# 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 $\frac{3}{7}$

3.2 3 $\frac{1}{9}$ - 1 $\frac{1}{2}$

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

e.g. 2 $\frac{4}{7}$ - 1 $\frac{6}{7}$

2 – 1 = 1 and

$\frac{4}{7}$ - $\frac{6}{7}$

( 4 – 6 --- this is not possible. Carry one whole: 1 = $\frac{7}{7}$ )

( 4 + 7 = 11 --- yes, 11 – 6 = 5)

Answer: $\frac{5}{7}$

• You could also reduce compound numbers to improper fractions and make the denominators similar.
• e.g.. $\frac{\text{18}}{7}-\frac{\text{13}}{7}=\frac{5}{7}$ (18 – 13 = 5: The denominators are the same. Subtract one numerator from the other.)

4. Do the following:

4.1 4 $\frac{1}{7}$ + 4 $\frac{\text{16}}{\text{42}}$

4.2 36 － 15 $\frac{6}{\text{11}}$

4.3 $\frac{1}{8}+0,\text{625}-\frac{3}{8}$

4.4 $4\frac{5}{\text{10}}+7\frac{1}{2}+6\frac{3}{4}$

4.5 7 $\frac{1}{3}$ - 4 $\frac{7}{8}$

4.6 7 a - $\frac{a}{4}$ a / 4

4.7 $\frac{9}{a}+\left(\frac{6}{\text{ab}}-\frac{3}{b}\right)$

4.8 - 6 + 2 $\frac{6}{7}$

4.9 5 - (4 $\frac{4}{9}$ + 2 $\frac{2}{3}$ )

4.10 3 $\frac{1}{3}$ a - 2 $\frac{1}{2}$ a

## [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 $\frac{1}{4}$ × 2 $\frac{1}{2}$ × 4

2. Division:

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

Use an example to explain this term.

e.g. $\frac{1}{3}÷\frac{2}{3}$

• 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 ÷ $\frac{8}{\text{11}}$

3.2 18 ÷ $\frac{7}{8}$

3.3 $\frac{5}{6}÷\frac{5}{2}$

3.4 -2 $\frac{2}{3}$ ÷ -1 $\frac{7}{9}$

3.5 6 $\frac{3}{4}$ mn ÷ -6 m 3

3.6 $\frac{-4\text{xy}}{3\text{ab}}÷\frac{-2x}{3a}$ -

## 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. $\pi$ 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. $\frac{a}{b}$ ; b ≠ 0

$\sqrt{2}$

3.1 Q

• Q 1

4.

  0 $\sqrt{1}$ $\sqrt{3}$ $\sqrt[3]{9}$ $\sqrt[3]{8}$ 2,47 $\sqrt{1,\text{45}}$ $\sqrt{}$ $\sqrt{}$ Rational √ √ √ √ √ √ Irrational √ √ √ √
• 1 + $\sqrt{4}$ ; -4
• $\frac{-2}{3}$ ; 12 $\frac{1}{5}$
• $\sqrt{9+4}$ ; $\frac{1+\sqrt{2}}{\sqrt{2}}$

6. Equal in value

7. $\frac{4}{\text{14}}$ = $\frac{6}{\text{24}}$ 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
• $\frac{6,\text{000}}{7}$ = 0,8571 . . . ≈ 0,86
• $\frac{7,\text{000}}{9}$ = 0,777 . . . = 0, $\stackrel{}{7}$ or 0,8
• 6 $\frac{8}{\text{1000}}$ = 6 $\frac{1}{\text{125}}$
• 4 $\frac{\text{65}}{\text{100}}$ = 4 $\frac{\text{13}}{\text{20}}$
• $\frac{\text{375}}{\text{1000}}$ = $\frac{3}{8}$
• 7 $\frac{\text{75}}{\text{1000}}$ = 7 $\frac{3}{\text{40}}$
• 13 $\frac{\text{65}}{\text{100}}$ = 13 $\frac{\text{13}}{\text{20}}$
• $\frac{\text{125}}{\text{1000}}$ = $\frac{1}{8}$

5.1 $\frac{7,\text{000}}{9}$ = 0, $\stackrel{}{7}$

5.2 -5,8 $\stackrel{}{3}$ $\frac{5,\text{000}}{6}$ = 0,8333 . . .

5.3 3, $\stackrel{}{1}$ $\stackrel{}{3}$ $\frac{\text{13},\text{0000}}{\text{99}}$ = 0,1313 . . .

7.1 $\frac{3}{9}$ = $\frac{1}{3}$

7.2 $\frac{\text{45}}{\text{99}}$ = $\frac{5}{\text{11}}$

7.3 $\frac{\text{23}}{\text{990}}$

7.4 $\frac{3}{\text{900}}$ = $\frac{1}{\text{300}}$

9. 0, $\stackrel{}{4}$ $\stackrel{}{5}$ = x

x = 0,4545 . . . 

100 x = 45,4545 . . .

• –  99 x = 45

x = $\frac{\text{45}}{\text{99}}$ = $\frac{5}{\text{11}}$

ACTIVITY 3

2.1 $\frac{\text{17}\mathrm{x5}}{\text{20}\mathrm{x5}}$ = 85%

2.2 $\frac{\text{19}}{\text{40}}$ x $\frac{\text{100}\text{}}{1}$ = 47,5%

2.3 $\frac{\text{38}\mathrm{x2}}{\text{50}\mathrm{x2}}$ = 76%

2.4 $\frac{\text{45}}{\text{60}}$ x $\frac{\text{100}\text{}}{1}$ = 75%

3.1 $\frac{\text{55}}{\text{100}}$ = $\frac{\text{11}}{\text{20}}$

3.2 $\frac{\text{15},5}{\text{100}}$ = 0,155 = $\frac{\text{155}}{\text{1000}}$ = $\frac{\text{31}}{\text{200}}$

3.3 $\frac{\text{33}}{\text{200}}$

3.4 $\frac{20}{\text{30 {}0}$ = $\frac{2}{\text{30}}$

4.a) $\frac{\text{33}}{\text{800}}$ x $\frac{\text{25500}}{1}$  1 052

b) $\frac{3}{5}$ x $\frac{\text{25500}}{1}$ = 15 300

c) $\frac{\text{85}}{\text{1000}}$ x $\frac{\text{25500}}{1}$ = 2 167,5  2 168

• (14,5) $\frac{\text{15300}}{\text{1052}}$ = $\frac{\text{7650}}{\text{526}}$ = $\frac{\text{3825}}{\text{263}}$
• 25 500 – 18 520 = 6 980

4.4

4.5 $\frac{3}{5}$ x $\frac{2}{1}$ = $\frac{6}{5}$ = $1\frac{1}{5}$

ACTIVITY 4

1.1 $\frac{\text{39}}{7}$

1.2 $\frac{\text{70}}{9}$

2. Numbers must be the same

3.1 $3\frac{4}{7}$

3.2 $2\frac{2-9}{\text{18}}$ = $1\frac{\text{20}-9}{\text{18}}$ = $1\frac{\text{11}}{\text{18}}$

4.1 $\frac{\text{29}}{7}$ + $\frac{\text{184}}{\text{42}}$ = $\frac{\text{174}+\text{184}}{\text{42}}$ = $\frac{\text{358}}{\text{42}}$ = $8\frac{\text{22}}{\text{42}}$ = $8\frac{\text{11}}{\text{21}}$

4.2 21 - $\frac{6}{\text{11}}$ = $\text{20}\frac{5}{\text{11}}$

• 0,125 + 0,625 – 0,375 = 0,375
• $\text{17}\frac{\text{10}+\text{10}+\text{15}}{\text{20}}$ = $\text{17}\frac{\text{35}}{\text{20}}$ = $\text{18}\frac{\text{15}}{\text{20}}$ = $\text{18}\frac{3}{4}$
• $3\frac{3-\text{21}}{\text{24}}$ = $2\frac{\text{11}}{\text{24}}$
•  $\frac{\text{28}{a}^{2}-a}{4}$

4.7+ $\left(\frac{6-3a}{\text{ab}}\right)$ = $\frac{9b+6-3a}{\text{ab}}$

4.8 $\frac{-6}{1}$ + $\frac{\text{20}}{7}$ = $\frac{-\text{42}+\text{20}}{7}$ = $\frac{-\text{22}}{7}$ = $-3\frac{1}{7}$

• 5 – $\left(6\frac{4+6}{9}\right)$ = 5 – $6\frac{\text{10}}{9}$ = 5 – $7\frac{1}{9}$

=– $\frac{\text{64}}{9}$

= $\frac{\text{45}-\text{64}}{9}$

= $\frac{-\text{19}}{9}$ = $-2\frac{1}{9}$

• $\frac{\text{10}a}{3}$ $\frac{5a}{2}$ = $\frac{\text{20}a-\text{15}a}{6}$

= $\frac{5a}{6}$

ACTIVITY 5

1. $\frac{5}{{}_{1}4}$ x $\frac{5}{2}$ x $\frac{{4}^{1}}{1}$ = $\frac{\text{25}}{2}$ = $\text{12}\frac{1}{2}$

3.1 $\frac{8}{1}$ ÷ $\frac{8}{\text{11}}$ = $\frac{{8}^{1}}{1}$ x $\frac{\text{11}}{{8}_{1}}$ = 11

3.2 $\frac{\text{18}}{1}$ x $\frac{8}{7}$ = $\frac{\text{144}}{7}$ = $\text{20}\frac{4}{7}$

3.3 $\frac{{5}^{1}}{{6}_{3}}$ x $\frac{{2}^{1}}{{5}_{1}}$ = $\frac{1}{3}$

3.4 $\frac{-{8}^{1}}{3{}_{1}\text{}}$ x $\frac{-{9}^{3}}{1{6}_{2}}$ = $\frac{3}{2}$ = $1\frac{1}{2}$

3.5 $\frac{2{7}^{9}\text{mn}}{4}$ x $\frac{1}{-6{}_{2}\text{}{m}^{3}}$ = $\frac{-9n}{{8m}^{2}}$

3.6 $\frac{-{4}^{2}\text{xy}}{3{}_{1}\text{}ab}$ x $\frac{3a}{-2x}$ = $\frac{2y}{b}$

how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
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I'm not sure why it wrote it the other way
Salomon
I got X =-6
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ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
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