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1.0 ARITHMETIC OPERATIONS
1.1
BASIC OPERATIONS
Recall:
The four basic mathematical operations are:
ADDITION
Adding two (or more) numbers means to find their sum (or total). The symbol used for
addition is '+'.
For example, 5 + 10 = 15
This is read as five plus ten is equal to fifteen or simply, five plus ten is fifteen.
Example 1
Find the sum of 9 and 8.
Solution:
9 + 8 = 17
ADDITION OF LARGE NUMBERS
To add large numbers, list them in columns and then add only those digits that have the
same place value.
Example 2
Find the sum of 5897, 78, 726 and 8569.
Solution:
Note:

Write the numbers in columns with the thousands, hundreds, tens and units lined
up.

7 + 8 + 6 + 9 = 30. Thus, the sum of the digits in the units column is 30. So, we
place 0 in the units place and carry 3 to the tens place.

The sum of the digits in the tens column after adding 3 is 27. So, we place 7 in
the tens place and carry 2 to the hundreds place.

The sum of the digits in the hundreds column after adding 2 is 22. So, we place 2
in the hundreds place and carry 2 to the thousands place.
SUBTRACTION
Subtracting one number from another number is to find the difference between them.
The symbol used for subtraction is '–'. This is known as the minus sign.
For example, 17 – 8 = 9
This is read as seventeen take away eight is equal to nine (or seventeen take away eight is
nine). Also, we can say that 17 minus 8 is 9.
Example 3
Subtract 9 from 16.
Solution:
16 – 9 = 7
SUBTRACTION OF LARGE NUMBERS
To subtract large numbers, list them in columns and then subtract only those digits that
have the same place value.
Example 4
Find the difference between 7064 and 489.
Solution:
Note:

Use the equals addition method or the decomposition method.

Line up the thousands, hundreds, tens and units place values for the two numbers
when placing the smaller number below the larger number as shown above.
MULTIPLICATION
Multiplication means times (or repeated addition). The symbol used for multiplication
is '×'.
For example, 7 × 2 = 14
This is read as seven times two is equal to fourteen or simply, seven times two is
fourteen.
To multiply a large number with another number, we write the numbers vertically and
generally multiply the larger number with the smaller number.
Note:
A product is the result of the multiplication of two (or more) numbers.
Example 5
Calculate 765 × 9.
Solution:
Write the smaller number, 9, under the larger number, 765, and then calculate the
multiplication.
Note:

9 × 5 = 45. So, place 5 units in the units column and carry the 4 (i.e. four tens) to
the tens column.

Calculate 9 × 6 and then add 4 to give 58 (i.e. 58 tens). Then place 8 in the tens
column and carry 5 to the hundreds column.

Finally multiply 7 by 9 and add 5 to give 68 (i.e. 68 hundreds). Write this number
down as shown above.
Remember:

To multiply two large numbers, write the numbers vertically with the larger
number generally being multiplied by the smaller number which is called the
multiplier.

We use the 'times table' to find the product of the larger number with each digit in
the multiplier, adding the results.

Remember to add a zero for every place value after the multiplying digit. For
example, if the multiplying digit is in the hundreds column, add two zeros for the
tens column and for the units column.
Example 6
Calculate 38 × 70.
Solution:
Note:

Multiplying 38 by 70 is quicker than multiplying 70 by 38 as 70 contains a zero.

A zero is placed in the units column. Then we calculate 7 × 38 as shown above.
Example 7
Calculate 385 × 500.
Solution:
Note:

Multiplying 385 by 500 is quicker than multiplying 500 by 385 as 500 contains
two zeros.

A zero is placed in the units column and also the tens column. Then we calculate
5 × 385 as shown above.
Example 8
Calculate 169 × 68.
Solution:
Note:

To multiply 169 by 68, place 68 below 169.

Then we calculate 8 × 169 and 60 × 169 as shown above.
DIVISION
Division 'undoes' multiplication and involves a number called the dividend being
'divided' by another number called the divisor. The symbol used for division is '÷'.
Example 9
Solution:
Example 10
Solution:
Note:

As division is the inverse of multiplication, start by dividing 4 into the column
furthest to the left.

6 ÷ 4 = 1 and 2 is the remainder.

Clearly, the remainder 2 is 200 (i.e. 20 tens); and we can carry this into the tens
column to make 29.

Now, 29 ÷ 4 = 7 with a remainder of 1. Clearly, the remainder of 1 is 10 (i.e. 10
units) and we carry this into the units column to make 12.

Finally, 12 ÷ 4 = 3.
Example 11
Solution:
Summary

The four basic mathematical operations are:

Adding two (or more) numbers means to find their sum (or total).

Subtracting one number from another number is to find the difference between
them.

Multiplication means times (or repeated addition). A product is the result of the
multiplication of two (or more) numbers.
Division 'undoes' multiplication
ORDER OF OPERATIONS
To evaluate an expression containing more than one operation, we follow this order of
operations as we work through the expression from left to right:
Note that 'of ' means multiplication.
Remember:
BODMAS
Note:
After removing any brackets in an expression, work through the expression from left to
right applying whichever operation comes first out of:

'of ', 'division' and 'multiplication'

'addition' and 'subtraction'
Alternatively, you may do all:

'of ', 'division' and 'multiplication' together

'addition' and 'subtraction' together
Example 12
Solution:
MULTIPLES
The multiples of a number are its products with the natural numbers 1, 2, 3, 4, 5, ....
So, the multiples of 8 are 8, 16, 24, 32, and so on.
Note:
The multiples of a number are obtained by multiplying the number by each of the natural
numbers.
Example 13
Write down the first five multiples of 9.
Solution:
The multiples of 9 are obtained by multiplying 9 with the natural numbers 1, 2, 3, 4, 5 …
So, the first five multiples of 9 are 9, 18, 27, 36 and 45.
COMMON MULTIPLES
Common multiples are multiples that are common to two or more numbers.
E.g. Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, …
Multiples of 3 are 3, 6, 9, 12, 15, 18, …
So, common multiples of 2 and 3 are 6, 12, 18, …
Example 14
Find the common multiples of 3 and 4.
Solution:
Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, …
Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, …
So, the common multiples of 3 and 4 are 12, 24, 36, …
LOWEST COMMON MULTIPLE
The lowest common multiple (LCM) of two or more numbers is the smallest common
multiple.
E.g. Multiples of 8 are 8, 16, 24, 32, …
Multiples of 6 are 6, 12, 18, 24, …
In general:
To find the lowest common multiple (LCM) of two or more numbers, list the multiples of
the larger number and stop when you find a multiple of the other number. This is the
LCM.
Example 15
Find the lowest common multiple of 2 and 5.
Solution:
List the multiples of 5 and stop when you find a multiple of 2.
Multiples of 5 are 5, 10, …
Multiples of 2 are 2, 4, 6, 8, 10, …
FACTORS
A factor of a given number is a whole number that divides exactly into the given
number.
So, 4 is a factor of 12 as it divides exactly into 12, and 3 is also a factor of 12.
Note:
If a number can be expressed as a product of two whole numbers, then the whole
numbers are called factors of that number.
So, the factors of 12 are 1, 2, 3, 4, 6 and 12.
COMMON FACTORS
Common factors are factors that are common to two or more numbers.
Example 16
Find the common factors of 10 and 20.
Solution:
So, the common factors of 10 and 20 are 1, 2, 5 and 10.
Example 17
Find the common factors of 22 and 33.
Solution:
So, the common factors of 22 and 33 are 1 and 11.
HIGHEST COMMON FACTOR
The highest common factor (HCF) of two (or more) numbers is the largest common
factor.
So, the common factors of 8 and 12 are 1, 2 and 4; and 4 is the largest common factor.
Setting out:
Often, we set out the solution as follows:
Example 18
Find the highest common factor of 16 and 32.
Solution:
The highest common factor (HCF) of two numbers (or expressions) is the largest
number (or expression) which is a factor of both.
Consider the highest common factor of 24 and 40.
The common factors of 24 and 40 are 2, 4, and 8. So, the highest common factor is 8.
Note:
The highest common factor is the product of the common prime factors.
The highest common factor of algebraic expressions is useful in factorisation; and it is
the product of the common prime factors, which includes both common numerical and
algebraic factors.
Example 19
Solution:
Example 20
Solution:
PRIME NUMBERS
A prime number has only two different factors, 1 and itself.
So, 13 is a prime number since it has only two different factors, 1 and 13.
COMPOSITE NUMBERS
A composite number is a number that has more than two factors.
So, 10 is a composite number as it has more than two factors.
Note:
1 is considered neither a prime number nor a composite number as it has 1 factor only. A
prime number has 2 different factors.
Example 21
State which of the following numbers are a prime:
a. 6
b. 19
Solution:
PRIME FACTORS
Prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, …
A prime factor is a factor of a number that is also a prime number.
To express a whole number as a product of prime factors, try prime numbers in order of
their magnitude. That is, first try to find out whether 2 is a factor of the given whole
number or not. If 2 is a factor of the whole number, it will divide into it exactly and we
can write the whole number as the product of 2 and another number. If 2 is a prime, try 2
again to see if we can further write the whole number as a product involving two 2s. If
not, try the next prime, 3, and so on. Stop when the number has been expressed as the
product of prime numbers.
Example 22
Express 90 as a product of prime numbers.
Solution:
Alternative solution:
We can use a factor tree to express 90 as a product of prime numbers.
Example 23
Express 200 as a product of prime numbers.
Solution:
APPLICATION OF PRIME NUMBERS
Prime factors are used to find the highest common factor (HCF) and the lowest common
multiple (LCM) of two (or more) large numbers.
Highest Common Factor (HCF) by Prime Factors
The HCF of two (or more) numbers is the product of common prime factors.
Example 24
Find the HCF of 300 and 375.
Solution:
Organise the above information as shown below and circle the prime factors that are
common to both numbers. The HCF is the product of common prime factors.
Note:

300 = 4 × 75 and 375 = 5 × 75.

75 is the largest factor common to 300 and 375.
Lowest Common Multiple (LCM) by Prime Factors
The LCM of two (or more) numbers is calculated as follows:

Express the numbers as a product of prime factors.

Circle all of the prime factors of the smaller of the two numbers.

Circle any prime factors of a larger number that have not already been circled for
the smaller number (or smaller numbers if you are looking for the LCM of more
than two numbers).

The LCM is the product of the circled prime factors.
These steps are better understood by reading the following examples.
Example 25
Find the LCM of 300 and 375.
Solution:
Organise the above information as shown below and circle all of the prime factors of the
smaller number. Then circle any prime factors of the larger number that have not already
been circled in the smaller number. The LCM is the product of the circled prime factors.
Note:

The product of prime factors of 375 has three 5s but the product of prime factors
of 300 has only two 5s. Since we circled two 5s for 300, we must circle the extra
5 for 375 as shown above.

1500 is a multiple of 300 as 5 × 300 = 1500.
1500 is a multiple of 375 as 4 × 375 = 1500.
Example 26
Find the LCM of 6 and 8.
Solution:
Organize the above information and circle all of the prime factors of the smaller number
as shown below. Then circle any prime factors of the larger number that have not already
been circled in the smaller number. The LCM is the product of circled prime factors.
Finding a Pattern
By observation from Example 24, we find that:
In general:
The lowest common multiple of two (or more) numbers can be computed as follows:

Express each number as a product of its prime factors using powers.

Then circle the prime factors to the highest power from the given numbers Find
the product of each prime factor to its highest power from the given numbers.
This product represents the LCM of the numbers.
Example 27
Find the LCM of 9, 40 and 48.
Solution:
Organize the above information and circle each prime factor with its highest power as
shown below. The highest power of 2 is 24, the highest power of 3 is 32 and the highest
power of 5 is 51. So, circle 24, 32 and 51. The LCM is the product of the prime factors to
the highest powers.
FRACTIONS AND DECIMALS
FRACTIONS
PROPER FRACTION
If the numerator is smaller than the denominator, then the fraction is said to be a proper
fraction.
Note:
Proper fractions are smaller than 1.
Improper Fraction
If the numerator is greater than the denominator, then the fraction is said to be an
improper fraction.
Note:
Improper fractions are greater than 1.
Equivalent Fractions
We notice that:
In general:
The numerator of a fraction and denominator of a fraction can be multiplied by the same
number without altering the value of the fraction.
In general:
The numerator of a fraction and denominator of a fraction can be divided by the same
number without altering the value of the fraction.
Example 1
Solution:
We find, by trial and error, that 7 divides into both the numerator and denominator.
Setting out:
Usually, we set out the solution as follows:
Note:
Mixed Numbers
Example 2
Solution:
In this example,

37 is the dividend

5 is the divisor

7 is the quotient

2 is the remainder
Example 3
Solution:
ADDITION AND SUBTRACTION OF FRACTIONS
We use the following steps to add or subtract fractions:

Change any mixed numbers to improper fractions.

Find the lowest common multiple of the denominators.

Express all fractions with the same denominator.

Perform addition or subtraction using the numerators to obtain the numerator of
the answer; and then simplify the fraction, if possible.

Write the answer either as an improper fraction or as a mixed number.
Example 4
Solution:
MULTIPLICATION OF FRACTIONS
We adopt the following steps to multiply fractions:

Change any mixed numbers to improper fractions.

Cancel any factors common to both the numerator and denominator.

Multiply the remaining terms in the numerator and in the denominator.

Write the answer either as a improper fraction or as a mixed number.
Example 5
Solution:
In general:
DIVISION OF FRACTIONS
We adopt the following steps to divide one fraction by another:

Change mixed numbers into improper fractions.

Change the ÷ into × and multiply the first fraction by the reciprocal of the second
fraction.

Remember to cancel any common factors and write the answer either as a
improper fraction or as a mixed number.
Example 6
Solution:
In general:
ADDITION AND SUBTRACTION OF DECIMALS
Knowing how to add and subtract decimals is important in life. For example, when you
handle money at a shop, the bank or post office, you should use your knowledge of
decimals to calculate the money and any change exchanged.
ADDITION OF DECIMALS
To add decimal numbers, insert zeros in empty decimal place values so that all of the
numbers have the same number of decimal places and write the numbers such that their
decimal points are below one another.
Example 7
Calculate 5.84 + 8 + 12.79.
Solution:
Note:
The decimal point in the answer is lined up with the decimal points in the numbers set out
vertically.
Subtraction of Decimals
To subtract a small decimal number from a larger decimal number, write them down with
the larger one on top and the decimal points underneath one another. Then calculate the
subtraction as you would for whole numbers and line up the decimal point in the answer.
Example 8
Rewrite 3.67 – 1.83 in columns and then calculate.
Solution:
Example 9
Rewrite 83.47 – 57.684 in columns and then calculate.
Solution:
Note:
To add (or subtract) decimals, always fill empty place values with zeros so that all of the
numbers have the same number of decimal places.
Example 10
Rewrite 24 – 8.327 in columns and then calculate.
Solution:
Note:
24 = 24.000
MULTIPLICATION OF DECIMALS
To multiply decimal numbers:

Ignore the decimal points and multiply the digits.

Then count the number of decimal places in each of the numbers being multiplied
which will be the total number of decimal places in the answer.

Use this total number of decimal places to place a decimal point in the answer.
Example 11
Calculate 0.8 × 0.9.
Solution:
Note:

There is 1 decimal place in 0.8 and 1 decimal place in 0.9 and thus, 2 decimal
places in the answer.

First ignore the decimal points and multiply 8 by 9. Then include the 2 decimal
places in the answer.
Example 12
Calculate 0.78 × 0.5.
Solution:
Note:

There are 2 decimal places in 0.78 and 1 decimal place in 0.5 and thus, 3 decimal
places in the answer.

First ignore the decimal points and multiply 78 by 5. Then include the 3 decimal
places in the answer.
Example 13
Calculate 3.24 × 0.67.
Solution:
Note:

There are 2 decimal places in 3.24 and 2 decimal places in 0.67 and thus, 4
decimal places in the answer.
First ignore the decimal points and multiply 324 by 67. Then include the 4 decimal
places in the answer.
DIVISION OF DECIMALS
To divide a decimal number by another decimal number:

Make the divisor a whole number by moving the decimal point in the divisor to
the right until it is a whole number.

Then move the decimal point in the dividend to the right by the same number of
places as the decimal point was moved to make the divisor a whole number.

Finally divide the new dividend by the new divisor.
Example 14
Solution:
Alternatively, we can divide as follows:
Note:
Example 15
Solution:
Alternatively, we can divide as follows:
ROUNDING OFF
Decimals numbers often contain more decimal places than we need in daily life. For
example, a builder may have calculated that 20.37 cubic metres of concrete is needed to
make a driveway, but actually only needs a figure to the nearest tenth of a cubic metre to
make the order.
From the number line, it is clear that 20.37 is closer to 20.4. So, the amount ordered
would be 20.4 cubic metres. The builder has rounded off the calculated figure to one
decimal place. We say that the amount is rounded upwards since 20.4 is greater than
20.37.
Likewise, a builder may have calculated that 20.34 cubic metres of concrete is needed to
make a driveway, but actually only needs a figure to the nearest tenth of a cubic metre to
make the order.
From the number line, it is clear that 20.34 is closer to 20.3. So, the amount ordered
would be 20.3 cubic metres. The builder has rounded off the calculated figure to one
decimal place. We say that the amount is rounded downward since 20.3 is less than
20.34.
To round off a decimal:

Look at the digit to the right of the required decimal place.

If the digit is less than 5, round down by ignoring the trailing digits.

If the digit is 5 or more, round up by increasing the required decimal place by 1
and remove the trailing digits.
Example 16
Round 8.9463 kilograms to:
a. one decimal place
b. two decimal places
c. three decimal places
d. the nearest gram
Solution:
a. The digit, 4, to the right of the required place is less than 5. So, round down by
ignoring the digits after the first decimal place.
b. The digit, 6, to the right of the required place is more than 5. So, round up by
increasing the second decimal place by 1 and ignoring the trailing digits.
c. The digit, 3, to the right of the required place is less than 5. So, round down by
ignoring the fourth decimal place.
The answer needs to be correct to 3 decimal places.
RECURRING DECIMALS
So far, we have considered divisions with a limited number of decimal places in the
quotient (i.e. answer).
These are examples of terminating decimals.
Sometimes when dividing, the division will never stop as there is always a remainder.
It is clear that if 8 is divided by 3, then the sixes in the answer never stop. This is an
example of a recurring decimal.
This is written as:
The dot above 6 means that it is repeated indefinitely (i.e. forever).
An alternative notation involves placing a bar above the repeating digit(s) in the quotient
(i.e. answer).
Example 17
Solution:
We notice that the remainder is always 2. So, the digit in the quotient will continue to be
6.
Example 18
Solution:
We notice that the digits 5, 7, 1, 4, 2 and 8 begin to repeat.
This is written by placing a dot over the first and the last recurring digit.
Alternatively, we can write it by placing a bar above the whole repeating set of digits
CHAPTER 3 INTEGERS
THE NUMBER LINE
A number line is a line on which numbers are represented in ascending order.
This number line can be extended to the left to represent numbers which are smaller than
0. Such numbers are called negative numbers.
For example, –1 (called negative one or minus one) is positioned 1 unit to the left of 0.
So, –1 is less than zero.
Similarly, –2 (called negative two or minus two) is 2 less than zero
and
–3 (called negative three or minus three) is 3 less than zero
and so on.
From the preceding discussion we can define the number line as follows:
A line on which the numbers on both sides of zero are represented is said to be a number
line (or directed number line).
Positive numbers are represented on the right of the zero and negative numbers are
represented on the left of the zero as shown in the above diagram.
Example 1
Write down the elevation information of the following places on a number line. Use the
minus sign (–) to represent elevation below sea level.
a. Mt. Kosciusko (Australia)
b. Bangalore (India)
c. Death Valley (USA)
1000 m above sea level
85 m below sea level
d. Qattara Depression (Egypt)
Solution:
2230 m above sea level
392 m below sea level
Example 2
Write down the temperature information for the following places on a number line and
use the minus sign (–) to indicate any temperature that is below zero.
a. Toronto (Canada)
–14ºC
b. Melbourne (Australia)
c. London (UK)
–6ºC
d. New Delhi (India)
Solution:
30ºC
20ºC
INTEGERS
Negative numbers are called negative integers and positive numbers are called positive
integers. Zero is neither a negative nor positive integer.
The set of integers consists of negative integers, zero and positive integers. So:
The set of integers = {…, –3, –2, –1, 0, +1, +2, +3, … }
It is common practice to represent positive integers as 1, 2, 3, 4, …
Often we use a number line to represent the set of integers.
It is clear that a number on the number line is always larger than the number to the left of
it. For example:
Example 3
Represent the integers –5, –1, 2 and 4 on a number line.
Solution:
Example 4
Represent the integers that are greater than –3 and less than 3 on a number line.
Solution:
Example 5
Insert the symbol > between the following pairs of integers:
Solution:
Use a number line to answer this question.
Example 6
Insert the symbol < between the following pairs of integers:
Solution:
Use a number line to answer this question.
ADDITION AND SUBTRACTION OF INTEGERS
Addition of Positive Integers
Consider the addition of 2 + 3.
The plus sign, +, tells us to face the positive direction.
So, to evaluate 2 + 3, start at 2, face the positive direction and move 3 units forwards.
This suggests that:
Positive integers can be added like natural numbers.
Addition of Negative Integers
Consider the addition of (–2) + (–3).
The plus sign, +, tells us to face the positive direction.
So, to evaluate (–2) + (–3), start at –2, face the positive direction and move 3 units
backwards.
Note:
We can write (–2) + (–3) as –2 + –3
Subtracting a Positive Integer from a Negative Integer
Consider the value of (–2) – (3).
The minus sign, –, tells us to face the negative direction.
So, to evaluate (–2) – (3), start at –2, face the negative direction and move 3 units
forwards.
We notice that:
That is:
This suggests that:
Adding a negative integer is the same as subtracting a positive integer.
From the above discussion, we can state that:
Negative integer are added like natural numbers; but place a minus sign, –, in front of the
sum.
Example 7
Find the value of:
Solution:
Addition of a Positive Integer and a Negative Integer
Consider the addition of 3 + (–7).
The plus sign, +, tells us to face the positive direction.
So, to evaluate 3 + (–7), start at 3, face the positive direction and move 7 units
backwards.
Note:
3 + (–7) is often written as 3 – 7.
To find the value of 3 – 7, first ignore the signs and subtract the smaller number, 3, from
the larger number, 7, and put the sign, –, of the larger number, 7, in front of the
difference. That is:
Example 8
Find the value of:
Solution:
Subtracting a Negative Integer from a Positive Integer
Consider the subtraction of 2 – (–3).
The minus sign, –, tells us to face the negative direction.
So, to evaluate 2 – (–3), start at 2, face the negative direction and move 3 units
backwards.
This suggests that:
To take away a negative integer, add its opposite which is a positive integer.
Example 9
Find the value of:
Solution:
Note:
If two minus signs are side by side, then the two minus signs become a plus sign (i.e. a
positive).
MULTIPLICATION OF INTEGERS
The Product of Two Positive Integers
Consider the product 3 × 4.
This suggests that:
The product of two positive integers is a positive integer.
Example 10
Find the value of:
Solution:
The Product of Two Negative Integers
Consider the product –3 × – 4.
This suggests that:
The product of two negative integers is a positive integer.
Example 11
Find the value of:
Solution:
The Product of a Positive Integer and a Negative Integer
Consider the product 4 × –3.
This suggests that:
The product of a positive integer and a negative integer is a negative integer.
Example 12
Find the value of:
Solution:
Just to recap the foregoing discussion:

The product of two numbers with like (i.e. the same) signs is always positive.
The product of two numbers with unlike (i.e. different) signs is always
negative
DIVISION OF INTEGERS
Division Involving Two Positive Integers
This suggests that:
Division involving two positive integers results in a positive answer.
Example 13
Find the value of:
Solution:
Division Involving Two Negative Integers
This suggests that:
Division involving two negative integers results in a positive answer.
Example 14
Find the value of:
Solution:
Division Involving a Positive Integer and a Negative Integer
This suggests that:
Division involving a positive integer and a negative integer results in a negative answer.
Example 15
Find the value of:
Solution:
Just to recap the foregoing discussion:

Division involving two numbers with like (i.e. the same) signs always results in a
positive answer.
Division involving two numbers with unlike (i.e. different) signs always
results in a negative answer.
CHAPTER 4 ALGEBRA
PRONUMERALS
A pronumeral is a letter that is used to represent a number (or numeral) in a problem.
For example, the formula for the area of a rectangle is:
Area of a rectangle = length × width
If A represents the area of the rectangle, l represents the length of the rectangle and w
represents the width of the rectangle, then we can write the formula for the area of the
rectangle as follows:
A=l×w
In this formula, the letters A, l and w are called pronumerals
Note:
A pronumeral is a letter used in a problem to represent the measurement of a quantity.
We often choose the first letter of the name of a quantity.
For example, the measurement of the base of a triangle will be represented by b.
MULTIPLICATION OF A PRONUMERAL BY A NUMBER
Note:
When a number and a pronumeral are multiplied together, then the multiplication sign is
usually omitted.
Example 1
Write the following in a simpler form:
Solution:
Note:
In part d above, we let 1m = m. When you see a pronumeral like m without a number
preceding it, the number 1 is understood to precede the pronumeral.
Note:
When multiplying numbers with pronumerals, we always write the numbers in front of
the letters.
Example 2
Write the following in a simpler form:
Solution:
COEFFICIENT OF A PRONUMERAL
The number in front of the pronumeral represents how many lots of the pronumeral there
are. This number is said to be the coefficient of the pronumeral.
For example, in 10x the coefficient of x is 10 and in 5y the coefficient of y is 5.
Terms
The product of a number and a pronumeral forms a term. Terms often include
pronumerals but may also be a number called a constant term.
Examples of terms include 4a, c, 5, 9, 5b, 8p, 15q, 20r, 26, n, 14x, 40y, 56z and 28v.
Note the following:

The product of a number and different pronumerals is also called a term.
E.g. 4ab, 5bc, 8pqr, 14xy, 40yz, 56xyz and 28uv are all terms.

The fraction formed by a number and a pronumeral or pronumerals is also called a
term.
Example 3
State the coefficient of the pronumeral:
a. 12x
b. 25m
Solution:
a. The coefficient of x is 12.
b. The coefficient of m is 25.
MULTIPLICATION OF A TERM BY A NUMBER
EXPRESSIONS
An expression consists of terms that are written with arithmetical signs between them
which include the addition, subtraction, division and multiplication signs.
Like Terms
Terms with the same pronumeral (or pronumerals) are called like terms.
Unlike Terms
Terms with different pronumerals are called unlike terms.
Addition or Subtraction of Like Terms
If the dollars are represented by the pronumeral d, then the statement
5 dollars + 10 dollars = 15 dollars
can be described by
5d + 10d = 15d
From the above discussion, we can state that:
Only like terms can be added (or subtracted) to simplify an expression.
Example 4
Solution:
Example 5
Solution:
Multiplication of Pronumerals
If two pronumerals are multiplied together, then the multiplication sign is omitted.
Similarly, if two pronumerals are multiplied the order does not matter. That is:
This is called the Commutative Law for Multiplication.
Example 6
Solution:
Note:
It is customary to write the multiplication of pronumerals in alphabetical order, as
illustrated in parts c and d of Example 6.
Multiplication of Terms
Example 7
Solution:
Further Like Terms
Recall that:
Like terms are those terms that contain the same pronumeral or pronumerals.
For example, 8x and 10x are like terms because they contain the same pronumeral in x.
Likewise, 8xy and 10xy are like terms because they contain the same pronumerals in xy.
Clearly, 8xz and 10yz are not like terms because although both terms contain the
pronumeral z, the other pronumeral in each term is different. Terms with different
pronumerals are called unlike terms.
Further Addition (or Subtraction) of Like Terms
Recall that:
Algebraic expressions containing like terms may be simplified by adding (or subtracting)
the like terms.
Note that the expression 3x + 2xy + 5xyz can not be simplified because it consists of three
unlike terms.
Example 8
Solution:
Example 9
Solution:
Example 10
Solution:
Division of a Term by a Number
If the amount of 10 dollars is divided evenly between 2 boys, then each boy receives 5
dollars. This can be written as:
Note:
So when a number divides an algebraic term, the divisor is divided into the coefficient of
the pronumeral.
Example 11
Simplify the following:
Solution:
Example 12
Solution:
Setting out:
Often, we set out the solution as follows:
Example 13
Simplify the following:
Solution:
Note:
Chapter 5 Equations
Equations
Equations enable us to describe complex problems in simple terms. They are built with
numbers, pronumerals and an equal sign.
It is clear that the number sentence
5 + 10 = 15
is an equation.
If the value of the pronumeral x is 5, then x can take the place of 5; and we can write this
equation as
x + 10 = 15
This is an equation containing the pronumeral x. The value of the pronumeral x is 5.
Example 1
Describe each of the following equations in words:
Solution:
Example 2
Write an equation to represent each of the following statements:
a. When I add 7 to a number, the answer is 16.
b. When I subtract 8 from a number, the answer is 23.
c. When I multiply a number by 9, the answer is 27.
d. When I divide a number by 5, the answer is 29.
Solution:
Remember:
An equation is a statement consisting of number(s) and pronumeral(s) that are linked
together with an equal sign.
EQUATIONS AND A PAIR OF SCALES
Recall that:
An equation is a statement that contains an equal sign.
Consider the simple equation
x=5
Visualise this equation as a balanced pair of scales with x and 5 measured in kilograms.
If we add 3 kg to the scale on the left-hand side, the scales will balance as long as we add
3 kg to the scale on the right-hand side. That is, x + 3 = 8.
Also, if we subtract the same weight, say 3 kg, from each side of the balance, the scales
will remain balanced. That is, x – 3 = 2.
If we double the weight in the scale on the left-hand side, the scales will balance as long
as we double the weight in the scale on the right-hand side. That is, 2x = 10.
Also, if we halve the weight in each scale of the balance, the scales will remain
balanced.
Solving Equations
Solving an equation means to find the value of a pronumeral that makes a statement
true.
In the preceding section, we observed that:
An equation behaves like a pair of balanced scales. The scales remain balanced as long
as we do the same thing to both scales.
This suggests that to solve an equation, we can do the same thing to both sides of an
equation. That is:




The same number can be subtracted from both sides of an equation.
The same number can be added to both sides of an equation.
Both sides of an equation can be divided by the same number.
Both sides of an equation can be multiplied by the same number.
We will now consider equations involving addition, subtraction, multiplication and
division.
Operations such as +, –, × and ÷ are used to build an equation. To solve an equation, we
use inverse (i.e. opposite) operations such that the pronumeral is the only term remaining
on the left-hand side.
Equations Involving Addition
The inverse operation of + is –. So, to solve an equation involving addition, we undo the
addition by subtracting the same number from both sides.
Example 3
Solve the equation x + 6 = 14.
Solution:
Note:
6 is added to x. So, we undo the addition by subtracting 6 from both sides.
Check:
Equations Involving Subtraction
The inverse operation of – is +. So, to solve an equation involving subtraction, we undo
the subtraction by adding the same number to both sides.
Example 4
Solve the equation x – 9 = 17.
Solution:
Note:
9 is subtracted from x. So, we undo the subtraction by adding 9 to both sides.
Check:
Equations Involving Multiplication
The inverse operation of × is ÷. So, to solve an equation involving multiplication, we
divide both sides of the equation by the same number.
Example 5
Solve the equation 8x = 72.
Solution:
Note:
x is multiplied by 8. So, we undo the multiplication by dividing both sides by 8.
Check:
Equations Involving Division
The inverse operation of ÷ is ×. So, to solve an equation involving division, we multiply
both sides of the equation by the same number.
Example 6
Solution:
Note:
x is divided by 6. So, we undo the division by multiplying both sides by 6.
Check:
Remember:






An equation is a statement that contains an equal sign.
To solve an equation, we do the same thing to both sides of the equation.
The same number can be subtracted from both sides of an equation.
The same number can be added to both sides of an equation.
Both sides of an equation can be divided by the same number.
Both sides of an equation can be multiplied by the same number.
Key Terms
Problem Solving 1
Linear equations help us to solve word problems. First, we assume that the number we
are trying to find is represented by a pronumeral. Then the problem given in words is
translated into an equation which is solved using the methods we have learned for solving
equations. Finally, we write the answer in words.
Example 7
A number is added to 85 and the result is 172. Find the number.
Solution:
Let the number be x.
So, the number is 87.
Check:
Remember:
To solve a word problem:




Read the problem.
Assume that the unknown is x or another suitable pronumeral.
Write an equation using the information provided in the problem.
Solve the equation by using inverse operations.
Write your answer in words.
EQUATIONS INVOLVING TWO OR MORE OPERATIONS
To solve an equation involving two or more operations, start by carrying out an inverse
operation on the number that is furthest away from x.
Example 8
Solve 7x + 4 = 25.
Solution:
The number 4 is furthest away from x. The inverse of + 4 is – 4. So, subtract 4 from
both sides.
Problem Solving 2
Example 9
If three times a number, when diminished by 4, equals 17, what is the number?
Solution:
Let x be the number.
Three times x is 3x, and diminishing this by 4 gives 3x – 4, which we are told equals 17.
So, the number is 7.
EQUATIONS WITH THE PRONUMERAL ON BOTH SIDES
When a pronumeral is on both sides of an equation, remove the pronumeral term from the
right-hand side of the equation by using inverse operations. Then continue to use inverse
operations to solve the equation for the pronumeral.
Example 10
Solution:
Problem Solving 3
Example 11
If 7 less than three times a number is 9 more than the number, what is the number?
Solution:
Let x be the number.
Seven less than three times x is 3x – 7, and 9 more than x is x + 9.
So, the number is 8.
Check:
Equations Containing Brackets
To solve the equation containing brackets, we may proceed as follows:

Remove the brackets by using the Distributive Law.
Collect the pronumeral terms on the left-hand side of the equation and the

numerical terms on the right-hand side of the equation by doing the same thing to
both sides of the equation.
Example 12
Solution:
Example 13
Solution:
Problem Solving 4
Example 14
Find the width of a rectangular paddock whose length is 60 m and perimeter is 220 m.
Solution:
So, the width of the paddock is 50 m.
Equations Containing Fractions
To solve equations containing fractions:



Find the lowest common multiple of the denominators which is known as the
lowest common denominator (LCD).
Remove the fractions by multiplying both sides of the equation by the LCD.
Solve the equation for the unknown pronumeral by performing the same
operations to both sides of the equation.
Example 15
Solution:
To solve equations containing fractions:



Find the lowest common multiple of the denominators which is known as the
lowest common denominator (LCD).
Remove the fractions by multiplying both sides of the equation by the LCD.
Solve the equation for the unknown pronumeral by performing the same
operations to both sides of the equation.
Example 15
Solution:
Lowest common multiple of 3 and 1 is 3. So, we multiply both sides by 3 to obtain:
Example 16
Solution:
Lowest common multiple of 8 and 3 is 24. So, we multiply both sides by 24 to obtain:
Problem Solving 5
Read the problem carefully and describe it by an equation. Then solve the equation and
write the answer as a sentence.
Example 17
The sum of three consecutive numbers is 87. What are the numbers?
Solution:
Let the smallest number be n.
The word consecutive means one after the other. So, the next two consecutive numbers
after n will be n + 1 and n + 2.
So, the three consecutive numbers are 28, 29 and 30
Problem Solving Unit
Problem 5.1 The Fake Coin
There are 11 coins that looks identical but one of the coins is fake. The fake coin weighs
less than the true coins. Find the minimum number of times you will have to use a
balance to isolate the fake coin from the others.
Briefly explain the procedure used in your own words.