Grade 9 Mathematics Study Guide

Grade 9 is the last year of the Senior Phase. Before you move on to deeper algebra and geometry, this chapter makes sure your number foundation is rock solid, and then puts those numbers to work where it matters most: money. You will revise the properties of whole numbers, use prime factors to find the HCF and LCM, work with ratio, rate and proportion, and solve financial problems involving profit, discount, VAT, interest, hire purchase and exchange rates.

1.1 Properties of whole numbers

The properties below are the rules that allow you to rearrange a calculation into an easier form. In Grade 9 you must know them by name and recognise them in algebra as well.

• Commutative property: the order of the numbers does not change the answer for addition and multiplication. 8 + 5 = 5 + 8 and 4 × 7 = 7 × 4. Subtraction and division are not commutative: 10 - 4 ≠ 4 - 10.
• Associative property: the grouping of the numbers does not change the answer for addition and multiplication: (25 × 4) × 7 = 25 × (4 × 7) = 100 × 7 = 700.
• Distributive property: multiplication distributes over addition and subtraction: a(b + c) = ab + ac. For example 7 × 103 = 7 × (100 + 3) = 700 + 21 = 721.
• Identity elements: 0 is the identity element for addition (12 + 0 = 12) and 1 is the identity element for multiplication (12 × 1 = 12). Multiplying any number by 0 gives 0, and division by 0 is undefined.

1.2 Multiples, factors, HCF and LCM

A multiple of a number is the answer you get when you multiply it by a whole number. A factor divides exactly into a number, with no remainder. Every whole number greater than 1 can be written as a product of prime factors in exactly one way. Use the ladder method: divide repeatedly by the smallest prime number that works.

84 = 2 × 2 × 3 × 7 = 2² × 3 × 7   and   126 = 2 × 3 × 3 × 7 = 2 × 3² × 7

• HCF (highest common factor): multiply the prime factors the numbers share, each at its lowest power. HCF of 84 and 126 = 2 × 3 × 7 = 42.
• LCM (lowest common multiple): multiply all the prime factors that appear, each at its highest power. LCM of 84 and 126 = 2² × 3² × 7 = 252.

Quick check: HCF × LCM = 42 × 252 = 10 584, and 84 × 126 = 10 584. The product of two numbers is always equal to the product of their HCF and LCM.

1.3 Ratio, rate and proportion

A ratio compares quantities measured in the same unit; the ratio itself has no unit. Simplify a ratio exactly like a fraction: 24 : 36 = 2 : 3. A rate compares quantities with different units, such as km/h or rand per kilogram.

Direct proportion: when the one quantity doubles, the other doubles too. If 3 kg of apples cost R54, then 5 kg cost (54 ÷ 3) × 5 = R90.

Indirect (inverse) proportion: when the one quantity increases, the other decreases so that the product stays constant. If 6 workers can paint a hall in 10 days, the whole job is 6 × 10 = 60 worker-days. With only 4 workers it takes 60 ÷ 4 = 15 days.

1.4 Percentages and money: profit, loss, discount and VAT

A business buys stock at a cost price and sells it at a selling price. Profit = selling price - cost price. If the selling price is lower, the business makes a loss. Percentage profit or loss is always calculated on the cost price.

A discount is a reduction on the marked price. A 20% discount on R880 means you pay 80% of the price: R880 × 0,80 = R704.

VAT (value added tax) in South Africa is 15%. To add VAT, multiply by 1,15. To remove VAT, divide by 1,15.

A school bag costs R460 excluding VAT: R460 × 1,15 = R529 including VAT. A kettle costs R690 including VAT: R690 ÷ 1,15 = R600 excluding VAT.

1.5 Simple interest

Interest is the price of money: the bank pays you interest when you save, and you pay interest when you borrow. With simple interest the interest is calculated on the original amount (the principal) only, so it is the same amount every year.

Formula: \( A = P(1 + i \times n) \) where P is the principal, i is the yearly interest rate written as a decimal, n is the number of years and A is the final amount. The interest on its own is P × i × n.

1.6 Compound interest

With compound interest the interest you earn is added to the principal, and the next year you earn interest on the interest as well. This is how bank savings, investments and most loans really work, and it is new in Grade 9.

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

The chart shows R1 000 invested at 10% per year. The two methods give the same amount after year 1, but the compound bars pull ahead in years 2 and 3 because interest is earned on interest.

1.7 Hire purchase and exchange rates

With hire purchase (HP) you take the goods home now and pay them off in monthly instalments. HP always costs more than the cash price, because simple interest and fees are added. Compare the total HP price with the cash price before you decide.

An exchange rate tells you what one unit of a foreign currency costs in rand. If 1 US dollar = R18,50 (an illustrative rate, not the live rate), then a game that costs 120 dollars will cost 120 × 18,50 = R2 220. A weaker rand makes imported goods more expensive.

In summary

• The commutative, associative and distributive properties let you rearrange calculations; division by 0 is undefined.
• HCF = the shared prime factors at their lowest powers; LCM = all the prime factors at their highest powers.
• Ratios compare like units, rates compare different units, and in indirect proportion the product of the two quantities stays constant.
• Percentage profit is calculated on the cost price. VAT is 15%: multiply by 1,15 to add it, divide by 1,15 to remove it.
• Simple interest grows on the principal only: A = P(1 + i × n). Compound interest grows on the interest too: A = P(1 + i)ⁿ.
• The total hire purchase price = deposit + all the instalments; always compare it with the cash price.