Grade 9
Whole Numbers
Whole Numbers
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Properties of Natural numbers
Properties of Whole numbers
Properties of Integers
Properties of Rational numbers
Properties of Irrational numbers
Calculations using whole numbers
Estimating
Long division
Rounding off and compensating
Multiples and factors
Ratio and proportion
Direct and indirect proportion
Financial problems about profit, loss, discount, VAT, budgets, accounts, loans
Simple interest and hire purchase
Compound interest
Rentals
Exchange rates and commission
Terminology
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Natural numbers
Whole numbers
Integers
Rational numbers
Irrational numbers
Multiples and factors
Prime numbers and composite numbers
Squares, cubes, square roots and cube roots
Ratio and proportion
Direct and indirect proportion
Simple interest and hire purchase
Compound interest
Rentals
Exchange rates and commission
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Grade 9
Whole Numbers
Properties of Numbers
Real numbers:
A set of real numbers include all rational and irrational numbers. The letter ℝ is used to
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denote the set of real numbers. ℝ = {4; ; 0; 𝜋; √3; 2,57; −8; 0, 3̇ }
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Rational numbers:
The set of rational numbers is made up of all numbers that can be written in the form
𝑎
𝑏
𝑖𝑛𝑡𝑒𝑔𝑒𝑟
where 𝑎 and 𝑏 are both integers and 𝑏 cannot be equal to zero i.e. 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 (where the
denominator cannot be zero). The letter ℚ is used to denote the set of rational numbers.
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ℚ = { ; 6; 0,4; 0, 3̇7̇; 0; −1 }
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The subset of rational numbers includes Natural numbers, whole numbers and integers.
Natural numbers
The set of natural numbers are whole positive numbers that start at 1 and go on to infinity.
The letter ℕ is used to denote the set of natural numbers. ℕ = {1, 2, 3, 4, 5, … }
Whole numbers
The set of whole numbers is also known as counting numbers. These are whole positive
numbers that start at zero. The letter ℕ0 is used to denote the set of whole numbers. ℕ0 =
{0; 1; 2; 3; … }
Integers
The set of integers consist of all whole numbers. This includes positive and negative whole
numbers and zero. The letter ℤ denotes the set of integers. ℤ = {… −
3; −2; −1; 0; 1; 2; 3 … }
Irrational numbers
𝑎
This is the set of numbers that cannot be written in the form , where 𝑎 and 𝑏 are integers
𝑏
𝑖𝑛𝑡𝑒𝑔𝑒𝑟
: 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
This set consists of

Non-recurring and non-terminating decimal fractions

Square roots and cube roots of numbers that are not whole numbers e.g. √7, √4

𝑃𝑖 (𝜋) = 3,141592654 (𝑛𝑜𝑡
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22
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𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑎 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟)
Irrational numbers are denoted by the letterℚ′ . ℚ′ = {𝜋; √5; 4,92748623 … . }
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Grade 9
Whole Numbers
Fractions
This set includes all common fractions, improper fractions, and mixed numbers. Fractions
can be either positive or negative. There is no specific letter used to denote fractions so the
letter ℚ must be used to show that they are rational numbers.
Decimal Fractions
Decimal fractions that are finite and/or recur belong to the set or rational numbers and
hence must be denoted by the symbol ℚ.
Decimal fractions that are non-recurring or non-terminating belong to the set or irrational
numbers and hence must be denoted by the symbol ℚ′ .
Below is a representation of the number System
𝑅𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 (ℝ)
𝐼𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 (ℚ′ )
𝑅𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 (ℚ)
𝐼𝑛𝑡𝑒𝑔𝑒𝑟𝑠 (ℤ)
𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠
𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠 (𝑤ℎ𝑖𝑐ℎ 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠)
𝑧𝑒𝑟𝑜
𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 (ℕ)
(𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠)
𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠
𝑣𝑎𝑙𝑔𝑎𝑟 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠
𝑝𝑟𝑜𝑝𝑒𝑟
𝑖𝑚𝑝𝑟𝑜𝑝𝑒𝑟
𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠
𝑑𝑒𝑐𝑖𝑎𝑚𝑙 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠
𝑚𝑖𝑥𝑒𝑑
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Grade 9
Whole Numbers
Factors
Factors are the set of numbers that a given number can be divided by to obtain a natural
number (No remainders). E.g. the factors of 12 are 𝐹12 = {1; 2; 3; 4; 6; 12}
Multiples
Multiples are the set of numbers that is obtained when a given number is multiplied by
various integers (Times table). E.g. the multiples of 3 are 𝑀3 = {3; 6; 9; 12}
Prime numbers
A prime number is a number that only has two factors,
one and itself. E.g. 5 𝐹5 = {1; 5}
𝑵𝑩: 𝑹𝒆𝒎𝒆𝒎𝒃𝒆𝒓 𝒕𝒉𝒂𝒕 𝟏 𝒊𝒔 𝒏𝒆𝒊𝒕𝒉𝒆𝒓 𝒂 𝒑𝒓𝒊𝒎𝒆 𝒏𝒖𝒎𝒃𝒆𝒓 𝒏𝒐𝒓 𝒂
𝒄𝒐𝒎𝒑𝒐𝒔𝒊𝒕𝒆 𝒏𝒖𝒎𝒃𝒆𝒓 𝒂𝒔 𝒊𝒕 𝒉𝒂𝒔 𝒐𝒏𝒍𝒚 𝒐𝒏𝒆 𝒇𝒂𝒄𝒕𝒐𝒓
Composite numbers
A composite number is a number that has three or more factors. E.g. 6 𝐹6 = {1; 2; 3; 6}
Square numbers
Square numbers are numbers which are derived from multiplying a number by itself (
squaring the number). E.g. 4 is a square number as 22 = 4
Square Root
Square root of a number n is a number which, when multiplied by itself, results in the
number n. E.g. 2 is a square root of 4
Cubic numbers
Cubed numbers are numbers which are derived from multiplying a number by itself twice.
E.g. 8 is a cubed number as 2 × 2 × 2 = 8 (23 = 8).
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Grade 9
Whole Numbers
Exercise 1
1.
Below is a representation of the real number system.
1.1
Complete the representation by filling in the words that have been left out
(Whole, Integers, Rational, Real, Irrational)
1.2
Give an example of the numbers that will be found in each set below.
𝑁𝑎𝑡𝑢𝑟𝑎𝑙
1; 2; 3
2.
From the list of numbers select examples for the following:
−
3.
3
5
−1
2.1
Natural numbers
2.2
Whole numbers
2.3
Integers
2.4
Rational numbers
1
4
0
7
−6
6
1
4,67
√9
15
0
20
5
List the following numbers one below each other. Next to each number, write the
category (ℕ, ℕ0 , ℤ, ℚ, ℚ′ ) to which it belongs (there could be more than one):
2
7
−6
25
√
16
3
− √8
0,35
2𝜋
1
2
5
3,6
4
(√3)2
2√8
3
√9
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4.
5.
Whole Numbers
Write down two examples of each oft eh following kinds of numbers
4.1
An even natural number greater than 8
4.2
An odd number 𝑥, where 𝑥 < 11 and 𝑥 is a prime number
4.3
A square number greater than 30
4.4
A rational number between 1 and 1 2
4.5
An integer 𝑥 where 𝑥 < 0 and 𝑥 > −5
4.6
A positive rational number that is not a fraction
4.7
A negative rational number that is not a fraction
4.8
The factors of 16
4.9
The ninth multiple of 8
4.10
A number that is a perfect square as well as a perfect cube
1
Copy and complete the table by making a tick if the number is an element of the set
of a cross if it is not
Number
14
2,8
4
−
2
√
ℕ
ℕ0
ℤ
ℚ
ℝ
16
2
1
4
−10
𝜋
−5
6.
Answer the following questions
6.1
Find the middle number between 98 and 142.
6.2
6.3
𝐹𝑖𝑙𝑙 𝑖𝑛 <, >, =∶
6.4
6.5
Write in scientific notation 1 274 000
6.6
13 + 9
6.7
𝐶𝑜𝑚𝑝𝑙𝑒𝑡𝑒: 4; 16; 64; ___; ____ ; ____
6.8
Calculate 3 ÷ 1 6
6.9
Write in standard form: 1,362 × 104 =
6.10
−17,3 ____ − 15,9
32
33
20
50
8
= 20 =
2
10
5
2
4
𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 (2)2 + 42
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Whole Numbers
Calculation Techniques
There are many different ways to solve problems. Some of these techniques include,
-Long division:
Place value is very important when carrying out long division
-long multiplication: Place value is very important when carrying out long multiplication.
-estimating
In many cases, exact numbers are not necessary or even desirable. In these circumstances
approximations are given instead. The most common types of approximations are:
-rounding: this is done for large numbers where the number itself is rounded to the
required multiple of ten, i.e. nearest 10, nearest 100, nearest tenth, or nearest 100 th.
-rounding off: Count up to the required number of places, then look at the number
immediately after it, to see if you need to round up. If the number immediately after is
5 or greater, round up.
-decimal places (d.p.): This refers to the numbers after the decimal point. Very often a
calculator gives us an answer that has too many decimal places. We can correct the
answer to a given number of decimal places.
-significant figures (s.f.): Consider the number 24,67. The digit 2 is the most significant
as it has the highest value, 20. The 4 is the next most significant. 6 is the next
significant and 7 is the least significant. We can therefore say that the number is given
to four significant figures. If the same number is rounded off to 3 significant figures
then the third significant figure is 6 therefore the answer is 24,6. With numbers less
than 1, the first significant figure is the first digit which is not a zero.
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Whole Numbers
-Don’t forget to round off your answer to two decimal unless otherwise stated.
-Don’t forget:
Brackets Of Division Multiplication Addition and Subtraction (BODMAS)
Exercise 2
Use the above techniques to answer the following questions:
1.
Evaluate the following.
1.1
5+8×3−6
1.2
15 + 45 ÷ 3 − 12
1.3
6×4−3×8
1.4
15 ÷ 3 + 2 × 7
2.
The product of two numbers is 72, and their sum is 18. What are the two numbers?
3.
How many days are there in 42 weeks?
4.
Work out 368 × 49
5.
Work out 7 835 ÷ 23 give you answer to 1 decimal place
6.
Copy these equivalent fractions and fill in the blanks.
24
4
60
=
= =
=
36 12
30
7.
Work out the following: 851 × 27.
8.
Calculate: 6843 ÷ 19
9.
Round off the following numbers to the degree of accuracy shown in brackets:
9.1
10.
11.
12.
2841 (𝑛𝑒𝑎𝑟𝑒𝑠𝑡 100)
9.2 7286 (𝑛𝑒𝑎𝑟𝑒𝑠𝑡 10) 9.3 48 765 (𝑛𝑒𝑎𝑟𝑒𝑠𝑡 1000)
Round off the following numbers to the number of decimal places shown in brackets
10.1
3.84 (1 𝑑. 𝑝. )
10.2
6.783 (2 𝑑. 𝑝. )
10.3
0.0077 (3 𝑑. 𝑝. )
10.4
0.8527 (2 𝑑. 𝑝. )
10.5
23.532 (1 𝑑. 𝑝. )
10.6234.2344 (3 𝑑. 𝑝. )
Round off the following numbers to the number of significant figures shown in
brackets:
11.1
3.23 (1 𝑠. 𝑓. )
11.2
2,4234 (2𝑠. 𝑓. )
11.3 923.234 (5 𝑠. 𝑓. )
11.4
23.342 (4 𝑠. 𝑓. )
11.5
0,002344 (2 𝑠. 𝑓. )
11.6
98,234 (3 𝑠. 𝑓. )
1 mile is 1760 yards. Estimate the number of yards in 11.5 miles.
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Whole Numbers
Solving Problems
Ratio
A ratio is used when we want to compare two or more values (quantities or measurements),
𝐸𝑥𝑎𝑚𝑝𝑙𝑒: 1: 2 𝑤𝑒 𝑟𝑒𝑎𝑑 𝑖𝑡 𝑎𝑠 one is to two. 𝑇ℎ𝑖𝑠 𝑚𝑒𝑎𝑛𝑠 𝑡ℎ𝑎𝑡 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑜𝑛𝑒
𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑤𝑒 ℎ𝑎𝑣𝑒 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑣𝑎𝑙𝑢𝑒, 𝑤𝑒 ℎ𝑎𝑣𝑒 𝑡𝑤𝑜 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑣𝑎𝑙𝑢𝑒.
Rate
A rate involves the comparison of two values, where each value by be a quantity or
measurement expressed in different units. The units are very important and must be kept as
without units, a rate is meaningless
A very useful rate is that of speed. The following formulas can be used:
𝒔𝒑𝒆𝒆𝒅 =
𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆
𝒕𝒊𝒎𝒆
𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆 = 𝒔𝒑𝒆𝒆𝒅 × 𝒕𝒊𝒎𝒆
𝒕𝒊𝒎𝒆 =
𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆
𝒔𝒑𝒆𝒆𝒅
𝐸. 𝑔.What distance would a bus cover if it travelled for 4 hours at an average speed of 100 𝑘𝑚 ∙ ℎ−1
Direct Proportion
𝑥
𝑦
𝑥
𝑦

𝑥 is directly proportional to 𝑦 if = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 , in other words = 𝑐

𝑥 and 𝑦 are directly proportional if, as the value of 𝑥 increases the value of 𝑦 increases in the
same proportion, and as the value of 𝑥 decreases the value of 𝑦 decreases in the same
proportion.
Directly proportional relationships can be represented on a graph as a straight line

Indirect Proportion
𝑐

𝑥 is indirectly or inversely proportional to 𝑦 if 𝑥 × 𝑦 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. In other words 𝑦 = 𝑥

𝑥 and 𝑦 are indirectly proportional if, as the value of 𝑥 increases the value of 𝑦 decreases
and as the value of 𝑥 decreases the value of 𝑦 increases
An indirectly proportional relationship is represented by a non-linear curve

Exchange Rates
An exchange rate gives you a monetary value of one currency, per unit of another currency.
Exchange rates fluctuate continuously.
𝐸. 𝑔.
£1 = 𝑅12, 45
𝑇ℎ𝑖𝑠 𝑚𝑒𝑎𝑛𝑠 𝑡ℎ𝑎𝑡 𝑜𝑛𝑒 𝐸𝑢𝑟𝑜 𝑤𝑖𝑙𝑙 𝑐𝑜𝑠𝑡 𝑅12, 45
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Whole Numbers
Exercise 3
1.
A light flashes every 6 minutes and a bell rings every 8 minutes. If the light flashes as
the bell is ringing, how long will it be until this happens again?
2.
A tree of height 42m casts a shadow of 28m. How high is a tree that casts a shadow of
30m?
3.
A rectangular box has a volume of 14 𝑐𝑚3. What will the volume of the box be if the
length, width and height are doubled?
4.
The rainwater on a flat glass roof measuring 40m by 15m flows into a rectangular rank
that measures 10𝑚 × 2,5𝑚 × 1,5𝑚. If the tank is empty, how much rainfall (in
millimetres) is needed to fill the tank?
5.
A fruit seller earns R30 per say plus 30c per packet of fruit sold. How much will the
fruit seller earn after selling 40 packets of fruit in a day?
6.
Mr Wong sells apples at 3 for R3.50 and pears for 4 for R4.50. Which fruit is cheaper?
7.
On a trip by car to Cape Town, I covered 238 km between Bloemfontein and Colesberg
in 2 hours and 312 km from Colesberg to Beaufort West in 3 hours. Which was the
fastest trip?
8.
It takes 15 workers 9 days to clean all the windows of a skyscraper. How long will it
take for 5 workers?
9.
Susie is 30 years younger than her mothers. In 10 years time the sum of their ages will
be 80 years. How old is Susie?
10.
A shop owner buys 400 bicycles with a listed price of R600 each from a manufacturer.
He receives a trade discount and pays only R192 000. Calculate the percentage trade
discount he received.
11.
The price of a large packet of chips is R5, 99. If the price is increased by 15%, how
much will you pay for the packet of chips? Round off your answer to the nearest cent.
12.
A ruler 30 cm long is broker into two parts into the ratio 8:7. How long are the two
parts?
13.
A photocopying machine is capable of making 50 copies each minute.
13.1
If four identical copies are used simultaneously how long would it take to
make a total of 50 copies?
13.2
How many copiers would be needed to make 6000 copies in 15 minutes?
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14.
Whole Numbers
It takes 16 hours for three bricklayers to build a wall.
14.1
Calculate how long it would take for eight bricklayers to build a similar wall.
14.1
Is the above relationship directly proportional or indirectly proportional?
15.
A bricklayer ears R5040,00 in 20 hours and a carpenter earns R6000,00 in 25 hours.
Who is paid at the higher rate?
16.
A bookshelf is filled with books of the same thickness, but the number of books
depends on their thickness.
Thickness (mm)
Number of books
17.
18.
10
60
15
40
20
30
30
20
40
15
50
12
12
50
16.1
What is the length of the bookcase?
16.2
Is the above relationship directly proportional or indirectly proportional?
Peter computers that he has 66m of tape on the reel which plays for 11 min. at the
faster speed.
17.1
How many meters are left after 8 minutes?
17.2
Is the above relationship directly proportional or indirectly proportional?
At the local nursery Jane buys 12 roses for R90.
18.1
What would she pay for 63?
18.2
Is the above relationship directly proportional or indirectly proportional?
.
Financial Maths
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Whole Numbers
Simple interest
Interest is paid only on the amount invested. This amount is also called capital
or the principal amount. The case of a loan, interest is charged on the full
amount of the loan over the period taken to repay the loan. The following
formula can be used.
𝑨 = 𝑷(𝟏 + 𝒊. 𝒏)
-where: A is the accumulated or total amount. In the case of an investment,
this is the amount invested plus the interest earned for the use of the money.
In the case of a loan, A is the total amount you have to pay back, with interest
included.
-𝑃 is the amount invested or borrowed.
-𝑖 is the simple interest rate per annum. Per annum means ‘per year’ and is
abbreviated as
𝑝. 𝑎.
-𝑛 is the number of years that teh money has been invested or borrowed.
Hire Purchase:
Is when a buyer purchases an item using monthly instalments and only owns
the item once it has been fully paid for. Hire purchase is always done using
Simple interest.
Commission
Commission is money that is paid to a seller for goods, services or property sold. The
commission charged is usually a percentage of the price of the item being sold.
Rentals
Rental is the money charged for the use of property or for the hire of vehicles or goods.
Compound interest:
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Whole Numbers
The interest paid on most savings and investments is not simple interest. It is a more
complicated type of interest. In the case of compound interest the interest is added to the
capital at the end of each period. The following formula can be used
𝑟
𝐴 = 𝑃(1 + 100)𝑛
𝑟
𝐴 = 𝑃(1 + 𝑖)𝑛 𝑤ℎ𝑒𝑟𝑒 𝑖 = 100
-𝐴 is the future value of the money invested or borrowed, or the total accumulated value
which includes P and the incurred amount.
-𝑃 is the present value or the initial amount invested or borrowed and is also called the
capital or the principal amount
-𝑖 is the interest rate per annum (it must be converted to a fraction
-𝑟 is the interest rate.
-𝑛 is the number of years for which the money has been invested or borrowed.
Exercise 4
1.
Mr. Ranchod invests R25 500 in a scheme where compound interest is calculated at
8% p.a. He invests this for 3 years. How much compound interest does he receive?
2.
If Mr Ranchod had to invest this amount using simple interest, what would his simple
interest be?
3.
A father is 30 years older than his son. Three years ago he was four times as old as his
son. How old is the father?
4.
Six less han eight times a certain number is equal to 50. What is this number?
5.
Mrs Maseko borrows R8 200 at 14% p.a. simple interest to pay a deposit on a car. She
wins the Lotto and pays the money back after 6 months. What is her total repayment
amount?
6.
The length of a rectangle is 8 cm linger than its breadth. The perimeter is 40 cm. What
is the area of the rectangle?
7.
Nosipho borrows R2 400 from a money lender to buy a new bed. The loan is for 3
years at an interest rate of 18,5% p.a. simple interest. What is the total amount
Nosipho has to pay back.
8.
A cricketer scores 65 runs consisting of fours, sixes, and single runs. He hits an equal
amount of fours and sixes and three times as many singles as fours. How many of each
did he hit?
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Grade 9
Whole Numbers
9.
I buy a car on hire-purchase agreement. The car costs R56 000. I pay a 25% deposit.
The remaining balance is paid off at 12% p.a. simple interest over a period of 4 years.
The amount has to be paid back in 48 equals monthly instalment. What is the monthly
instalment?
10.
Tumi invests R10 000 for one month at 7 2 % simple interest p.a., how much interest
1
would he earn?
11.
1
If Tumi invests R10 000 for 5 years at 7 2 % simplest interest p.a., how much interest
would he earn?
12.
A farmer produces 1200 bags of potatoes at the end of a certain year. He plans to
increase his production by 10% per annum from then on. How many bags of potatoes
will he be producing after 3 years? Round off your answer to the nearest bag.
13.
Sipho wishes to purchase a new car costing R 225 000. His father will lend him all the
money at 12% simple interest p.a. for 5 years. The bank will lend him all the money at
16% compound interest p.a. for 3 years. Which would be the better deal?
14.
Calculate the compound interest on R17 500 in the following cases:
15.
16.
14.1
at 12,5% for 3 years.
14.2
at 20,5 % for 3 years.
14.3
at 17,75 % for 4 years.
Nikki moves into her new flat and buys R 7 500 worth of furniture. The company
charges finance rates of 20 % p.a. for 3 years at simple interest
15.1
Calculate the interest over 3 years.
15.2
Determine the monthly instalments if she is to pay if all off in 36 months.
15.3
What would she pay back in total if the finance charges were 20% p.a.
compound interest.
Which would be a better investment over 3 years: R25 000 invested at 16% p.a.
compound interest, or R 25 000 invested at 17,5% p.a. simple interest.
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