Semi-annually: \( R = 5\% \), \( n = 2 \)

\[ A = 1000(1.05)^2 = 1000 \times 1.1025 = 1102.50 \] \[ CI = 1102.50 – 1000 = 102.50 \text{ Birr} \]
Question 8
Fill in the Blank

In a hire purchase agreement, if the cash price is 20,000 Birr, the deposit is 4,000 Birr, and 12 installments of 1,500 Birr are paid, the extra amount paid is ________

Answer: 4,000 Birr

\[ \text{HP Price} = 4000 + 12 \times 1500 = 4000 + 18000 = 22000 \] \[ \text{Extra} = 22000 – 20000 = 4000 \text{ Birr} \]
Question 9
Fill in the Blank

A discount of 15% on a marked price of 2,000 Birr gives a selling price of ________

Answer: 1,700 Birr

\[ SP = 2000 \times (1 – 0.15) = 2000 \times 0.85 = 1700 \text{ Birr} \]
Question 10
Fill in the Blank
See also  Statistics: Detailed Notes & Exam Questions | Grade 12 Mathematics Unit 3

If a person earns 9,000 Birr gross monthly with a 600 Birr allowance, the taxable income is ________ and it falls in the ________ tax bracket.

Answer: 8,400 Birr; 7,801 – 10,900 (30%)

\[ \text{Taxable Income} = 9000 – 600 = 8400 \text{ Birr} \]

This falls in the bracket 7,801 – 10,900 which is taxed at 30%.

Section C: Short Answer Questions

Question 11
Short Answer

Explain the difference between simple interest and compound interest with a numerical example using 5,000 Birr at 10% for 3 years.

Simple Interest:

\[ I = \frac{5000 \times 10 \times 3}{100} = 1500 \text{ Birr}, \quad A = 6500 \text{ Birr} \]

Compound Interest:

\[ A = 5000(1.10)^3 = 5000 \times 1.331 = 6655 \text{ Birr} \] \[ CI = 6655 – 5000 = 1655 \text{ Birr} \]

Difference: CI − SI = 1655 − 1500 = 155 Birr

The key difference: In SI, interest is always calculated on the original 5000 Birr. In CI, each year’s interest is added to the principal before calculating the next year’s interest, so the interest “compounds” (grows on itself). CI is always greater than SI for positive rates and time > 1 year.

Question 12
Short Answer

Why is it wrong to remove 15% VAT from a price of 2,300 Birr by calculating \( 2300 – 0.15 \times 2300 \)? What is the correct method?

Wrong method: \( 2300 – 0.15 \times 2300 = 2300 – 345 = 1955 \) Birr

Correct method: \( \frac{2300}{1.15} = 2000 \) Birr

Why it’s wrong: The 15% VAT is calculated on the PRE-VAT price (2000), not on the final price (2300). When you do \( 0.15 \times 2300 \), you’re calculating 15% of 2300, which is larger than the actual VAT. The actual VAT is \( 0.15 \times 2000 = 300 \), not 345. Always divide by 1.15 to “undo” the VAT.

Question 13
Short Answer

A shopkeeper marks goods 40% above cost price and gives 25% discount. Does he make a profit or loss? Find the percentage.

Let CP = 100

MP = 100 × 1.40 = 140

SP = 140 × 0.75 = 105

Since SP (105) > CP (100), it’s a profit.

\[ \text{Profit \%} = \frac{5}{100} \times 100\% = 5\% \]

Section D: Step-by-Step Calculation Questions

Question 14
Calculation

A person borrows 25,000 Birr from a bank at 12% compound interest per year. He repays 10,000 Birr at the end of the first year and 15,000 Birr at the end of the second year. Does he fully repay the loan? If not, how much more does he owe?

End of Year 1:

\[ A_1 = 25000(1.12) = 28000 \text{ Birr} \]

He pays 10,000 Birr. Balance = \( 28000 – 10000 = 18000 \) Birr

End of Year 2:

\[ A_2 = 18000(1.12) = 20160 \text{ Birr} \]

He pays 15,000 Birr. Balance = \( 20160 – 15000 = 5160 \) Birr

No, he does not fully repay. He still owes 5,160 Birr.

Question 15
Calculation

Calculate the income tax for a person with monthly gross income of 8,500 Birr and a tax-free allowance of 600 Birr. Then find the net monthly income and the net annual income.

Taxable Income = 8500 − 600 = 7900 Birr

This falls in the bracket 7,801 – 10,900 at 30%:

\[ \text{Tax} = 1385 + 30\% \times (7900 – 7800) \] \[ = 1385 + 0.30 \times 100 \] \[ = 1385 + 30 = 1415 \text{ Birr per month} \]
\[ \text{Net Monthly Income} = 8500 – 1415 = 7085 \text{ Birr} \] \[ \text{Net Annual Income} = 7085 \times 12 = 85020 \text{ Birr} \]
Question 16
Calculation

A washing machine has a cash price of 22,000 Birr. Under hire purchase: deposit = 15% of cash price, 24 monthly installments of 950 Birr. Find: (a) the deposit, (b) the HP price, (c) the extra amount paid, (d) the flat rate of interest per year.

(a) Deposit = 15% of 22000 = \( 0.15 \times 22000 = 3300 \) Birr

(b) HP Price = 3300 + (24 × 950) = 3300 + 22800 = 26100 Birr

(c) Extra = 26100 − 22000 = 4100 Birr

(d) Balance after deposit = 22000 − 3300 = 18700 Birr. Time = 24 months = 2 years.

\[ \text{Flat Rate} = \frac{4100 \times 100}{18700 \times 2} = \frac{410000}{37400} \approx 10.96\% \]
Question 17
Calculation

A wholesaler buys an item for 800 Birr and sells it to a retailer at a profit of 20%. The retailer sells it to a customer at a profit of 25% on his cost. If the customer pays 15% VAT on top of the retail price, how much does the customer pay in total?

Wholesaler’s SP to retailer = \( 800 \times 1.20 = 960 \) Birr (retailer’s CP)

Retailer’s SP = \( 960 \times 1.25 = 1200 \) Birr

Customer pays (including 15% VAT):

\[ \text{Total} = 1200 \times 1.15 = 1380 \text{ Birr} \]

The customer pays 1,380 Birr.

Question 18
Calculation

Which is a better investment for 2 years: (A) 12% simple interest per year, or (B) 10% compound interest per year compounded annually? Support your answer with calculations using a principal of 50,000 Birr.

Option A: 12% SI for 2 years

\[ I = \frac{50000 \times 12 \times 2}{100} = 12000 \] \[ A = 50000 + 12000 = 62000 \text{ Birr} \]

Option B: 10% CI for 2 years

\[ A = 50000(1.10)^2 = 50000 \times 1.21 = 60500 \text{ Birr} \]

Comparison: 62000 > 60500, so Option A (12% simple interest) gives a higher return for 2 years.

However, note that for longer periods, compound interest at a lower rate will eventually overtake simple interest at a higher rate. The “crossover” depends on the specific rates and time.

Question 19
Calculation

A trader buys 200 items at 150 Birr each. He sells 60% of them at a profit of 25% and the remaining at a loss of 10%. Find his total profit or loss and the overall profit/loss percentage.

Total CP = \( 200 \times 150 = 30000 \) Birr

First batch (60% = 120 items):

\[ \text{CP}_1 = 120 \times 150 = 18000 \] \[ \text{SP}_1 = 18000 \times 1.25 = 22500 \]

Second batch (40% = 80 items):

\[ \text{CP}_2 = 80 \times 150 = 12000 \] \[ \text{SP}_2 = 12000 \times 0.90 = 10800 \]

Total:

\[ \text{Total SP} = 22500 + 10800 = 33300 \] \[ \text{Total CP} = 30000 \] \[ \text{Profit} = 33300 – 30000 = 3300 \text{ Birr} \] \[ \text{Profit \%} = \frac{3300}{30000} \times 100\% = 11\% \]
Question 20
Calculation

Find the compound interest on 20,000 Birr at 8% per year compounded quarterly for 1.5 years.

Quarterly: \( R = \frac{8}{4} = 2\% \) per quarter, \( n = 4 \times 1.5 = 6 \) quarters

\[ A = 20000\left(1 + \frac{2}{100}\right)^6 = 20000(1.02)^6 \] \[ = 20000 \times 1.126162 = 22523.24 \text{ Birr} \] \[ CI = 22523.24 – 20000 = 2523.24 \text{ Birr} \]
Question 21
Calculation

A man saves 3,000 Birr at the end of each year and deposits it at 8% compound interest per year. How much will he have after 4 years? (This is called an annuity problem.)

Each deposit earns interest for the remaining years:

  • 1st deposit (end of year 1): earns interest for 3 years → \( 3000(1.08)^3 = 3779.14 \)
  • 2nd deposit (end of year 2): earns interest for 2 years → \( 3000(1.08)^2 = 3499.20 \)
  • 3rd deposit (end of year 3): earns interest for 1 year → \( 3000(1.08) = 3240 \)
  • 4th deposit (end of year 4): earns interest for 0 years → \( 3000 \)
\[ \text{Total} = 3000(1.08)^3 + 3000(1.08)^2 + 3000(1.08) + 3000 \] \[ = 3000[(1.08)^3 + (1.08)^2 + 1.08 + 1] \] \[ = 3000[1.259712 + 1.1664 + 1.08 + 1] \] \[ = 3000 \times 4.506112 = 13518.34 \text{ Birr} \]

He will have approximately 13,518.34 Birr after 4 years.

Using the annuity formula: \( FV = P \times \frac{(1+r)^n – 1}{r} = 3000 \times \frac{(1.08)^4 – 1}{0.08} = 3000 \times \frac{1.360489 – 1}{0.08} = 3000 \times 4.506112 = 13518.34 \). Same answer!

Question 22
Calculation

A company offers two salary options to an employee:

Option A: Starting salary 6,000 Birr/month with annual increase of 10%

Option B: Starting salary 7,000 Birr/month with annual increase of 5%

Which option is better after 3 years? Show calculations for the monthly salary at the end of year 3 for each option.

Option A: This is like compound growth on salary.

\[ \text{Salary after 3 years} = 6000(1.10)^3 = 6000 \times 1.331 = 7986 \text{ Birr/month} \]

Option B:

\[ \text{Salary after 3 years} = 7000(1.05)^3 = 7000 \times 1.157625 = 8103.38 \text{ Birr/month} \]

After 3 years, Option B gives a higher monthly salary (8103.38 vs 7986 Birr).

Note: Even though Option A has a higher percentage increase, Option B starts at a significantly higher base. This shows that the starting point matters a lot in compound growth!

Question 23
Calculation

A person has two options to pay for a 36,000 Birr item:

Option 1: Pay cash and get a 5% cash discount

Option 2: Pay no deposit and 36 monthly installments of 1,100 Birr

Which option is cheaper and by how much?

Option 1 (Cash with discount):

\[ \text{Discount} = 36000 \times 0.05 = 1800 \] \[ \text{Price} = 36000 – 1800 = 34200 \text{ Birr} \]

Option 2 (Installments):

\[ \text{Total} = 36 \times 1100 = 39600 \text{ Birr} \]

Difference: \( 39600 – 34200 = 5400 \) Birr

Option 1 is cheaper by 5,400 Birr.

The installment option costs 5,400 Birr more — that’s effectively the interest cost of paying over time instead of upfront.

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