Grade 8 Whole numbers

Number facts and properties

The number facts to know by heart, and the rules that let you rearrange a sum to make it easier.

commutative

You may swap the order of two numbers when you add or multiply. $3+5$ and $5+3$ both give $8$.

associative

You may choose which pair to work out first when you add or multiply.

distributive

Multiplying a bracket means multiplying every number inside it.

identity element

The number that leaves another number unchanged: $0$ for adding, $1$ for multiplying.

undefined

Has no answer at all. Dividing by $0$ is undefined.

Knowing your times tables saves time in every other topic. $7\times 8=56$ should come without counting. Each fact carries a division fact with it: $56÷8=7$ and $56÷7=8$.

Adding and multiplying obey two handy rules. The commutative property lets you swap the order of two numbers. The associative property lets you choose which pair to work out first. Subtracting and dividing follow neither.

The distributive property splits a bracket open. To work out $6\times 47$, split $47$ into $40+7$. Then $6\times 40=240$ and $6\times 7=42$. Every number inside gets multiplied, not just the first one.

Adding $0$ changes nothing. Multiplying by $1$ changes nothing. Each one is called an identity element, one for adding and one for multiplying. Dividing by $0$ is not like that. It is undefined, which means no answer exists.

Rules to remember

• $a+b=b+a$
• $a(b+c)=ab+ac$
• $a+0=a$ and $a\times 1=a$

Examples

Traps

• Calling every rearrangement commutative, because both properties seem to change the order. Commutative swaps two numbers. Associative changes which pair you work out first.
• Swapping the numbers in a subtraction, so $8-3$ is treated as the same as $3-8$. Only adding and multiplying may be swapped. $8-3=5$, and $3-8$ is not $5$.
• Multiplying only the first number in the bracket, so $4\times (12+9)$ becomes $48+9$. Multiply every number inside: $4\times 12+4\times 9=84$.
• Writing $0$, or the number itself, as the answer to a division by $0$. Dividing by $0$ is undefined. There is no number you can write down.

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Calculate and check

Working a sum out in the right order, setting it out neatly, and proving your answer is right.

order of operations

The agreed order for working a sum out: brackets, then multiply and divide, then add and subtract.

estimate

A quick, rough answer. You use it to see whether your real answer is sensible.

quotient

The answer to a division. In $84÷12=7$, the quotient is $7$.

remainder

What is left over when a division does not come out exactly.

inverse operation

The operation that undoes another one. Subtracting undoes adding.

The order of operations is the order everyone agrees to use. Without it, one sum would give many answers. Brackets come first. Then the multiplying and dividing, from left to right. Adding and subtracting come last. So $7+3\times 4=19$, not $40$.

In column work, line the digits up by place value. Units go under units and tens under tens. When a column comes to $10$ or more, carry into the next column. Column multiplication is the distributive property set out in rows.

Long division works from the left. Ask how many times the smaller number goes into each part. Write that digit on top, subtract, then bring the next digit down. If it does not go in, write $0$ on top. The quotient grows across the top.

Make an estimate before you calculate. Round the numbers to something easy and work with those instead. If your real answer is nowhere near the estimate, you have gone wrong somewhere.

Rounding off and compensating is a mental trick. To add $198$, add $200$ and then take the extra $2$ away. Check your work with the inverse operation. If $37+48=85$, then $85-48$ must bring you back to $37$.

Rules to remember

• Order of operations: brackets, then $\times$ and $÷$, then $+$ and $-$

Examples

Traps

• Working left to right and ignoring that multiplication and division come before addition and subtraction. Do multiplication and division first, then addition and subtraction. Brackets always go first of all.
• Rounding down whenever the digit you look at is a $5$. A digit of $5$ rounds up. To the nearest hundred, $1250$ becomes $1300$.
• Leaving a remainder that is bigger than the number you divided by. The remainder must always be smaller. If it is not, your quotient is one too small.
• Checking by doing the same sum the same way, which repeats the same mistake. Use the inverse operation instead. Undo a subtraction by adding your answer back on.

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Primes, LCM and HCF

Breaking a number into its building blocks, and using them to find what two numbers share.

factor

A whole number that divides into another with no remainder. $3$ is a factor of $12$.

multiple

What you get when you multiply a number by $1$, $2$, $3$ and so on.

prime number

A whole number with exactly two different factors: itself and $1$.

prime factorisation

Writing a number as a product of primes only, such as $12=2\times 2\times 3$.

LCM

The lowest common multiple: the smallest number that both numbers divide into.

HCF

The highest common factor: the biggest number that divides into both.

The factors of $12$ are $1$, $2$, $3$, $4$, $6$ and $12$. Each one divides in exactly. The multiples of $6$ run $6$, $12$, $18$, $24$ and on forever.

A prime number has exactly two factors, so $2$, $3$, $5$, $7$ and $11$ are prime. $1$ has only one factor, so it is not prime. Odd does not mean prime either: $51=3\times 17$.

Prime factorisation breaks a number down until only primes are left. Keep dividing by the smallest prime that fits. Stop when nothing can be split any further. So $120=2^3\times 3\times 5$.

The LCM is the smallest number that both numbers go into. Count multiples of the bigger number and test each one. The LCM of $6$ and $18$ is $18$, because $18$ is already a multiple of $6$.

The HCF is the biggest number that goes into both. List the factors of each and pick the largest one they share. The HCF of $18$ and $24$ is $6$.

Rules to remember

• Every whole number above $1$ is a product of primes, in exactly one way.

Examples

Traps

• Counting $1$ as a prime number. A prime needs two different factors. $1$ has only itself, so it is not prime.
• Assuming every odd number is prime, so $51$ and $91$ get called prime. Test the small primes first. $51=3\times 17$ and $91=7\times 13$.
• Multiplying the two given numbers together to find the LCM, even when they share a factor. Check for a shared factor first. The LCM of 6 and 18 is 18, not 108.
• Multiplying every prime factor when finding the HCF, not only the shared ones. Use just the primes both numbers have. For $48$ and $36$ that gives $12$.

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Ratio and rate

Sharing an amount fairly, and working out how much one of something costs or covers.

ratio

A comparison of two amounts of the same kind. $3:4$ means $3$ parts to $4$ parts.

rate

A comparison of two amounts of different kinds, such as rands for every litre.

unit rate

The amount for exactly one of something, like the price of one litre.

A ratio compares two amounts of the same kind, and the order matters. If sugar to flour is $2:5$, then $5:2$ says something else. To share in a ratio, add the numbers first. That total tells you how many parts the whole is cut into.

You can also increase or decrease an amount in a given ratio. To increase R$60$ in the ratio $5:3$, work out $\frac{5}{3}\times 60=100$, so R$100$. The bigger number goes on top when the amount must grow.

A rate compares two amounts of different kinds. Speed is a rate, in kilometres per hour. Price is a rate, in rands per kilogram. The word per means for every one.

Find the unit rate first, then scale it up. If $5$ litres of paraffin cost R$90$, then $90÷5=18$, so one litre is R$18$. Now $8$ litres cost $8\times 18=144$, so R$144$.

Rules to remember

• One part is the whole divided by the total number of parts.
• A unit rate is the total divided by the number of units.

Examples

Traps

• Dividing the total by one of the ratio numbers instead of by the total number of parts. Add the ratio numbers first to get the number of parts. Divide the total by that, then multiply.
• Writing the two shares the other way round to the order given in the question. In $4:5$, the amount named first gets $4$ parts.
• Multiplying by the rate when the question needs you to divide by it. Ask what one unit costs or covers. Work out the unit rate first, then scale.
• Using minutes with a speed given in kilometres per hour. Change the time into hours first. Sixty minutes make one hour.

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Financial maths

Money questions: what you make, what you save, what you owe, and what a thing really costs.

percentage

A number out of $100$. $15\%$ means $15$ out of every $100$.

profit

What is left over when you sell something for more than it cost you.

discount

An amount taken off the marked price, so you pay less.

VAT

Value Added Tax, added on to a price. In South Africa it is $15\%$.

simple interest

Money charged for a loan, worked out on the amount borrowed each year.

hire purchase

Paying a deposit now and the rest in monthly instalments.

Nearly every money question here is a percentage of an amount. Write the percentage over $100$ and multiply. So $15\%$ of R$240$ is $\frac{15}{100}\times 240=36$, which is R$36$.

Profit is the selling price less what you paid. A loss is the other way round. A discount comes off the marked price and VAT goes on to it. Work both out on the original price.

Simple interest is charged on the amount borrowed, every year. R$2000$ at $10\%$ gives $\frac{10}{100}\times 2000=200$ for one year. Over three years that is $3\times 200=600$, so R$600$.

Hire purchase means a deposit now and instalments later. Add the deposit to all the instalments to find the total paid. That total is always more than the cash price.

A budget lists money coming in and money going out. Sort every figure into income or expense before you add. An exchange rate is a rate too. It says how many rand you need for one dollar.

Rules to remember

• To find $p\%$ of an amount, multiply the amount by $\frac{p}{100}$.
• The hire purchase price is the deposit plus all the instalments.
• Interest for several years is one year's interest times the number of years.

Examples

Traps

• Giving the discount or VAT amount as the final answer instead of the new price. Read the question again. If it asks what someone pays, add or subtract the discount from the original price.
• Working the discount out on the price you get after the discount. The percentage is always taken of the original marked price.
• Working out one year of interest and stopping there. Multiply one year's interest by the number of years first.
• Multiplying by the exchange rate when the question needs you to divide. Check which way you are going. Rand into dollars divides, dollars into rand multiplies.

Learn and practise “Financial maths” in the app →