Consumer Arithmetic

Welcome to this comprehensive guide on Consumer Arithmetic for CXC/CSEC Mathematics! This lesson covers all aspects of Consumer Arithmetic as required by the 2024-2025 CXC syllabus.

Introduction to Consumer Arithmetic

Consumer Arithmetic involves mathematical calculations related to money matters in everyday life. This branch of mathematics is crucial as it helps us make informed financial decisions.

Consumer Arithmetic is the branch of mathematics that deals with calculations involving financial transactions such as profits, losses, interest, taxes, discounts, and other consumer-related calculations.

Percentages in Consumer Arithmetic

Percentages form the foundation of many consumer arithmetic calculations.

Converting Between Fractions, Decimals, and Percentages

Example 1: Convert the following:

a) 0.35 to a percentage

0.35 × 100% = 35%

b) 3/4 to a percentage

3/4 × 100% = 75%

c) 62% to a decimal

62% ÷ 100 = 0.62

d) 45% to a fraction in its simplest form

45% = 45/100 = 9/20

Finding Percentages of Quantities

To find p% of a quantity Q: p% × Q

Example 2: Find 12% of $850.

12% of $850 = 0.12 × $850 = $102

Percentage Increase and Decrease

New amount after p% increase = Original amount × (1 + p/100)

New amount after p% decrease = Original amount × (1 - p/100)

Example 3: The price of a laptop increases by 15%. If the original price was $1200, what is the new price?

New price = $1200 × (1 + 15/100) = $1200 × 1.15 = $1380

Profit and Loss

Cost Price (CP) and Selling Price (SP)

Cost Price (CP) is the price at which a retailer purchases goods from a supplier.

Selling Price (SP) is the price at which the retailer sells the goods to customers.

Calculating Profit and Loss

Profit = SP - CP (when SP > CP)

Loss = CP - SP (when CP > SP)

Profit % = (Profit/CP) × 100%

Loss % = (Loss/CP) × 100%

Example 4: A merchant buys a television for $500 and sells it for $650.

a) Calculate the profit.

Profit = SP - CP = $650 - $500 = $150

b) Calculate the profit as a percentage of the cost price.

Profit % = (Profit/CP) × 100% = ($150/$500) × 100% = 30%

Example 5: A shopkeeper bought a chair for $200 and sold it for $170.

a) Calculate the loss.

Loss = CP - SP = $200 - $170 = $30

b) Calculate the loss as a percentage of the cost price.

Loss % = (Loss/CP) × 100% = ($30/$200) × 100% = 15%

Calculating SP Given CP and Profit/Loss Percentage

For p% profit: SP = CP × (1 + p/100)

For p% loss: SP = CP × (1 - p/100)

Example 6: A merchant buys a product for $80 and wants to make a profit of 25%. What should be the selling price?

SP = CP × (1 + profit%/100) = $80 × (1 + 25/100) = $80 × 1.25 = $100

Calculating CP Given SP and Profit/Loss Percentage

For p% profit: CP = SP ÷ (1 + p/100)

For p% loss: CP = SP ÷ (1 - p/100)

Example 7: A merchant sells a product for $150 with a profit of 20%. What was the cost price?

CP = SP ÷ (1 + profit%/100) = $150 ÷ (1 + 20/100) = $150 ÷ 1.2 = $125

Discount

Understanding Discount

Discount is a reduction in the marked price (MP) or listed price of an item.

Marked Price (MP) is the original price of an item before any discount is applied.

Sale Price (SP) is the final price after the discount has been applied.

Discount = Discount Rate × Marked Price

Sale Price (SP) = Marked Price (MP) - Discount

OR

Sale Price (SP) = MP × (1 - Discount Rate)

Example 8: A shirt is marked at $60 and is being sold at a discount of 25%. Calculate:

a) The discount amount

Discount = 25% × $60 = 0.25 × $60 = $15

b) The sale price

Sale Price = $60 - $15 = $45

OR

Sale Price = $60 × (1 - 0.25) = $60 × 0.75 = $45

Successive Discounts

Successive Discounts refer to multiple percentage discounts applied one after another.

For successive discounts of p% and q%:

Single equivalent discount rate = p + q - (p×q/100)%

Final price = MP × (1 - p/100) × (1 - q/100)

Example 9: A store offers a 20% discount on all items, followed by an additional 10% discount on the reduced price during a special sale. A jacket originally costs $100.

a) Calculate the final price after both discounts.

Final price = $100 × (1 - 20/100) × (1 - 10/100) = $100 × 0.8 × 0.9 = $72

b) Calculate the single equivalent discount.

Single equivalent discount = 20 + 10 - (20×10/100) = 30 - 2 = 28%

Check: $100 × (1 - 28/100) = $100 × 0.72 = $72 ✓

Tax

Value Added Tax (VAT)

Value Added Tax (VAT) is a consumption tax placed on products when value is added at a stage of production and at the point of retail sale.

To add VAT to a price: Final Price = Original Price × (1 + VAT Rate)

To find the original price before VAT: Original Price = Final Price ÷ (1 + VAT Rate)

Example 10: A refrigerator costs $1200 exclusive of VAT. If the VAT rate is 15%, calculate the price inclusive of VAT.

Price inclusive of VAT = $1200 × (1 + 15/100) = $1200 × 1.15 = $1380

Example 11: A television costs $575 including VAT at 15%. What is the price before VAT?

Price before VAT = $575 ÷ (1 + 15/100) = $575 ÷ 1.15 = $500

Other Taxes

Example 12: A car valued at $30,000 is imported into a country where there is an import duty of 25% and VAT of 15% (calculated on the value including import duty). Calculate the total cost including all taxes.

Import duty = 25% × $30,000 = $7,500

Value including import duty = $30,000 + $7,500 = $37,500

VAT = 15% × $37,500 = $5,625

Total cost = $37,500 + $5,625 = $43,125

Interest

Simple Interest

Simple Interest is calculated only on the initial principal amount, and does not compound.

Simple Interest (I) = Principal (P) × Rate (R) × Time (T)

Where R is expressed as a decimal and T is in years

Amount (A) = P + I = P(1 + R×T)

Example 13: Find the simple interest on $5,000 invested for 3 years at 6% per annum.

I = P × R × T = $5,000 × 0.06 × 3 = $900

Amount after 3 years = $5,000 + $900 = $5,900

Compound Interest

Compound Interest is calculated on the initial principal and also on the accumulated interest of previous periods.

Amount (A) = P(1 + R)^T

Where P is principal, R is rate per period (as a decimal), and T is number of periods

Compound Interest = A - P

Compound Interest vs Simple Interest Value Time Simple Interest Compound Interest

Example 14: Calculate the compound interest on $10,000 invested for 2 years at 8% per annum, compounded annually.

A = P(1 + R)^T = $10,000(1 + 0.08)^2 = $10,000 × 1.08^2 = $10,000 × 1.1664 = $11,664

Compound Interest = A - P = $11,664 - $10,000 = $1,664

Compound Interest with Different Compounding Periods

A = P(1 + R/n)^(n×T)

Where n is the number of times interest is compounded per year

Example 15: Calculate the amount when $5,000 is invested for 3 years at 6% per annum, compounded quarterly.

A = P(1 + R/n)^(n×T) = $5,000(1 + 0.06/4)^(4×3) = $5,000(1 + 0.015)^12 = $5,000 × 1.015^12 = $5,000 × 1.1956 = $5,978

Depreciation

Straight-Line Depreciation

Depreciation is the decrease in value of an asset over time.

Straight-Line Depreciation assumes that an asset loses value at a constant rate each year.

Annual Depreciation = (Initial Value - Salvage Value) ÷ Useful Life

Value after T years = Initial Value - (Annual Depreciation × T)

Example 16: A machine costs $20,000 and has a salvage value of $2,000 after 6 years. Calculate:

a) The annual depreciation using the straight-line method

Annual Depreciation = ($20,000 - $2,000) ÷ 6 = $18,000 ÷ 6 = $3,000 per year

b) The value of the machine after 4 years

Value after 4 years = $20,000 - ($3,000 × 4) = $20,000 - $12,000 = $8,000

Reducing Balance Depreciation

Reducing Balance Depreciation (also called declining balance method) assumes that an asset loses a fixed percentage of its current value each year.

Value after T years = Initial Value × (1 - R)^T

Where R is the annual rate of depreciation as a decimal

Example 17: A car worth $25,000 depreciates at 15% per year using the reducing balance method. What will be the value of the car after 3 years?

Value after 3 years = $25,000 × (1 - 0.15)^3 = $25,000 × 0.85^3 = $25,000 × 0.614125 = $15,353.13

Currency Exchange

Currency Conversion

Exchange Rate is the value of one currency for the purpose of conversion to another.

Amount in Foreign Currency = Amount in Local Currency ÷ Exchange Rate

Amount in Local Currency = Amount in Foreign Currency × Exchange Rate

Example 18: If the exchange rate is $1 US = $2.50 EC, calculate:

a) How many Eastern Caribbean dollars can be obtained for $200 US?

Amount in EC$ = Amount in US$ × Exchange Rate = $200 × $2.50 = $500 EC

b) How many US dollars are equivalent to $1250 EC?

Amount in US$ = Amount in EC$ ÷ Exchange Rate = $1250 ÷ $2.50 = $500 US

Hire Purchase and Installment Plans

Understanding Hire Purchase

Hire Purchase is a method of buying goods where the buyer pays a deposit followed by installments over time.

Cash Price is the price of the item if purchased outright with one payment.

Deposit is the initial down payment.

Installment is the regular payment made after the deposit.

Total Hire Purchase Price = Deposit + (Number of Installments × Installment Amount)

Hire Purchase Interest = Total Hire Purchase Price - Cash Price

Example 19: A washing machine has a cash price of $1,200. It can be purchased by paying a deposit of $300 followed by 12 monthly installments of $85 each.

a) Calculate the total hire purchase price.

Total HP Price = Deposit + (Number of Installments × Installment Amount)

Total HP Price = $300 + (12 × $85) = $300 + $1,020 = $1,320

b) Calculate the hire purchase interest.

HP Interest = Total HP Price - Cash Price = $1,320 - $1,200 = $120

c) Express the interest as a percentage of the cash price.

Interest as percentage = (HP Interest ÷ Cash Price) × 100% = ($120 ÷ $1,200) × 100% = 10%

Utility Bills and Everyday Finances

Understanding Utility Bills

Utility Common Units Pricing Structure
Electricity Kilowatt-hour (kWh) Fixed charge + Rate per kWh (often tiered)
Water Cubic meters or gallons Fixed charge + Rate per unit
Telephone Minutes, data (MB/GB) Monthly fee + Usage charges
Internet Data (MB/GB) Fixed monthly fee or tiered by usage

Example 20: An electricity bill has a fixed charge of $15 plus a rate of $0.20 per kWh for the first 100 kWh and $0.30 per kWh for any additional usage. Calculate the total bill for a household that used 250 kWh in a month.

Fixed charge: $15

First 100 kWh: 100 × $0.20 = $20

Remaining 150 kWh: 150 × $0.30 = $45

Total bill: $15 + $20 + $45 = $80

Rates and Property Taxes

Rates are taxes paid on property to local authorities, usually calculated based on the property's value.

Example 21: A property is valued at $250,000. If the property tax rate is 0.5% of the property value per year, what is the annual property tax payment?

Annual property tax = 0.5% × $250,000 = 0.005 × $250,000 = $1,250

Banking and Finance

Savings Accounts

Savings Account is a bank account that earns interest on the deposited money.

Example 22: A savings account pays 3% interest per annum, compounded monthly. If $5,000 is deposited in the account, how much will it be worth after 2 years?

A = P(1 + R/n)^(n×T) = $5,000(1 + 0.03/12)^(12×2) = $5,000(1 + 0.0025)^24 = $5,000 × 1.0025^24 = $5,000 × 1.0617 = $5,308.50

Loans and Mortgages

Loan is a sum of money borrowed that is expected to be paid back with interest.

Mortgage is a loan used to purchase property, with the property serving as collateral.

Example 23: A loan of $20,000 is to be repaid in equal monthly installments over 4 years at an interest rate of 9% per annum. Using the formula for monthly payment:

Monthly Payment = P × R × (1 + R)^n ÷ ((1 + R)^n - 1)

Where P is principal, R is monthly interest rate (annual rate ÷ 12), and n is number of payments

Monthly Payment = $20,000 × (0.09/12) × (1 + 0.09/12)^48 ÷ ((1 + 0.09/12)^48 - 1) = $20,000 × 0.0075 × 1.0075^48 ÷ (1.0075^48 - 1) ≈ $498.57

Investment

Simple and Compound Interest Investments

Investment is the allocation of money with the expectation of generating income or profit over time.

Example 24: Compare the returns on $10,000 invested for 5 years at 7% per annum using:

a) Simple interest

Interest = P × R × T = $10,000 × 0.07 × 5 = $3,500

Final amount = $10,000 + $3,500 = $13,500

b) Compound interest (annually)

A = P(1 + R)^T = $10,000(1 + 0.07)^5 = $10,000 × 1.07^5 = $10,000 × 1.4026 = $14,026

The compound interest investment yields $526 more than the simple interest investment.

Stocks and Shares

Stocks/Shares represent ownership in a company.

Dividend is a payment made by a company to its shareholders, usually as a distribution of profits.

Yield is the dividend expressed as a percentage of the share price.

Dividend Yield = (Annual Dividend per Share ÷ Share Price) × 100%

Return on Investment = ((Selling Price - Purchase Price + Dividends) ÷ Purchase Price) × 100%

Example 25: An investor buys 200 shares at $15 per share. The annual dividend is $0.60 per share.

a) Calculate the amount invested.

Amount invested = Number of shares × Price per share = 200 × $15 = $3,000

b) Calculate the annual dividend income.

Annual dividend income = Number of shares × Dividend per share = 200 × $0.60 = $120

c) Calculate the dividend yield.

Dividend yield = (Annual Dividend per Share ÷ Share Price) × 100% = ($0.60 ÷ $15) × 100% = 4%

Real-World Applications and Problem Solving

Budgeting

Budget is a financial plan for a defined period, often one year, that balances income with expenditure.

Example 26: A family has a monthly income of $4,500. They allocate their budget as follows: Housing (30%), Food (25%), Transportation (15%), Utilities (10%), Savings (10%), and Miscellaneous (10%). Calculate the amount allocated to each category.

Housing: 30% × $4,500 = $1,350

Food: 25% × $4,500 = $1,125

Transportation: 15% × $4,500 = $675

Utilities: 10% × $4,500 = $450

Savings: 10% × $4,500 = $450

Miscellaneous: 10% × $4,500 = $450

Monthly Budget Allocation Housing (30%) Food (25%) Transport (15%) Utilities (10%) Savings (10%) Misc (10%)

Comparing Financial Options

Example 27: A refrigerator can be purchased in three ways:

Option 1: Cash price of $800

Option 2: Hire purchase with $200 deposit and 12 monthly installments of $55 each

Option 3: Credit card payment with 18% APR, paying $75 per month

Which option is the most economical?

Option 1: $800

Option 2: Total cost = $200 + (12 × $55) = $200 + $660 = $860

Option 3: Assuming the full amount is charged to the credit card and minimum payments of $75 are made:

Approximate time to pay off: 12 months (with last payment less than $75)

Approximate total including interest: $872

Option 1 (cash price) is the most economical.

Self-Assessment Questions

Question 1

A dress is marked at $80 and is sold at a discount of 15%. Calculate:

a) The discount amount

b) The sale price

a) Discount amount = 15% × $80 = 0.15 × $80 = $12

b) Sale price = $80 - $12 = $68

Question 2

A merchant buys a television for $350 and sells it for $455. Calculate the profit percentage.

Profit = SP - CP = $455 - $350 = $105

Profit percentage = (Profit/CP) × 100% = ($105/$350) × 100% = 30%

Question 3

Find the simple interest on $2,500 invested for 4 years at 5% per annum.

I = P × R × T = $2,500 × 0.05 × 4 = $500

Question 4

A car worth $40,000 depreciates in value by 20% in the first year and 15% in the second year using the reducing balance method. Calculate the value of the car after 2 years.

Value after first year = $40,000 × (1 - 0.20) = $40,000 × 0.80 = $32,000

Value after second year = $32,000 × (1 - 0.15) = $32,000 × 0.85 = $27,200

Question 5

A store offers a 30% discount on all items, followed by an additional 10% discount on the reduced price during a special sale. Calculate the single equivalent discount percentage.

Single equivalent discount = 30 + 10 - (30×10/100) = 40 - 3 = 37%

Verification: Original price × (1 - 0.30) × (1 - 0.10) = Original price × 0.70 × 0.90 = Original price × 0.63

This represents a 37% discount: Original price × (1 - 0.37) = Original price × 0.63

Question 6

The price of a computer is $1,500 exclusive of VAT. If the VAT rate is 17.5%, calculate the price inclusive of VAT.

Price inclusive of VAT = $1,500 × (1 + 17.5/100) = $1,500 × 1.175 = $1,762.50

Question 7

Calculate the compound interest on $6,000 invested for 3 years at 4.5% per annum, compounded quarterly.

A = P(1 + R/n)^(n×T) = $6,000(1 + 0.045/4)^(4×3) = $6,000(1 + 0.01125)^12 = $6,000 × 1.01125^12

A = $6,000 × 1.1435 = $6,861

Compound interest = $6,861 - $6,000 = $861

Question 8

A washing machine has a cash price of $900. It can be purchased using a hire purchase plan with a deposit of $180 followed by 10 monthly installments of $78 each.

a) Calculate the total hire purchase price.

b) Calculate the hire purchase interest.

c) Express the interest as a percentage of the cash price.

a) Total HP price = $180 + (10 × $78) = $180 + $780 = $960

b) HP interest = Total HP price - Cash price = $960 - $900 = $60

c) Interest as percentage = ($60/$900) × 100% = 6.67%

Question 9

If the exchange rate is $1 US = J$150, calculate:

a) How many Jamaican dollars can be obtained for $350 US?

b) How many US dollars are equivalent to J$75,000?

a) Amount in J$ = Amount in US$ × Exchange rate = $350 × J$150 = J$52,500

b) Amount in US$ = Amount in J$ ÷ Exchange rate = J$75,000 ÷ J$150 = $500 US

Question 10

A sofa costs $1,200 including VAT at 20%. What is the price before VAT?

Price before VAT = Price including VAT ÷ (1 + VAT rate) = $1,200 ÷ 1.20 = $1,000

Glossary of Terms

Cost Price (CP)
The price at which a retailer buys goods from a supplier.
Selling Price (SP)
The price at which a retailer sells goods to customers.
Profit
The financial gain when SP > CP (Profit = SP - CP).
Loss
The financial loss when CP > SP (Loss = CP - SP).
Profit Percentage
Profit expressed as a percentage of the cost price = (Profit/CP) × 100%.
Loss Percentage
Loss expressed as a percentage of the cost price = (Loss/CP) × 100%.
Marked Price (MP)
The original price of an item before any discount is applied.
Discount
A reduction in the marked price of an item.
Discount Rate
The percentage by which the marked price is reduced.
Sale Price
The price after discount has been applied (MP - Discount).
Successive Discounts
Multiple percentage discounts applied one after another.
Value Added Tax (VAT)
A consumption tax placed on products when value is added.
Principal
The original amount of money invested or borrowed.
Interest
The fee paid for borrowing money or the return received on invested money.
Simple Interest
Interest calculated only on the initial principal.
Compound Interest
Interest calculated on both the initial principal and the accumulated interest of previous periods.
Depreciation
The decrease in value of an asset over time.
Straight-Line Depreciation
Asset loses value at a constant amount each year.
Reducing Balance Depreciation
Asset loses a fixed percentage of its current value each year.
Exchange Rate
The value of one currency for the purpose of conversion to another.
Hire Purchase
A method of buying goods by paying a deposit followed by installments.
Installment
Regular payment made after the initial deposit in a hire purchase agreement.
Dividend
A payment made by a company to its shareholders.
Yield
The dividend expressed as a percentage of the share price.

Summary

Consumer Arithmetic is a fundamental part of mathematics that deals with financial calculations in everyday life. Key areas covered in this lesson include:

Mastering these concepts will enable you to make informed financial decisions in your personal and professional life, as well as prepare you for success in your CXC/CSEC examinations.