What you'll learn
Consumer arithmetic is the most directly useful topic in CXC CSEC Mathematics, and one of the most heavily weighted. It deals with the calculations people actually perform: working out wages, understanding a payslip, comparing a hire purchase agreement with paying cash, calculating interest on savings and loans, converting currency, and reading a utility bill. Because the contexts are drawn from Caribbean life, the questions are wordy, and the main skill after the arithmetic itself is identifying which calculation a situation calls for. By the end of this guide you should be able to calculate percentages in every direction including reverse percentages, work out wages and salaries with overtime, compute simple and compound interest, analyse hire purchase and mortgage arrangements, handle profit, loss, discount and taxes, and convert between currencies in both directions.
Key terms and definitions
Principal — the original sum of money invested or borrowed
Simple interest — interest calculated only on the original principal
Compound interest — interest calculated on the principal together with interest already earned
Amount — the total value of principal plus interest
Hire purchase — buying by paying a deposit followed by regular instalments, with the total exceeding the cash price
Deposit — the initial payment in a hire purchase or mortgage agreement
Gross pay — total earnings before any deductions
Net pay — earnings after deductions, often called take-home pay
Overtime — hours worked beyond the basic week, usually paid at a higher rate
Basic rate — the normal hourly rate of pay
Mark-up — the amount added to the cost price to give the selling price
Discount — a reduction in the marked price
Rate of exchange — the value of one currency expressed in terms of another
VAT — value added tax, a percentage added to the price of goods and services
Core concepts
Percentages in every direction
Almost every part of this topic reduces to a percentage calculation, so the three directions must be fluent.
To find a percentage of a quantity, convert the percentage to a decimal and multiply. To find 15 per cent of 480, calculate 0.15 × 480 = 72.
To increase or decrease by a percentage, use a multiplier. An increase of 12 per cent uses the multiplier 1.12; a decrease of 12 per cent uses 0.88. This is faster and less error-prone than finding the change and adding or subtracting separately.
To express one quantity as a percentage of another, divide and multiply by 100.
Reverse percentages are the hardest of the three and are frequently tested. If a price after a 20 per cent increase is 360 dollars, the original is not 360 less 20 per cent. The 360 represents 120 per cent of the original, so the original is 360 ÷ 1.2 = 300 dollars. Whenever a question gives the value after a change and asks for the value before, divide by the multiplier rather than reversing the operation.
Wages and salaries
A salary is a fixed annual amount, usually paid monthly, so the monthly figure is the annual salary divided by 12 and the weekly figure is the annual salary divided by 52.
A wage is paid by the hour. Basic pay is the basic rate multiplied by the basic hours.
Overtime is paid at a higher rate, described as time and a half, meaning 1.5 times the basic rate, or double time, meaning twice the basic rate. Overtime pay is the overtime hours multiplied by the basic rate multiplied by the overtime factor.
Gross pay is basic pay plus overtime plus any bonus or commission. Commission is usually a percentage of sales.
Deductions are then subtracted to give net pay. Typical deductions include income tax, National Insurance or social security contributions, pension contributions and union dues. Net pay equals gross pay minus total deductions.
Piece work pays a fixed amount per item produced, so the pay is the number of items multiplied by the rate per item.
Simple interest
Simple interest is calculated only on the original principal, so the interest earned is the same each year.
The interest equals the principal multiplied by the rate multiplied by the time, with the rate expressed as a percentage and the result divided by 100. Writing it as I = PRT ÷ 100 is the standard form.
The amount is the principal plus the interest.
The formula rearranges to find any of the four quantities, and questions asking for the rate or the time are common. To find the rate, multiply the interest by 100 and divide by the product of principal and time.
Time must be in years. A period of 8 months is 8 ÷ 12 years, and a period of 18 months is 1.5 years. Failing to convert is a frequent source of error.
Compound interest
Compound interest is calculated on the principal together with the interest already earned, so the interest increases each year.
The amount after n years equals the principal multiplied by the multiplier raised to the power n, where the multiplier is 1 plus the rate expressed as a decimal. For 5 per cent over 3 years, the amount is the principal multiplied by 1.05 cubed.
The compound interest itself is the amount minus the original principal. Candidates routinely give the amount when the question asked for the interest, so read the final line of the question carefully.
Compound interest always exceeds simple interest over the same period at the same rate, and the difference grows with time. A question may ask for that difference, which requires calculating both.
The same method handles depreciation, where the value falls each year. A car depreciating at 15 per cent per year has a multiplier of 0.85, and its value after 4 years is the original value multiplied by 0.85 to the power 4.
Hire purchase
Hire purchase allows a buyer to take goods immediately and pay over time, at a cost higher than the cash price.
The total hire purchase price equals the deposit plus the product of the number of instalments and the instalment amount.
The extra cost is the hire purchase price minus the cash price, and questions often ask for this extra amount as a percentage of the cash price.
The deposit is frequently given as a percentage of the cash price, so it must be calculated first before anything else in the question can be attempted.
Mortgages and loans
A mortgage is a long-term loan for property. The amount borrowed is the purchase price minus the deposit.
Questions typically ask for the deposit, the amount borrowed, the total repaid over the term, and the total interest paid. The total repaid is the monthly payment multiplied by the number of months, and the interest is that total minus the amount borrowed.
The number of months is the number of years multiplied by 12, which is a step candidates often omit.
Profit, loss, discount and taxes
Profit is selling price minus cost price; loss is cost price minus selling price.
Percentage profit or loss is always calculated on the cost price, not the selling price. This is the most commonly misapplied rule in the topic.
Discount is a reduction on the marked price, and the sale price is the marked price multiplied by one minus the discount as a decimal.
Value added tax is added to the price, so the final price is the pre-tax price multiplied by one plus the rate as a decimal. Where a question gives the price including tax and asks for the price before tax, divide by the multiplier — another reverse percentage.
Property tax and income tax questions often involve a tax-free allowance, which must be subtracted from the income before the tax rate is applied. Taxable income is gross income minus allowances.
Currency conversion
The rate of exchange states how much of one currency equals one unit of another.
To convert from the foreign currency to the local one, multiply by the rate. To convert back, divide by the same rate.
Deciding which operation to use causes more errors than the arithmetic. A reliable check is to ask whether the answer should be larger or smaller than the starting figure, given the relative value of the two currencies, and to confirm your answer matches.
Some questions include a commission charged by the bank, which is a percentage deducted from the amount before or after conversion, and the question will specify which.
Utility bills
An electricity or water bill typically has a fixed charge plus a charge per unit consumed.
The units consumed are the present meter reading minus the previous reading. The consumption charge is the units multiplied by the rate per unit, and the total bill is the fixed charge plus the consumption charge, with any tax added at the end.
Reading the meter figures in the correct order is essential, since reversing them gives a negative consumption.
Worked examples
Example 1: Wages with overtime and deductions (5 marks)
A worker is paid a basic rate of 18.50 dollars per hour for a 40-hour week, with overtime at time and a half. In one week she works 47 hours. Deductions total 15 per cent of gross pay. Calculate her net pay.
Basic pay is 40 × 18.50 = 740.00 dollars.
Overtime hours are 47 − 40 = 7. The overtime rate is 18.50 × 1.5 = 27.75 dollars per hour, so overtime pay is 7 × 27.75 = 194.25 dollars.
Gross pay is 740.00 + 194.25 = 934.25 dollars.
Deductions are 15 per cent of gross pay, which is 0.15 × 934.25 = 140.14 dollars to the nearest cent.
Net pay is 934.25 − 140.14 = 794.11 dollars.
Example 2: Comparing simple and compound interest (5 marks)
Find the difference between the simple interest and the compound interest on 8,000 dollars invested for 3 years at 6 per cent per annum.
For simple interest, I = PRT ÷ 100 = (8000 × 6 × 3) ÷ 100 = 1,440.00 dollars.
For compound interest, the multiplier is 1.06. The amount is 8000 × 1.06 cubed. Calculating 1.06 cubed gives 1.191016, so the amount is 8000 × 1.191016 = 9,528.13 dollars to the nearest cent.
The compound interest is 9,528.13 − 8,000 = 1,528.13 dollars.
The difference is 1,528.13 − 1,440.00 = 88.13 dollars. Note that the compound interest is the larger, as expected.
Example 3: Hire purchase against cash (5 marks)
A refrigerator has a cash price of 3,200 dollars. Under hire purchase a deposit of 25 per cent is required, followed by 18 monthly instalments of 165 dollars. Calculate the hire purchase price and express the extra cost as a percentage of the cash price.
The deposit is 25 per cent of 3,200, which is 0.25 × 3200 = 800 dollars.
The instalments total 18 × 165 = 2,970 dollars.
The hire purchase price is 800 + 2,970 = 3,770 dollars.
The extra cost is 3,770 − 3,200 = 570 dollars.
As a percentage of the cash price, this is (570 ÷ 3200) × 100 = 17.8 per cent to one decimal place.
Common mistakes and how to avoid them
The most frequent error is calculating percentage profit on the selling price. It is always calculated on the cost price.
Reverse percentage questions are routinely attempted by subtracting the percentage instead of dividing by the multiplier. If the figure given is after the change, divide.
In compound interest questions, many candidates give the amount when the interest was asked for. The interest is the amount minus the principal.
Students often forget to convert time into years in the simple interest formula, using 18 instead of 1.5 for eighteen months.
In hire purchase questions, a deposit expressed as a percentage is sometimes omitted or added twice. Calculate it first and note it clearly.
Finally, in currency conversion, multiplying instead of dividing is common. Check whether the answer should be larger or smaller before accepting it.
Exam technique for "Consumer Arithmetic"
Read the question twice and underline exactly what is being asked for — the interest or the amount, the gross pay or the net pay, the total price or the extra cost. Most lost marks in this topic come from answering a slightly different question.
Set the working out in labelled steps. These questions carry method marks generously, and an arithmetic slip costs far less when the method is visible.
Work in dollars and cents throughout and round only at the end, giving money answers to two decimal places unless told otherwise.
Use multipliers rather than calculating increases and decreases separately. It is faster and reduces the number of steps where an error can occur.
Check every answer for reasonableness. A hire purchase price below the cash price, a net pay above gross pay, or compound interest below simple interest each signal an error immediately.
Quick revision summary
Percentages underpin the whole topic: multiply by a decimal to find a percentage, use multipliers such as 1.12 and 0.88 for increases and decreases, and divide by the multiplier for reverse percentages where the figure given is after the change. Gross pay is basic pay plus overtime at time and a half or double time, plus any commission; net pay is gross pay minus deductions. Simple interest is principal times rate times time divided by 100, with time in years; compound interest uses the principal multiplied by the multiplier raised to the power of the number of years, and the interest is the amount minus the principal. Depreciation uses the same method with a multiplier below 1. Hire purchase price is deposit plus the total of the instalments, and the extra cost is usually expressed as a percentage of the cash price. Percentage profit and loss are always calculated on the cost price. Currency is converted by multiplying or dividing by the exchange rate, checked by asking whether the answer should be larger or smaller. Utility bills combine a fixed charge with units consumed, found by subtracting the previous meter reading from the present one.