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  4. To differentiate between rational

To differentiate between rational and irrational numbers  (Page 3/3)

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2. What is of cardinal importance before attempting to add or subtract fractions?

3. Show whether you are able to do the following:

3.1 8 - 4 3 7 size 12{ { { size 8{3} } over { size 8{7} } } } {}

3.2 3 1 9 size 12{ { { size 8{1} } over { size 8{9} } } } {} - 1 1 2 size 12{ { { size 8{1} } over { size 8{2} } } } {}

  • Note this : The denominators must be similar when you add fractions together or subtract them from one another.

e.g. 2 4 7 size 12{ { { size 8{4} } over { size 8{7} } } } {} - 1 6 7 size 12{ { { size 8{6} } over { size 8{7} } } } {}

2 – 1 = 1 and

4 7 size 12{ { { size 8{4} } over { size 8{7} } } } {} - 6 7 size 12{ { { size 8{6} } over { size 8{7} } } } {}

( 4 – 6 --- this is not possible. Carry one whole: 1 = 7 7 size 12{ { { size 8{7} } over { size 8{7} } } } {} )

( 4 + 7 = 11 --- yes, 11 – 6 = 5)

Answer: 5 7 size 12{ { { size 8{5} } over { size 8{7} } } } {}

  • You could also reduce compound numbers to improper fractions and make the denominators similar.
  • e.g.. 18 7 − 13 7 = 5 7 size 12{ { { size 8{"18"} } over { size 8{7} } } - { { size 8{"13"} } over { size 8{7} } } = { { size 8{5} } over { size 8{7} } } } {} (18 – 13 = 5: The denominators are the same. Subtract one numerator from the other.)

4. Do the following:

4.1 4 1 7 size 12{ { { size 8{1} } over { size 8{7} } } } {} + 4 16 42 size 12{ { { size 8{"16"} } over { size 8{"42"} } } } {}

4.2 36 － 15 6 11 size 12{ { { size 8{6} } over { size 8{"11"} } } } {}

4.3 1 8 + 0, 625 − 3 8 size 12{ { { size 8{1} } over { size 8{8} } } +0,"625" - { { size 8{3} } over { size 8{8} } } } {}

4.4 4 5 10 + 7 1 2 + 6 3 4 size 12{4 { { size 8{5} } over { size 8{"10"} } } +7 { { size 8{1} } over { size 8{2} } } +6 { { size 8{3} } over { size 8{4} } } } {}

4.5 7 1 3 size 12{ { { size 8{1} } over { size 8{3} } } } {} - 4 7 8 size 12{ { { size 8{7} } over { size 8{8} } } } {}

4.6 7 a - a 4 size 12{ { { size 8{a} } over { size 8{4} } } } {} a / 4

4.7 9 a + 6 ab − 3 b size 12{ { { size 8{9} } over { size 8{a} } } + left ( { { size 8{6} } over { size 8{ ital "ab"} } } - { { size 8{3} } over { size 8{b} } } right )} {}

4.8 - 6 + 2 6 7 size 12{ { { size 8{6} } over { size 8{7} } } } {}

4.9 5 - (4 4 9 size 12{ { { size 8{4} } over { size 8{9} } } } {} + 2 2 3 size 12{ { { size 8{2} } over { size 8{3} } } } {} )

4.10 3 1 3 size 12{ { { size 8{1} } over { size 8{3} } } } {} a - 2 1 2 size 12{ { { size 8{1} } over { size 8{2} } } } {} a

Activity 1.5

Multiplication and division of rational numbers (fractions)

[lo 1.2.6, 1.6.2]

  • You did this in grade 7 – let's refresh the memory.

1. Multiplication:

  • Important : Write all compound numbers as fractions.Then do crosswise cancellation.

Try the following:

  • 1 1 4 size 12{ { { size 8{1} } over { size 8{4} } } } {} × 2 1 2 size 12{ { { size 8{1} } over { size 8{2} } } } {} × 4

2. Division:

  • The reciprocal plays an important role in the division of fractions.

Use an example to explain this term.

e.g. 1 3 ÷ 2 3 size 12{ { { size 8{1} } over { size 8{3} } } div { { size 8{2} } over { size 8{3} } } } {}

  • Both numbers are fractions
  • Change ÷ to the × sign and obtain the reciprocal of the denominator (fraction following the ÷ sign).
  • Do cancellation as with multiplication.

3. Do the following:

3.1 8 ÷ 8 11 size 12{ { { size 8{8} } over { size 8{"11"} } } } {}

3.2 18 ÷ 7 8 size 12{ { { size 8{7} } over { size 8{8} } } } {}

3.3 5 6 ÷ 5 2 size 12{ { { size 8{5} } over { size 8{6} } } div { { size 8{5} } over { size 8{2} } } } {}

3.4 -2 2 3 size 12{ { { size 8{2} } over { size 8{3} } } } {} ÷ -1 7 9 size 12{ { { size 8{7} } over { size 8{9} } } } {}

3.5 6 3 4 size 12{ { { size 8{3} } over { size 8{4} } } } {} mn ÷ -6 m 3

3.6 − 4 xy 3 ab ÷ − 2x 3a size 12{ { { size 8{ - 4 ital "xy"} } over { size 8{3 ital "ab"} } } div { { size 8{ - 2x} } over { size 8{3a} } } } {} -

Assessment

Learning outcomes(LOs)
LO 1
Numbers, Operations and RelationshipsThe learner will be able to recognise, describe and represent numbers and their relationships, and to count, estimate, calculate and check with competence and confidence in solving problems.
Assessment standards(ASs)
We know this when the learner:
1.2 recognises, classifies an represents the following numbers to describe and compare them:
1.2.2 decimals, fractions and percentages;
1.2.5 additive and multiplicative inverses;
1.2.6 multiples and factors;
1.2.7 irrational numbers in the context of measure­ment (e.g. π size 12{π} {} and square and cube roots of non-perfect squares and cubes);
1.3 recognises and uses equivalent forms of the rational numbers listed above;
1.6 estimates and calculates by selecting and using operations appropriate to solving problems that involve:
1.6.1 rounding off;
1.6.2 multiple operations with rational numbers (including division with fractions and decimals);
1.7 uses a range of techniques to perform calculations, including:
1.7.1 using the commutative, associative and distributive properties with rational numbers;
1.7.2 using a calculator;
1.9 recognises, describes and uses:
1.9.1 algorithms for finding equivalent fractions;
1.9.2 the commutative, associative and distributive properties with rational numbers (the expecta­tion is that learners should be able to use these properties and not necessarily to know the names of the properties).

Memorandum

ACTIVITY 1

1. Natural numbers

Counting numbers

Integers

Real numbers

2. a b size 12{ { {a} over {b} } } {} ; b ≠ 0

2 size 12{ sqrt {2} } {}

3.1 Q

  • Q 1

4.

size 12{ {2} wideslash {7} } {} 0 1 size 12{ sqrt {1} } {} 3 size 12{ sqrt {3} } {} 9 3 size 12{ nroot { size 8{3} } {9} } {} 8 3 size 12{ nroot { size 8{3} } {8} } {} 2,47 1, 45 size 12{ sqrt {1,"45"} } {} size 12{ sqrt { {4} wideslash {8} } } {} size 12{ sqrt { {"16"} wideslash { sqrt {9} } } } {}
Rational √ √ √ √ √ √
Irrational √ √ √ √
  • 1 + 4 size 12{ sqrt {4} } {} ; -4
  • − 2 3 size 12{ { { - 2} over {3} } } {} ; 12 1 5 size 12{ { {1} over {5} } } {}
  • 9 + 4 size 12{ sqrt {9+4} } {} ; 1 + 2 2 size 12{ { {1+ sqrt {2} } over { sqrt {2} } } } {}

6. Equal in value

7. 4 14 size 12{ { {4} over {"14"} } } {} = 6 24 size 12{ { {6} over {"24"} } } {} etc

  • Proper fraction
  • Inproper fraction
  • Mixed number
  • Decimal number
  • Recurring decimal number
  • Percentage

ACTIVITY 2

1. 2,15

  • 0,625
  • 3,25
  • 5,75
  • 2,875
  • 6, 000 7 size 12{ { {6,"000"} over {7} } } {} = 0,8571 . . . ≈ 0,86
  • 7, 000 9 size 12{ { {7,"000"} over {9} } } {} = 0,777 . . . = 0, 7 size 12{ {7} cSup { size 8{ cdot } } } {} or 0,8
  • 6 8 1000 size 12{ { {8} over {"1000"} } } {} = 6 1 125 size 12{ { {1} over {"125"} } } {}
  • 4 65 100 size 12{ { {"65"} over {"100"} } } {} = 4 13 20 size 12{ { {"13"} over {"20"} } } {}
  • 375 1000 size 12{ { {"375"} over {"1000"} } } {} = 3 8 size 12{ { {3} over {8} } } {}
  • 7 75 1000 size 12{ { {"75"} over {"1000"} } } {} = 7 3 40 size 12{ { {3} over {"40"} } } {}
  • 13 65 100 size 12{ { {"65"} over {"100"} } } {} = 13 13 20 size 12{ { {"13"} over {"20"} } } {}
  • 125 1000 size 12{ { {"125"} over {"1000"} } } {} = 1 8 size 12{ { {1} over {8} } } {}

5.1 7, 000 9 size 12{ { {7,"000"} over {9} } } {} = 0, 7 size 12{ {7} cSup { size 8{ cdot } } } {}

5.2 -5,8 3 size 12{ {3} cSup { size 8{ cdot } } } {} 5, 000 6 size 12{ { {5,"000"} over {6} } } {} = 0,8333 . . .

5.3 3, 1 size 12{ {1} cSup { size 8{ cdot } } } {} 3 size 12{ {3} cSup { size 8{ cdot } } } {} 13 , 0000 99 size 12{ { {"13","0000"} over {"99"} } } {} = 0,1313 . . .

7.1 3 9 size 12{ { {3} over {9} } } {} = 1 3 size 12{ { {1} over {3} } } {}

7.2 45 99 size 12{ { {"45"} over {"99"} } } {} = 5 11 size 12{ { {5} over {"11"} } } {}

7.3 23 990 size 12{ { {"23"} over {"990"} } } {}

7.4 3 900 size 12{ { {3} over {"900"} } } {} = 1 300 size 12{ { {1} over {"300"} } } {}

9. 0, 4 size 12{ {4} cSup { size 8{ cdot } } } {} 5 size 12{ {5} cSup { size 8{ cdot } } } {} = x

x = 0,4545 . . . 

100 x = 45,4545 . . .

  • –  99 x = 45

x = 45 99 size 12{ { {"45"} over {"99"} } } {} = 5 11 size 12{ { {5} over {"11"} } } {}

ACTIVITY 3

2.1 17 x5 20 x5 size 12{ { {"17"x5} over {"20"x5} } } {} = 85%

2.2 19 40 size 12{ { {"19"} over {"40"} } } {} x 100 1 size 12{ { {"100"%} over {1} } } {} = 47,5%

2.3 38 x2 50 x2 size 12{ { {"38"x2} over {"50"x2} } } {} = 76%

2.4 45 60 size 12{ { {"45"} over {"60"} } } {} x 100 1 size 12{ { {"100"%} over {1} } } {} = 75%

3.1 55 100 size 12{ { {"55"} over {"100"} } } {} = 11 20 size 12{ { {"11"} over {"20"} } } {}

3.2 15 , 5 100 size 12{ { {"15",5} over {"100"} } } {} = 0,155 = 155 1000 size 12{ { {"155"} over {"1000"} } } {} = 31 200 size 12{ { {"31"} over {"200"} } } {}

3.3 33 200 size 12{ { {"33"} over {"200"} } } {}

3.4 2 0 30 { 0 size 12{ { {2 { {0}}} over {"30 {"{0}}} } } {} = 2 30 size 12{ { {2} over {"30"} } } {}

4.a) 33 800 size 12{ { {"33"} over {"800"} } } {} x 25500 1 size 12{ { {"25500"} over {1} } } {} size 12{ approx } {} 1 052

b) 3 5 size 12{ { {3} over {5} } } {} x 25500 1 size 12{ { {"25500"} over {1} } } {} = 15 300

c) 85 1000 size 12{ { {"85"} over {"1000"} } } {} x 25500 1 size 12{ { {"25500"} over {1} } } {} = 2 167,5 size 12{ approx } {} 2 168

  • (14,5) 15300 1052 size 12{ { {"15300"} over {"1052"} } } {} = 7650 526 size 12{ { {"7650"} over {"526"} } } {} = 3825 263 size 12{ { {"3825"} over {"263"} } } {}
  • 25 500 – 18 520 = 6 980

4.4

4.5 3 5 size 12{ { {3} over {5} } } {} x 2 1 size 12{ { {2} over {1} } } {} = 6 5 size 12{ { {6} over {5} } } {} = 1 1 5 size 12{1 { {1} over {5} } } {}

ACTIVITY 4

1.1 39 7 size 12{ { {"39"} over {7} } } {}

1.2 70 9 size 12{ { {"70"} over {9} } } {}

2. Numbers must be the same

3.1 3 4 7 size 12{3 { {4} over {7} } } {}

3.2 2 2 − 9 18 size 12{2 { {2 - 9} over {"18"} } } {} = 1 20 − 9 18 size 12{1 { {"20" - 9} over {"18"} } } {} = 1 11 18 size 12{1 { {"11"} over {"18"} } } {}

4.1 29 7 size 12{ { {"29"} over {7} } } {} + 184 42 size 12{ { {"184"} over {"42"} } } {} = 174 + 184 42 size 12{ { {"174"+"184"} over {"42"} } } {} = 358 42 size 12{ { {"358"} over {"42"} } } {} = 8 22 42 size 12{8 { {"22"} over {"42"} } } {} = 8 11 21 size 12{8 { {"11"} over {"21"} } } {}

4.2 21 - 6 11 size 12{ { {6} over {"11"} } } {} = 20 5 11 size 12{"20" { {5} over {"11"} } } {}

  • 0,125 + 0,625 – 0,375 = 0,375
  • 17 10 + 10 + 15 20 size 12{"17" { {"10"+"10"+"15"} over {"20"} } } {} = 17 35 20 size 12{"17" { {"35"} over {"20"} } } {} = 18 15 20 size 12{"18" { {"15"} over {"20"} } } {} = 18 3 4 size 12{"18" { {3} over {4} } } {}
  • 3 3 − 21 24 size 12{3 { {3 - "21"} over {"24"} } } {} = 2 11 24 size 12{2 { {"11"} over {"24"} } } {}
  • {} 28 a 2 − a 4 size 12{ { {"28"`a rSup { size 8{2} } - a} over {4} } } {}

4.7+ ( 6 − 3a ab ) size 12{\( { {6 - 3a} over { ital "ab"} } \)} {} = 9b + 6 − 3a ab size 12{ { {9b+6 - 3a} over { ital "ab"} } } {}

4.8 − 6 1 size 12{ { { - 6} over {1} } } {} + 20 7 size 12{ { {"20"} over {7} } } {} = − 42 + 20 7 size 12{ { { - "42"+"20"} over {7} } } {} = − 22 7 size 12{ { { - "22"} over {7} } } {} = − 3 1 7 size 12{ - 3 { {1} over {7} } } {}

  • 5 – 6 4 + 6 9 size 12{ left (6 { {4+6} over {9} } right )} {} = 5 – 6 10 9 size 12{6 { {"10"} over {9} } } {} = 5 – 7 1 9 size 12{7 { {1} over {9} } } {}

=– 64 9 size 12{ { {"64"} over {9} } } {}

= 45 − 64 9 size 12{ { {"45" - "64"} over {9} } } {}

= − 19 9 size 12{ { { - "19"} over {9} } } {} = − 2 1 9 size 12{ - 2 { {1} over {9} } } {}

  • 10 a 3 size 12{ { {"10"a} over {3} } } {} – 5a 2 size 12{ { {5a} over {2} } } {} = 20 a − 15 a 6 size 12{ { {"20"a - "15"a} over {6} } } {}

= 5a 6 size 12{ { {5a} over {6} } } {}

ACTIVITY 5

1. 5 1 4 size 12{ { {5} over { {} rSub { size 8{1} } { {4}}} } } {} x 5 2 size 12{ { {5} over {2} } } {} x 4 1 1 size 12{ { { { {4}} rSup { size 8{1} } } over {1} } } {} = 25 2 size 12{ { {"25"} over {2} } } {} = 12 1 2 size 12{"12" { {1} over {2} } } {}

3.1 8 1 size 12{ { {8} over {1} } } {} ÷ 8 11 size 12{ { {8} over {"11"} } } {} = 8 1 1 size 12{ { { { {8}} rSup { size 8{1} } } over {1} } } {} x 11 8 1 size 12{ { {"11"} over { { {8}} rSub { size 8{1} } } } } {} = 11

3.2 18 1 size 12{ { {"18"} over {1} } } {} x 8 7 size 12{ { {8} over {7} } } {} = 144 7 size 12{ { {"144"} over {7} } } {} = 20 4 7 size 12{"20" { {4} over {7} } } {}

3.3 5 1 6 3 size 12{ { { { {5}} rSup { size 8{1} } } over { { {6}} rSub { size 8{3} } } } } {} x 2 1 5 1 size 12{ { { { {2}} rSup { size 8{1} } } over { { {5}} rSub { size 8{1} } } } } {} = 1 3 size 12{ { {1} over {3} } } {}

3.4 − 8 1 3 1 size 12{ { { - { {8}} rSup { size 8{1} } } over { { {3}}"" lSub { size 8{1} } } } } {} x − 9 3 1 6 2 size 12{ { { - { {9}} rSup { size 8{3} } } over { { {1}} { {6}} rSub { size 8{2} } } } } {} = 3 2 size 12{ { {3} over {2} } } {} = 1 1 2 size 12{1 { {1} over {2} } } {}

3.5 2 7 9 mn 4 size 12{ { { { {2}} { {7}} rSup { size 8{9} } ital "mn"} over {4} } } {} x 1 − 6 2 m 3 size 12{ { {1} over { - { {6}}"" lSub { size 8{2} } m rSup { size 8{3} } } } } {} = − 9n 8m 2 size 12{ { { - 9n} over {8m rSup { size 8{2} } } } } {}

3.6 − 4 2 xy 3 1 a b size 12{ { { - { {4}} rSup { size 8{2} } ital "xy"} over { { {3}}"" lSub { size 8{1} } { {a}}b} } } {} x 3 a − 2 x size 12{ { { { {3}} { {a}}} over { - { {2}} { {x}}} } } {} = 2y b size 12{ { {2y} over {b} } } {}

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Source:  OpenStax, Mathematics grade 8. OpenStax CNX. Sep 11, 2009 Download for free at http://cnx.org/content/col11034/1.1
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