Whole numbers (Grade 9) — Mathematics Grades 8–11

Number types, LCM and HCF, ratio and rate, and money maths: profit, interest and hire purchase.

Gr 9 Term 1 — Whole numbersTerm 1

Practise Whole numbers in the app →

What gets asked

Say whether a number is rational or irrational.
Find the LCM or HCF of two numbers using prime factors.
Solve a profit, loss, discount or VAT problem.
Calculate simple interest, a hire purchase instalment or compound interest.

You must be able to

Classify a number as natural, whole, an integer, rational or irrational.
Break numbers into prime factors to find their LCM or HCF.
Calculate profit, loss, discount and VAT on a sale.
Calculate simple interest, hire purchase instalments and compound interest.

Traps that cost marks

Swapping the LCM and HCF rules round. HCF uses the lowest power of each shared prime factor. LCM uses the highest power of every prime factor.
Working out profit percentage using the selling price. Profit percentage always uses the cost price as the base amount, never the selling price.
Using the simple interest method every year for compound interest. Compound interest is worked out on the new total each year, not on the original amount every time.

Worked example

Lindiwe borrows R1 500 from her stokvel at 10% simple interest a year, for 2 years. Calculate the total she must pay back.

Interest for one year is 10% of R1 500, which is R150.
Simple interest over 2 years is 2 × R150 = R300.
Total to pay back = R1 500 + R300 = R1 800.

The number system

Which family a number belongs to, and why one number can belong to several families at once.

natural number: A counting number: $1$ , $2$ , $3$ and so on. They never stop, and they never drop below $1$ .
whole number: A counting number, or $0$ .
integer: A whole number or its negative, such as $- 7$ , $0$ or $12$ .
rational number: Any number you can write as one integer divided by another, such as $\frac{3}{4}$ . The number underneath may not be $0$ .
irrational number: A number you cannot write that way. Its decimal runs on forever and never repeats.

The families sit inside one another. Every natural number is also a whole number. Every whole number is also an integer. Every integer is also a rational number. So one number carries several names at the same time.

To show a number is rational, write it as one integer over another. For $7$ , write $\frac{7}{1}$ . For $0, 25$ , write $\frac{25}{100}$ , which simplifies to $\frac{1}{4}$ . Every decimal that stops, and every decimal that repeats, can be written this way.

Irrational numbers are the ones left over. $2$ and $π$ are irrational. Their decimals run on forever without settling into a pattern. You can get close to them with a fraction, but never exactly.

$\frac{22}{7}$ is a rational number close to $π$ , and so is $3, 14$ . Neither one is $π$ itself. A shortcut you use in a calculation does not change what kind of number $π$ is.

Rules to remember

$7 = \frac{7}{1}$ , so every integer is rational
$0, 25 = \frac{1}{4}$ , so a decimal that stops is rational

Examples

Give every name that fits the number

- 6

Worked answer

Counting starts at $1$ , so $- 6$ is not a natural number.
Whole numbers are the counting numbers and $0$ , so $- 6$ is not one either.
Integers include the negatives, so $- 6$ is an integer.
It is rational too, because $- 6 = \frac{- 6}{1}$ .

Answer: integer, and rational number

The names build up. Once a number is an integer, it is automatically rational as well.

Show that

0, 45

is a rational number.

Worked answer

There are two digits after the comma, so use hundredths.
$0, 45 = \frac{45}{100}$ .
Divide top and bottom by $5$ : $\frac{45}{100} = \frac{9}{20}$ .
Both $9$ and $20$ are integers, so the number is rational.

Answer: $\frac{9}{20}$ , so it is rational

A decimal that stops always sits over a power of ten, and that is already a fraction.

A learner writes

π

3, 14

and says

π

is therefore rational. Is that right?

Worked answer

$3, 14$ can be written as $\frac{314}{100}$ , so $3, 14$ is rational.
But $3, 14$ is only close to $π$ ; the two are not the same number.
The decimal for $π$ runs on forever and never repeats.
So $π$ is an irrational number, and the learner is wrong.

Answer: No — $π$ is irrational

The rounded value you type into a calculator is a different number from the one it stands for.

Traps

Calling every number with a comma in it irrational. $0, 25$ is $\frac{1}{4}$ , so it is rational. A decimal that stops or repeats is always rational.
Treating $\frac{22}{7}$ as the exact value of $π$ . $\frac{22}{7}$ is a rational number that lies close to $π$ . No fraction is exactly $π$ .
Saying $5$ is not rational because it is not written as a fraction. Put it over $1$ : $5 = \frac{5}{1}$ . Every integer is a rational number.
Thinking each number gets one name only. The families are nested. $3$ is natural, whole, an integer and rational, all at once.

Learn and practise “The number system” in the app →

Calculate and estimate

The order you must work in, the two written methods, and the quick check that catches a slip.

order of operations: The order you must work in: brackets, then powers, then multiplying and dividing, then adding and subtracting.
place value: What a digit is worth because of where it stands. In 2 406 the 4 stands for four hundred.
quotient: The answer to a division. The quotient of $96$ and $4$ is $24$ .
estimate: A quick, rough answer worked out with easier numbers, used to check the real one.

Brackets first. Then powers. Then multiplying and dividing, working from left to right. Then adding and subtracting, again from left to right. Nothing about the order of operations changes from one sum to the next.

In columns, line the digits up by place value: units under units, tens under tens. When you multiply by the tens digit, write a $0$ in the units place first. That $0$ is what says you are multiplying by $30$ and not by $3$ .

Long division works one digit at a time. If the divisor will not go into the digit you are on, write a $0$ in the quotient and bring the next digit down. Leaving that $0$ out makes the answer ten times too small.

An estimate is your safety net. Swap each number for a nearby easy one, then do the easy sum in your head. Round both by about the same amount. Rounding one to the nearest ten and the other to the nearest thousand throws the estimate right out.

Rules to remember

$2 + 3 \times 4 = 14$ , not $20$
$(2 + 3) \times 4 = 20$

Examples

Calculate:

40 - 6 \times 5 + 2^{3}

Worked answer

Powers come first: $2^{3} = 8$ .
Then multiply: $6 \times 5 = 30$ .
Now work from left to right: $40 - 30 = 10$ .
Then add: $10 + 8 = 18$ .

Answer: $18$

Working straight across from the left would turn $40 - 6 \times 5$ into $170$ , before any adding.

Multiply in columns:

246 \times 32

Worked answer

Start with the $2$ units: $246 \times 2 = 492$ .
Now the $3$ tens. Write the place-holder $0$ down first.
$246 \times 30 = 7380$ .
Add the two lines: $492 + 7380 = 7872$ .

Answer: $7872$

The second line is ten times the first, and only the place-holder $0$ makes that true.

A school orders

7224

exercise books and packs them in boxes of

24

. Estimate the number of boxes, then work it out.

Worked answer

Estimate with easier numbers: $7200 \div 24 = 300$ .
$24$ into $72$ goes $3$ times, with nothing left over.
$24$ will not go into $2$ , so write $0$ and bring the $4$ down.
$24$ into $24$ goes once, so the quotient is $301$ .
That sits right next to the estimate, so it looks right.

Answer: $301$ boxes

The middle digit is the $0$ most learners drop, and the estimate is what would have caught it.

Traps

Working straight from the left, so $40 - 6 \times 5$ becomes $170$ . Multiplying is done first: $6 \times 5 = 30$ , and then $40 - 30 = 10$ .
Leaving out the place-holder $0$ when multiplying by the tens digit. The $3$ in $32$ stands for $30$ . Write the $0$ first, so the line reads $7380$ .
Skipping a step in long division when the divisor will not go in. Write a $0$ in the quotient and carry on. Every step must leave a digit behind.
Rounding one number to the nearest ten and the other to the nearest thousand. Round both by about the same amount, or the estimate is too far off to check anything.

Learn and practise “Calculate and estimate” in the app →

Factors and multiples

Break a number down into primes once, and both the HCF and the LCM fall straight out of it.

prime number: A number with exactly two different factors, $1$ and itself. $1$ is not prime, because it has only one factor.
prime factorisation: A number written as a product of prime numbers only, such as $12 = 2^{2} \times 3$ .
highest common factor: The biggest number that divides into all of them with nothing left over. It is written HCF.
lowest common multiple: The smallest number that all of them divide into. It is written LCM.

Every whole number bigger than $1$ is either prime, or a product of primes. Divide by $2$ as often as it goes, then by $3$ , then by $5$ , and carry on up. Stop only when every factor you are left with is prime.

$4$ and $9$ are not prime, so a prime factorisation that still shows them is not finished. Write $4$ as $2^{2}$ and $9$ as $3^{2}$ . And $1$ never belongs in the list.

Once both numbers are written as products of primes, both answers fall out. For the highest common factor, keep only the primes that appear in both lists, each at its lowest power, and multiply.

For the lowest common multiple, keep every prime that appears in either list, each at its highest power, and multiply. That is the one you need when two things repeat on different cycles.

Rules to remember

$12 = 2^{2} \times 3$ and $18 = 2 \times 3^{2}$
HCF: the lowest power of each shared prime, $2 \times 3 = 6$
LCM: the highest power of every prime, $2^{2} \times 3^{2} = 36$

Examples

Write

84

as a product of its prime factors.

Worked answer

$84 \div 2 = 42$ , and $42 \div 2 = 21$ .
$21 \div 3 = 7$ , and $7$ is prime, so stop.
$84 = 2 \times 2 \times 3 \times 7$ .
Gather the repeats: $84 = 2^{2} \times 3 \times 7$ .

Answer: $2^{2} \times 3 \times 7$

Every factor left at the end is prime, which is the only sign that you are finished.

Find the HCF and the LCM of

24

and

36

Worked answer

$24 = 2^{3} \times 3$ and $36 = 2^{2} \times 3^{2}$ .
Both lists contain a $2$ and a $3$ .
HCF: lowest power of each shared prime, $2^{2} \times 3 = 12$ .
LCM: highest power of every prime, $2^{3} \times 3^{2} = 72$ .

Answer: HCF $12$ , LCM $72$

One prime factorisation of each number answered both halves of the question.

Two taxis leave the rank together. One comes back every

8

minutes, the other every

12

minutes. When do they next leave together?

Worked answer

They leave together on a common multiple of $8$ and $12$ .
$8 = 2^{3}$ and $12 = 2^{2} \times 3$ .
Highest power of each prime: $2^{3} \times 3 = 24$ .
So they leave together again after $24$ minutes.

Answer: $24$ minutes

Two cycles meeting again is exactly what the lowest common multiple measures.

Traps

Swapping the LCM and HCF rules round. HCF uses the lowest power of each shared prime factor. LCM uses the highest power of every prime factor.
Stopping while a factor like $4$ or $9$ is still in the list. Neither is prime. Break them up: $4 = 2^{2}$ and $9 = 3^{2}$ .
Putting $1$ into the prime factorisation. $1$ has only one factor, so it is not a prime number. It also changes no product.
Assuming the LCM is always the two numbers multiplied together. That holds only when they share no prime factor. For $12$ and $18$ the LCM is $36$ , not $216$ .

Learn and practise “Factors and multiples” in the app →

Ratio, rate and proportion

Sharing in parts, comparing two different things, and knowing when more means less.

ratio: A comparison of amounts of the same kind, written like $3 : 5$ . It has no units.
rate: A comparison of two amounts of different kinds, such as R18 for a litre of milk, or 60 km in an hour.
direct proportion: When one amount doubles, the other doubles too. Twice as many loaves cost twice as much.
indirect proportion: When one amount doubles, the other halves. Twice as many workers take half the time.
commission: Money a seller or a bank keeps, worked out as a percentage of the amount handled.

A ratio compares amounts of the same kind. To share in a ratio, add the parts first. Divide the total by that number of parts to find one part, then multiply for each share.

A rate compares different kinds of amount, so it carries units. Speed is a rate: distance divided by time. An exchange rate is a rate too, comparing rands with another country's money.

Units must agree before you divide. One hour and $30$ minutes is $1, 5$ hours, because $30$ minutes is half an hour. It is never $1, 30$ hours.

In direct proportion the two amounts grow together, and the graph is a straight line. In indirect proportion one grows while the other shrinks, and the graph is a curve.

For an indirect proportion, find the fixed total first. Six workers for $10$ days is $60$ days of work altogether, and that total does not change when the number of workers does.

Rules to remember

speed $=$ distance $\div$ time
one hour and $30$ minutes is $1, 5$ hours, not $1, 30$ hours

Examples

A stokvel shares R2 400 between Thandi and Bongi in the ratio

3 : 5

. How much does each get?

Worked answer

Add the parts: $3 + 5 = 8$ .
One part is $2400 \div 8 = 300$ , so R300.
Thandi gets $3$ parts: $3 \times 300 = 900$ .
Bongi gets $5$ parts: $5 \times 300 = 1500$ .

Answer: Thandi R900, Bongi R1 500

The two shares add back up to R2 400, and that is the check worth doing every time.

Six bricklayers build a school boundary wall in

10

days. How long would

4

bricklayers take, working at the same rate?

Worked answer

Fewer workers means more days, so this is indirect proportion.
Find the fixed total: $6 \times 10 = 60$ days of work.
Share that among the $4$ workers: $60 \div 4 = 15$ .
So $4$ bricklayers take $15$ days.

Answer: $15$ days

Direct proportion would have made $4$ workers faster than $6$ , which cannot happen.

Thabo's cousin sends him

200

US dollars. The bank pays R19 for one dollar and charges

2%

commission. How many rands does Thabo get?

Worked answer

Dollars into rands, so multiply by the rate.
$200 \times 19 = 3800$ , which is R3 800.
The commission is $\frac{2}{100} \times 3800 = 76$ , so R76.
Take it off: $3800 - 76 = 3724$ .

Answer: R3 724

Two rates act one after the other, and the commission is worked out on the rand amount.

Traps

Writing one hour and $12$ minutes as $1, 12$ hours. $12$ minutes is $\frac{12}{60}$ of an hour, so the time is $1, 2$ hours.
Dividing the time by the distance to find a speed. Speed is distance divided by time. For 90 km in $1, 5$ hours, $90 \div 1, 5 = 60$ , so 60 km/h.
Treating every proportion question as a direct one. Ask what happens as the first amount grows. More workers means fewer days, so it is indirect.
Multiplying by the exchange rate when the question needs a division. Dollars into rands, multiply. Rands into dollars, divide: $3800 \div 19 = 200$ .

Learn and practise “Ratio, rate and proportion” in the app →

Financial maths

Profit, tax, budgets and interest all come down to one question: a percentage of what?

cost price: What the seller paid for the item.
selling price: What the buyer pays for it.
VAT: Value Added Tax. In South Africa $15%$ is added to the price of most goods.
principal: The amount of money borrowed or saved at the start.
simple interest: Interest worked out on the principal only, so the same amount is added each year.
compound interest: Interest worked out on the new total each year, so it grows faster.

Profit is the selling price minus the cost price. A loss is the other way round. Profit percentage is always worked out on the cost price, because that is the money that went in.

VAT is added on top of the price before tax. A $15%$ discount followed by a $15%$ increase does not bring you back to the start, because the two are taken of different amounts.

A budget has two sides: money coming in, and money going out. Keep them in separate columns and subtract at the end.

Simple interest is the same every year. The same formula finds the rate, the number of years or the principal, once you know the other three. It gives you the interest, never the total.

Compound interest is worked out afresh each year on the new total. Year one's interest joins the principal, and year two's interest is worked out on that bigger amount.

Rules to remember

interest $= \frac{P \times r \times n}{100}$
total $=$ principal $+$ interest

Examples

A spaza owner buys

12

loaves for R180 and sells them all for R216. Find the profit percentage.

Worked answer

Profit is $216 - 180 = 36$ , so R36.
The base is the cost price, R180.
$\frac{36}{180} \times 100 = 20$ .

Answer: $20%$

Using R216 as the base would have given about $17%$ , which is not the profit percentage.

Lindiwe borrows R1 500 from her stokvel and pays back R1 800 after

2

years. Find the simple interest rate per year.

Worked answer

The interest is $1800 - 1500 = 300$ , so R300.
Put the numbers in: $300 = \frac{1 500 \times r \times 2}{100}$ .
The right-hand side works out to $30 r$ , so $30 r = 300$ .
Divide by $30$ : $r = 10$ .

Answer: $10%$ a year

The R1 800 is the total, so it had to become interest before it went into the formula.

Sipho saves R2 000 at

10%

compound interest a year. How much is in the account after

2

years?

Worked answer

Year one interest: $\frac{10}{100} \times 2000 = 200$ , so R200.
The account now holds R2 200.
Year two interest is worked out on R2 200, not R2 000.
$\frac{10}{100} \times 2200 = 220$ .
New total: $2200 + 220 = 2420$ .

Answer: R2 420

Simple interest would have added R200 twice and lost the extra R20.

Traps

Working out profit percentage using the selling price. Profit percentage always uses the cost price as the base amount, never the selling price.
Using the simple interest method every year for compound interest. Compound interest is worked out on the new total each year, not on the original amount every time.
Giving the interest when the total to pay back was asked for. Add the interest onto the principal. R300 of interest on R1 500 means R1 800 to pay back.
Putting a wage into a budget alongside the shopping and the rent. A wage is money coming in. It belongs on the income side, not with the expenses.

Learn and practise “Financial maths” in the app →

Grade 9 Whole numbers

What gets asked

You must be able to

Traps that cost marks

Worked example

The number system

Rules to remember

Examples

Traps

Calculate and estimate

Rules to remember

Examples

Traps

Factors and multiples

Rules to remember

Examples

Traps

Ratio, rate and proportion

Rules to remember

Examples

Traps

Financial maths

Rules to remember

Examples

Traps

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