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Multiplication of Fraction
1. Convert all mixed numbers to improper fractions first.
2. Simply the fractions by ‘cancelling’ to see if anything will divide into any of the top numbers and also the bottom numbers.
3. Multiply the numerators, the the denominators together.
Example 1
8/9 x 3/4
= 2/3 x 1/1
= 2/3
Example 2
1 2/5 x 2/3
= 7/5 x 2/3
= 14/15
Example 3
2 1/3 x 3 1/5
= 5/2 x 16/5
= 1/1 x 8/1
= 8
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Decimal Places and Significant Figures
To round off a decimal
1. Include one extra digit for consideration.
2. Drop the extra digit if it is less than 5
3. If is is 5 or more, add 1 to the previous number before dropping the extra digit
Example
Express 2.8547
a) correct to 3 decimal place
2.855
b) correct to 2 decimal place
2.85
c) correct to 1 decimal place
2.9
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Significant figures
Example 1
2093 = 2000 (1 sf)
2093 = 2100 (2 sf)
Example 2
5.045 = 5 (1 sf)
5.045 = 5.0 (2 sf)
Example 3
0.0854 = 0.09 (1 sf)
0.0584 = 0.085 (2 sf)
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Algebra
Expansion of Algebraic Expressions
1. a(b+c) = ab + ac
2. a(b-c) = ab – ac
3. (a+b)^2 = a^2 + 2ab + b^2
4. (a-b)^2 = a^2 – 2ab + b^2
5. (a+b)(a-b) = a^2 – b^2
6. (a+b)(c+d) = ac + ad + bc + bd
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Method for solving simultaneous equations
1. Substitution Method
2. Elimination Method
3. Graphing Method
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Properties of Inequilities
Property 1
If a > b
a+k > b+k where k > 0
Example
7 > 5
7+2 > 5+2
9 > 7
Property 2
If a > b
a-k > b-k where k > 0
Example
7 5-2
5 > 3
Property 3
If a > b
a x m > b x m where m > 0
Example
If 8 > 6
8 x 3 > 6 x 3 where 3 > 0
24 > 18
Property 4
If a > b
a x n < b x n where n 6
8 x -5 > 6 x -5 where -5 > 0
-40 b
a/m > b/m where m > 0
Example
If 8 > 6
8/2 > 6/2 where 2 > 0
4 > 3
Property 6
If a > b
a/m < b/m where m 6
8/(-2) < 6/(-2) where -2 < 0
-4 < -3
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Rate, Ratio and Speed
1. A ratio is a comparison of like quantities measured in like units or quantities of the same kind. A ratio has no units.
2. If the number of children in a school is increased from 40 to 50, then the ratio of the number of present children to the number of previous children
= 50 : 40
=5 : 4
3. We say the number of children has been increased in the
ratio 5 : 4
4. For example,
A shopkeeper sold 72 t-shirts yesterday. The number of t-shirts he sold today decreased to 54. We say that the ratio of the number of t-shirts he sold had been decreased in the ratio 3 : 4. (Note : 54 : 72 = 3 : 4)
5. A rate expresses a relationship between two quantities of different kinds.
6. Average speed of a moving body is given by the formula:
Average speed = Total distance travelled /Total time taken
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Primes, Factors and Multiples
1. A prime number is a number which has only two different factors, 1 and the number itself.
For example, 2, 3, 5, 7, 11, 13 are prime numbers.
Note that “1” is not a prime number.
2. A number which has more than two different factors is a composite number.
For example, 4, 6, 8, 9, 10, 12 are composite numbers.
Note that “1” is not a composite number.
3. Factors of a number are whole numbers that multiply to give that number.
For example, 18 = 1 x 18
= 2 x 9
= 3 x 6
Hence, the factors of 18 are 1, 2, 3, 6 and 18.
4. The common factors of two or more numbers are the factors that the numbers have in common.
For example,
Factors of 18 = 1, 2, 3, 6, 18
Factors of 12 = 1, 2, 3, 4, 6, 12
Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
Hence, the common factors of 18, 12 and 24 are 1, 2, 3 and 6. The highest common factor (HCF) of 18, 12 and 24 is 6.
5. When a number is multiplied by a non – zero whole number, a multiple of the number is obtained.
For example,
Multiples of 4 : 4, 8, 12, 16, 20, 24, 28, 32 36, 40„
Multiples of 5 : 5, 10, 15, 20, 25, 30, 35, 40,
6. Common multiples of two numbers are numbers that are multiples of the two numbers.
For example,
The first two common multiples of 4 and 5 are 20 and 40.
It follows that the lowest common multiple (LCM) of 4 and 5 is 20.
7. The decomposition of a composite number into prime factors is known as prime factorisation.
For example,
60 = 2 x 2 x 3 x 5
8. A number, when multiplied by itself, gives the square of the number. For example, 72= 7 x 7 = 49
9. 1, 4, 9, 16, 25, 36 are squares of whole numbers. These numbers are called perfect squares.
10. The square root of a number, when multiplied by itself gives the number. For example, ‘144 = 12
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Integers, Rational Numbers and Real Numbers
1. Real numbers are made up of Rational and Irrational Numbers.
2. Irrational numbers are made up of non-recurring and non-terminating decimals.
Example 1/rt 2 , pie
3. Rational numbers are made up of fractions and integers.
4. Some examples of fractions include : 2/3 , 6/7
5. Integers.
Example 1, 5, -5, -8
Integers are made up of whole numbers and negative integers.
7. Negative integers are { —7, — 6, — 5, — 4, — 3, — 2}
8. Whole numbers are made up of positive integers and zero.
Whole numbers : {0,1,2,3,4,5,6,7}
9. Natural numbers are {1,2,3,4,5,6,7,8,}
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Approximation and Estimation
1. The following figures in a number are significant:
i. All non-zero figures
ii. All zeros between significant figures
iii. All zeros at the end of a decimal
2. The following figures in a number are not significant:
i. All zeros at the start of a decimal less than 1.
ii. All zeros at the end of a whole number may or may not be significant, depending on how the estimation is made
Time
1. 1 minute (min) = 60 seconds (s).
2. 1 hour (h) = 60 minutes (min)
Standard Form
1. A number in the standard form can be written as A x 10″, where A<10, and n is an integer
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Scales and Maps
1. If the linear scale of a map is 1 : x, it means that 1 cm on the map represents x cm on the actual piece of land.
2. The scale of a map can also be represented as a representative fraction (R.F.).
For example, the scale of 1 : 100 can also be expressed as 1/100
Similarly, if the R.F. is 1/50 , the scale will be 1: 50.
Note ; When we use R.F. , the numerator must always be 1.
3. From the scale of a map, we can also find the actual area of a site from its area on the map.
For example, if the scale of a map is 1cm to 5km,
then 1 cm2 on the map will represent (5km)^2 = 25km^2.
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Percentages
1. A percentage is a fraction whose denominator is 100 and we use % to represent percent.
2. In general, y percent is defined as y parts per hundred.
Y% = Y/100
3. A percentage is converted to a fraction or decimal by dividing its value by 100.
4. Final percentage x Original value = New value
5. Increase/Decrease = percent increase/decrease x original value
6. % Profit = Profit/(CostPrice) x 100%
% Loss = Loss/(CostPrice) x 100%
7. Discount = Original Selling Price – Sale Price
Percentage Discount = (Discount/Market Price) x 100%
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Simplification of Algebraic Expressions
1. In algebra, symbols like x, y, xy, z2 are used to represent numbers and variables.
2. We add or subtract the like terms by adding or subtracting the coefficients. For example:
3a + 5a – 7a = a
3. We do not add or subtract the coefficients of unlike terms. Hence, adding 3a and 5b gives 3a + 5b.
4. If an expression within brackets is multiplied by a number, each term within the
brackets must be multiplied by that number when the brackets are removed. For example:
8 (x + 2y) = 8x + 16y
Simplify the following expressions:
i. 6a + 2a – 4a
ii. 4b + 7b – a – 2b
iii. 7ab – 3ab + 3bc – 5/4 cb 2
iv. 4x – (2x – 3x)
Ans
i. 4a
ii.9b-a
111. 11/2ab – 1/2bc
iv. 5x
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Substitution into Algebraic Expressions
If a = 2, b=5, c = -3, find the value of each of the following:
i. a^2 + b^3 – 2c^3
ii. (2a – b + 4c) (2b – c^2)
Solution i
a^2 + b^3 – 2c^3
= 2^2 + 5^3- 2(-3)^3
= 4 + 125 – 2(-27)
= 183
Solution ii
(2a – b + 4c) (2b – c^2)
= [(2 x 2) – 5 + 4 (- 3)] [2 x 5 – (-3) (-3)]
= (4 – 5 – 12) (10 – 9)
= (-13 ) (1)
= -13
Solving Linear Equations in One Unknown
To solve an equation, we can add or subtract the same number to each side.
We can also multiply or divide each side by the same number.
Example 1 :
a — 6 = 7
(a— 6)+6 = 7+ 6
Hence, a = 13
Example 2 :
a+ 3 = 16
(a+ 3)— 3 = 16 —3
a = 13
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Factorisation
1. Factorisation of algebraic expressions can be done by
i. taking out common factors from all terms in the given expressions
ii. grouping the terms such that the new terms obtained have a common factor
iii. using the “cross” method for quadratic expressions
2. If two factors A and B are such that A x B = 0, then, either A = 0, or B = 0, or both A and B = 0. This principal is used to find the unknown in quadratic equations.
Example 1
16a^2 – 9
= (4a)^2 – 3^2
= (4a + 3)(4a – 3)
Example 2
d^2-1+4d+4
= (d-1)(d+1)+ 4(d+1)
=(d+1)(d-1 +4)
=(d + 1 )(d + 3)
Example 3
4xy + 8y – 5x – 10
= 4y(x + 2) – 5(x + 2)
= (x + 2)(4y – 5)
Example 4
3a^2 – a -10
= (3a + 5) (a – 2)
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