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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} } } {}

Questions & Answers

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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