PROPER AND IMPROPER FRACTIONS
A fraction is a portion of a whole that has been divided into equal parts.
A common fraction is written as ½ or ¼ or ¾.
The number at the top represents a whole number called the numerator and the number at the bottom represents a whole number called the denominator.
In proper fractions, the numerator of the fraction is smaller than the denominator.
In improper fractions, the numerator of the fraction is bigger than the denominator.
MIXED NUMBERS
Sometimes we write an improper fraction as a mixed number, for example:
We would write^{ 8}/_{5} as 1 ^{3}/_{5}
The mixed number has a whole number part and a fraction part.
CONVERTING FRACTIONS
To convert an improper fraction to a mixed number, simply divide the number by the denominator:
Example:
^{12}/_{5} = 12 ÷ 5 = 2 r 2
We write this as 2 ^{2}/_{5}
To convert a mixed number to an improper fraction, multiply the whole number by the denominator. Add the numerator to this. Write this answer as the numerator and keep the denominator the same.
Examples:
8 ½ = Multiply 8 by 2, and then add 1
This will give you a total of 17
The improper fraction will therefore be ^{17}/_{2 }
Equivalent fractions
SIMPLIFYING FRACTIONS
To simplify a fraction, you must reduce the fraction to its smallest form.
To do this, you need to divide both the numerator and the denominator by the same highest common factor.
Example:^{12}/_{30} = ∗
The highest number that can fit into both 12 and 30 is 6.
6 is therefore the highest common factor (HCF)
Divide the numerator and denominator by the highest common factor.
e.g. ^{12}/_{30}÷ ^{6}/_{6} = ^{2}/_{5}
SO: ^{12}/_{30}= ^{2}/_{5}
NB: A common fraction must always be written in the simplest form!
FRACTIONS OF QUANTITIES
When asked to work out a fraction of a quantity, use one of the following methods:
Method 1:
^{1}/_{10} of 30 (Bodmas Rule: “of “ becomes x)
= ^{1}/_{10} × ^{30}/_{1} (Multiply numerators, then denominators)
= ^{30}/_{10}(÷ ^{10}/_{10}) (Reduce answer to simplest form)
= ^{3}/_{1} = 3
Method 2:
^{1}/_{10} of 30 (÷by 10; x1)
= 3
GIVING PARTS OF QUANTITIES AS FRACTIONS
First change the amounts to the same unit of measurement.
Write both amounts as fractions.
Reduce the fraction to its simplest form.
Example: What fraction is 20c of R2?
R2 = 200c (Same unit of measurement)
= ^{20}/_{200} ÷ ^{20}/_{10} (Both amounts as fractions)
= ^{1}/_{10} (Simplest form)
ADDITION AND SUBTRACTION OF COMMON FRACTIONS
If the denominators are different, you must make them the same by finding the lowest common denominator.
Remember that when changing to the LCD, what you do to the bottom must be done to the top!
Also remember that you must always write your answer in the simplest form.
MULTIPLICATION OF FRACTIONS
If you are asked to multiply mixed numbers, first change these to improper fractions. Continue with the same method as before, ensuring that the answer is simplified.
Example:
^{3}/_{5} × ^{4}/_{1}
= ^{28}/_{5} × ^{17}/_{4}
= ^{119}/_{20}
= 23 ^{4}/_{5}
Exercise 18
Complete the following:
Exercise 19
Mixed Exercise:
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Profit is the surplus remaining after total costs are deducted from total revenue. Revenue means your income.
Profit can be calculated in different ways. Normally when we talk about a 10% profit, we calculate it on the cost price. We sometimes also refer to a 10% mark-up.
Example: If I sell a football which cost me R200,00 for R220,00, I made a 10% profit.
Loss is the excess of expenditure over income.
Discount is the amount deducted from the asking price before payment.
Remember that profit and loss do not only apply to businesses but also to your personal income.
Are you making a profit or a loss in these examples. How much profit or loss?
FINANCES - BUDGET
Do you know what a budget is? Can I have my own budget or is it only for adults?
Budget is the estimate of cost and revenues over a specific period.
Budget is like a scale where you try to balance your income and your expenses. Important: Your income should always outweigh your expenses.
Creating a budget is the most important step in controlling your money. The first rule of budgeting is: Spend less than you earn!
Example: If you received a R250,00 allowance (pocket money) per month and another R80 for your birthday, you cannot spend more than R330,00 for the entire month.
Net income is, like profit, the surplus remaining after all costs are deducted from total (gross) revenue. If the expenses exceed the income, we call it a shortage.
It is always a bright idea to SAVE for a RAINY day!
What is a loan? What is interest?
A loan is a sum of money that an individual or a company lends to an individual or a company with the objective of gaining profits from interest when the money is paid back.
Interest is the fee charged by a lender to a borrower for the use of borrowed money, usually expressed as an annual percentage of the amount borrowed, also called interest rate.
There are two kinds of interest: Simple and compound. Simple or flat rate interest is usually paid each year as a fixed percentage of the amount borrowed or lent at the start. With compound interest, you also pay interest on the interest!
The simple interest formula is as follows:
Interest = Principal x Rate x Time
Where:
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DECIMAL FRACTIONS
What is a decimal fraction?
If we have 9 units and we add 1 more, we now have a Ten.
Each place value on the left is 10 times bigger than the one on the right, e.g.
SO:
DECIMALS AND PLACE VALUE
The place value table can be represented as follows:
Place value is very important when working with decimals!
Exercise 21
Use the place value table (if you need to) to complete the following:
DECIMAL FRACTIONS AND ROUNDING OFF
Remember:
1st decimal place = tenths
2nd decimal place = hundredths
3rd decimal place = thousandths
WHOLE NUMBER = UNIT
The first digit to the right of decimal point is in the tenths place. The second digit to the right of decimal point is in the hundredths place. The third digit to the right of decimal point is in the thousandths place.
When rounding off a decimal, the rules for rounding off stay the same, i.e.
COMPARING AND ORDERING DECIMALS
When you compare decimal fractions, it is much easier to do so if the number of digits to the right of the decimal comma is the same in both decimal fractions.
We can always add 0’s to the right of the decimal fraction without changing its value.
For example, the number 2,367 = 2,36700
Examples:
Arrange the following decimal fractions in ascending order:
3,31; 3,301; 0,301; 3,4; 33,013; 3,41
Answer:
We first need to make sure that all the decimal numbers have the same number of digits to the right of the decimal number. We do this by adding zeros.
3,310; 3,301; 0,301; 3,400; 33,013; 3,410
We then look at the whole numbers and arrange these in ascending order, not worrying about the fractional part yet.
0,301; 2,210; 3,301; 3,400; 3,410; 33,013
Then we arrange the numbers with equal whole number parts in ascending order by looking at the fractional part of the decimal fractions and comparing them.
0,301; 3,301; 3,310; 3,400; 3,410; 33,013
CONVERTING TO COMMON FRACTIONS AND PERCENTAGES DENOMINATORS OF 10, 100 OR 1000
If you can, simply change the denominator to 100. What you do to the bottom, also do to the top.
E.g. ^{17}/_{50} = * 3^{1}/_{25} = *
^{17}/_{50} × ^{2}/_{2} = ^{34}/_{100} =^{76}/_{25} × ^{4}/_{4} = ^{304}/_{100}
= 34% = 304%
= 0,34 = 3,04
If the denominator cannot be changed to 100, simply multiply by ^{100}/_{1}
E.g. ^{19}/_{30}= *
= ^{19}/_{30} × ^{100}/_{1}
= ^{190}/_{3}
= 63,3
∴^{19}/_{30}= ^{63,3}/_{100}=63,3%
= 0,63
ADDITION AND SUBTRACTION OF DECIMALS
When adding or subtracting decimals, remember the following:
Example 1: Example 2:
142,7 + 6,395 + 12,42 15,8 – 2,345
142,700 15,800
6,395 - 2,345
+12,420 13,455
161,515
MULTIPLICATION OF DECIMALS- HORIZONTAL MULTIPLICATION
This is a mental process that can be carried out without showing the method.
This should be used for basic probems only.
Example:
6 x 0,02 → Ask what 6 x 2 is. Write the answer of 12, then count
= 12 how many spaces there are after the comma.
= 0,12 Insert the comma in the answer
Other examples:
0,7 x 0,3 = 0,21 1,5 x 0,3 = 0,45
0,08 x 0,2 = 0,016 0,004 x 0,003 = 0,000012
Can you see how we arrived at these answers? Discuss this is class.
Example 1: Example 2:
483,2 x 7 13,5 x 2,4
483,2 (1 place after the 13,5 (1 place after the comma
x 7 comma) x2,4 comma + 1 place comma)
3382,4 (1 place after the comma) 540 (leave out the comma)
+2700
32,40 insert comma after 2 numbers.
MULTIPLYING BY 10, 100 AND 1000
Study the examples below:
0,6 x 10 = 6 0,23 x 1000 = 230
0,145 x 100 = 14,5 0,002 x 104 = 20
You should see that when you multiply by 10, 100 or 1000 to make the number bigger, the number of times the comma “moves” is in direct relation to the number of zeroes there are in the number you are multiplying by:
i.e.
DIVISION OF DECIMALS. HORIZONTAL (SHORT) DIVISION
This mental process can be carried out without showing the method.
This should be used for basic problems only.
Example:
85,635 ÷ 9 = 9,515
Dividing by 10, 100 OR 1000
Study the examples below:
21,795 ÷ 10 = 2,1795
469,837 ÷ 1000 = 0,469837
3,46 ÷ 100 = 0,0346
78 346,27 ÷ 104 = 7,834627
You will notice that when dividing by 10, 100 or 1000 to make the number smaller, the number of times the comma “moves” is linked to the number of zeroes in the number you are diving by.
Dividing by multiples of 10, 100 OR 1000
When you multiplied by numbers of 10, 100 or 1000, you did the following:
71,246 x 30
= 71,246 x 10 x 3
= 712,46 x 3
= 2137,38
When you divide by multiples of 10, 100 or 1000, you follow the same procedure. However, this time you need to replace the X signs with ÷ signs because you are doing a division sum.
Example:
493,64 ÷ 700
= 496,64 ÷ 100 ÷ 7
= 4,9664 ÷ 7
= 0,705
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RATIO
RATE
A rate is used to compare the sizes of two or more quantities that use different units of measurement, e.g., hours (h), minutes (m), Rands (R), millimeters (mm), centimeters (cm), etc.
SHARING A “WHOLE” IN A GIVEN RATIO
Share R 2 250,00 in the ratio 3:2:1
CALCULATING PERCENTAGE INCREASE AND DECREASE
When increasing or decreasing a number by a given percentage, write the percentage out of 100 and multiply it by the given number.
Example: Increase R 1 500 by 25%
= ^{25}/_{100} x 1500 *Simplify / Cancel if possible
= R 375
Now add this amount to the original value:
i.e. R1500+R375
= R 1 875
Example: Decrease R 3 000 by 45%
= ^{45}/_{100} x 3000
= R 1 350
Decreased amount: R3000–R1350
= R1650
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A factor is a number that divides exactly into a whole number without any remainders. F10 = {1; 2; 5; 10}
A prime number has only 2 factors: 1 and itself. The number 2 is the first prime number. We say that 2 x 1 = 2. The number 2 is the only even prime number as all other numbers have more than two factors. The numbers 2; 3; 5; 7 and 11 are examples of prime numbers because they have only two factors, the number itself and 1.
e.g. F3 = {1; 3}
3 is therefore a prime number.
A multiple is the product of two natural numbers. For example, 24 is multiple of 8 and 3 because 8 x 3 =24. The number 24 is also a multiple of 12 and 2 because 12 x 2 = 24.
Multiplication Tables: M
Multiplication Tables. M7 = {7; 14; 21; 28 ….}
A composite number has more than 2 factors.
e.g. F20 = {1; 2; 4; 5; 10; 20}
Examples: Multiplication
Calculate 2310 x 35
Answers:
2310 x 35 = 2310 x (30 + 5)
= (231 x 30) + (2310 x 5) Distributive law
= (2310 x 5) + (2310 x 30) Commutative law
= 11550 =69 300
= 80850
In columns, it looks like this:
2310
x 35
11 550 2310 x 5, multiply by units
+ 69 300 2310 x 30, multiply by tens
80 850 Add the two products together
Division
When we divide large numbers, we use a method called long division.
Example:
453 This number is the answer
321√145413
1284 321 x 4 = 1284
1701 Subtract 1284 from 1454 and bring down the 1
1605 321 x 5 = 1605
963 Subtract 1605 from 1701 and bring down the 3
963 321 x 3 = 963
0 Subtract 963 from 963
Do these calculations. Show your method. Not just an answer.
The HCF and LCM are numbers that share the same factors. These are called common factors and you can find the highest common factor, HCF, of two or more numbers.
You can also find the lowest common multiple, LCM, of two or more numbers.
Example:
The multiples of 12 are 12 ; 24; 36; 48; 60; 72 ; 84; … and the multiples of 15 are 15; 30; 45; 60; 75; 90; …which means that the LCM of 12 and 15 is 60.
You can use the prime factor method for finding the LCM (or an HCF).
For example:
12 = 2 x 2 x 3 and 15 = 3 x 5 so the LCM is 2 x 2 x 3 x 5 which contains all possible prime factors of both numbers. The HCF of 12 and 15 is 3 as that is the highest factor common to both numbers.
Use prime factors to write numbers in the Factor tree method
Use prime factors to write numbers in exponential form (Ladder method)
72 | |
2 | 72 |
2 | 36 |
2 | 18 |
3 | 9 |
3 | 3 |
1 |
72 = 2 x 2 x 2 x 3 x 3
= 2^{3} x 3^{2}
Write in exponential form using only prime numbers as bases. (Ladder Method)
a | 6 and 9: | |
b | 14 and 18: | |
c | 30 and 24: | |
d | 15 and 10: |
a | 5 and 3: | |
b | 9 and 6: | |
c | 8 and 10: | |
d | 12 and 9: | |
e | 15 and 20: |
BODMAS stands for Brackets, Of, Division, Multiplication, Addition, and Subtraction.
BODMAS is the order of operation of a mathematical expression.
BODMAS is an acronym to remember the order of mathematical operations – the correct order in which to solve Mathematics problems.
Complete the sums below using BODMAS/BOMDAS.
No CALCULATORS are allowed. Show all working out.
PROBLEM SOLVING: BRAIN TEASER
Two lighthouse beacons can be seen from the top of a hill. These two beacons start flashing at the same time. One beacon flashes every 4 minutes and the other flashes every 9 minutes.
Calculate how long it will be before they both flash at the same time again. Use your 4 x and 9 x table to calculate.
There are special rules that apply to the number zero and the number one.
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All the positive numbers 1; 2; 3; 4; … are called the set of natural numbers. If we include 0 in the set of natural numbers, we get the set of counting numbers or whole numbers. We use numbers to add, subtract, multiply and divide. We can also write numbers in a particular order, from largest to smallest, e.g., 124; 1124; 5124; 9124. When we need to estimate, we can round off numbers to the nearest 5, 10, 100 or 1000. Whole numbers – or counting numbers are the numbers, 0; 1; 2; 3; 4; … and are represented by the symbol Nₒ.
Natural numbers – are whole numbers greater than or equal to 1: (1; 2; 3; 4; …) and are represented by the symbol N.
Rounding off to the nearest 5:
Look at the last digit of the number (the units digit) and round the number off to the closest number that 5 divides into.
1; 2 – “Move back to number ending in 0”
3; 4 – “Move forward to the number ending in 5”
6; 7 - “Move back to number ending in 5”
8; 9 - “Move forward to the number ending in 0”
Round off a number to the nearest 10:
When rounding off to the nearest 10, look at the units- digit.
Underline the Tens digit - 586
Look at the digit to the RIGHT of the Tens digit - 586
If this digit is 0, 1, 2, 3, or 4, the Tens stay the same. This is called rounding down. If this digit is 5, 6, 7, 8 or 9, round up. This is called rounding up.
586 rounded to the nearest 10 is 590.
We use the same method to round off to 100 (look at the tens digit) and 1000 (look at the hundreds digit)
For example: 465 784 rounded off to the nearest 10 is 465 780.
465 784 rounded to the nearest 100 is 465 800.
465 784 rounded to the nearest 1000 is 466 000.
Try this:
Round off 987 516 to:
Adding numbers is called finding the sum, and subtracting numbers is called finding the difference. Multiplying numbers is called finding the product and dividing numbers is called finding the quotient.
When you add or multiply numbers, the order of the numbers does not matter, for example: 4 + 5 = 5 + 4 and 4 x 5 = 5 x 4. This is called the commutative property of addition and multiplication.
The order in which you add or multiply numbers also does not matter, for example: (4+5) +6 = 4 + (5+6) and (4x5) x6 = 4 x (5x6). This is called the associative property of addition and multiplication.
When numbers in brackets are multiplied by a number in front of the brackets, each number inside the brackets is affected. This property of numbers works for addition and subtraction, for example: 4(5 + 6) = (4 x 5) + (4 x 6) or 6(5 – 4) = (6 x 5) - (6 x 4). This property is called the distributive property of multiplication.
Distributive property
What is the answer to 2(4 + 3)?
The "2" outside the brackets is multiplied onto everything that is inside the brackets.
Addition and subtraction are called inverse operations. If you add and subtract the same amount from a number, you end up back where you started. These operations have an effect on each other, for example: 856 + 12 – 12 = 856.
Multiplication and division are called inverse operations. If you multiply and divide a number by the same amount, you end up back where you started as the operations have an inverse effect on each other, for example: 524 x 12 ÷ 12 = 524.