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Fractions - 07

Module by: Siyavula Uploaders. E-mail the author

MATHEMATICS

Common fractions

EDUCATOR SECTION

Memorandum

18.1

ADDITION

1212 size 12{ { { size 8{1} } over { size 8{2} } } } {} + 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {} + 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {} + 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {} + 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}

1414 size 12{ { { size 8{1} } over { size 8{4} } } } {} + 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {} + 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {} + 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {} + 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {} + 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {}

3737 size 12{ { { size 8{3} } over { size 8{7} } } } {} + 3737 size 12{ { { size 8{3} } over { size 8{7} } } } {}

2323 size 12{ { { size 8{2} } over { size 8{3} } } } {} + 2323 size 12{ { { size 8{2} } over { size 8{3} } } } {} + 2323 size 12{ { { size 8{2} } over { size 8{3} } } } {}

2525 size 12{ { { size 8{2} } over { size 8{5} } } } {} + 2525 size 12{ { { size 8{2} } over { size 8{5} } } } {} + 2525 size 12{ { { size 8{2} } over { size 8{5} } } } {} + 2525 size 12{ { { size 8{2} } over { size 8{5} } } } {}

PRODUCT

2 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}

1 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}

6 7 6 7 size 12{ { { size 8{6} } over { size 8{7} } } } {} (1)

2

1 3535 size 12{ { { size 8{3} } over { size 8{5} } } } {}

b) numerators x numerators

denominators x denominators

d)

(i) 21102110 size 12{ { { size 8{"21"} } over { size 8{"10"} } } } {}

= 2 110110 size 12{ { { size 8{1} } over { size 8{"10"} } } } {}

(ii) 123123 size 12{ { { size 8{"12"} } over { size 8{3} } } } {}

= 4

(iii) 849849 size 12{ { { size 8{"84"} } over { size 8{9} } } } {}

= 9 1313 size 12{ { { size 8{1} } over { size 8{3} } } } {}

18.2 3 1 2 8

  1. a) (i) 15 x 4 (ii) 18 x 40

8 5 25 27

2 1 5 3

k = 3232 size 12{"" lSub { size 8{ {} rSub {} rSup {} } } lSup {} { {3} over {2} } } {} = 1 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}c = 16151615 size 12{ { { size 8{"16"} } over { size 8{"15"} } } } {} = 1 115115 size 12{ { { size 8{1} } over { size 8{"15"} } } } {}

7 4 3

18.3 b) (i) = 38 x 16 (ii) = 17 x 9

4 5 3 10

m = 28 n = 51105110 size 12{ { { size 8{"51"} } over { size 8{"10"} } } } {}

n = 51105110 size 12{5 { { size 8{1} } over { size 8{"10"} } } } {}

6

(iii) = 18 x 8

5 3

1

= 485485 size 12{ { { size 8{"48"} } over { size 8{5} } } } {}

p = 9 3535 size 12{ { { size 8{3} } over { size 8{5} } } } {}

19.1

a) 1

b) 1

c) 1

d) 1

19.2 Product is 1 every time

19.4 a) 20172017 size 12{ { { size 8{"20"} } over { size 8{"17"} } } } {}

b) 140140 size 12{ { { size 8{1} } over { size 8{"40"} } } } {}

c) 531531 size 12{ { { size 8{5} } over { size 8{"31"} } } } {}

d) 873873 size 12{ { { size 8{8} } over { size 8{"73"} } } } {}

19.5 c) 531531 size 12{ { { size 8{5} } over { size 8{"31"} } } } {} : First make an improper fraction ( 315315 size 12{ { { size 8{"31"} } over { size 8{5} } } } {} )

d) 873873 size 12{ { { size 8{8} } over { size 8{"73"} } } } {} : First make an improper fraction ( 738738 size 12{ { { size 8{"73"} } over { size 8{8} } } } {})

20. a) 1 2323 size 12{ { { size 8{2} } over { size 8{3} } } } {} x 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}

= 5353 size 12{ { { size 8{5} } over { size 8{3} } } } {} x 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}

= 5656 size 12{ { { size 8{5} } over { size 8{6} } } } {}m = 83, 3.3. size 12{ {3} cSup { size 8{ "." } } } {} cm

b) 5656 size 12{ { { size 8{5} } over { size 8{6} } } } {} x 1313 size 12{ { { size 8{1} } over { size 8{3} } } } {} = 518518 size 12{ { { size 8{5} } over { size 8{"18"} } } } {} m

= 27, 7.7. size 12{ {7} cSup { size 8{ "." } } } {} cm

22.

(a) 32

(b) 15

(c) 25

(d) 25

(e) 45

(f) 2

(g) 8

(h) 7

(i) 7

(j) 6

(k) 6

(l) 8

(m) 8

(n) 8

(o) 100

LEANER SECTION

Content

ACTIVITY: Multiplication of fractions [LO 1.7.3, LO 2.1.5]

18. MULTIPLICATION OF FRACTIONS

18.1 Multiplication of fractions with natural numbers

You already know that multiplication is repeated addition.

a) See if you can complete the following table:

Table 1
  SUM SKETCH REPEATED ADDITION PRODUCT
e.g. 5 × 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}   1 2 + 1 2 + 1 2 + 1 2 + 1 2 1 2 + 1 2 + 1 2 + 1 2 + 1 2 size 12{ { { size 8{1} } over { size 8{2} } } + { { size 8{1} } over { size 8{2} } } + { { size 8{1} } over { size 8{2} } } + { { size 8{1} } over { size 8{2} } } + { { size 8{1} } over { size 8{2} } } } {} 2 1 2 2 1 2 size 12{2 { { size 8{1} } over { size 8{2} } } } {}
  6 × 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {}   .................................................. .................
  2 × 3737 size 12{ { { size 8{3} } over { size 8{7} } } } {}   .................................................. .................
  3 × 2323 size 12{ { { size 8{2} } over { size 8{3} } } } {}   .................................................. .................
  4 × 2525 size 12{ { { size 8{2} } over { size 8{5} } } } {}   .................................................. .................

b) Look carefully at the completed table. Can you think of a shorter way/method to find the answers?

..........................................................................................................................................

..........................................................................................................................................

..........................................................................................................................................

c) TAKE NOTE!

You could also follow this method:

1. Write both numbers as fractions e.g. 6×14=61×146×14=61×14 size 12{6 times { { size 8{1} } over { size 8{4} } } = { { size 8{6} } over { size 8{1} } } times { { size 8{1} } over { size 8{4} } } } {}

2. Multiply the numerators: 6 × 1 = 6

3. Multiply the denominators: 1 × 4 = 4

4. Simplify the answer: 64=124=11264=124=112 size 12{ { { size 8{6} } over { size 8{4} } } =1 { { size 8{2} } over { size 8{4} } } =1 { { size 8{1} } over { size 8{2} } } } {}

d) Calculate:

(i) 7×3107×310 size 12{7 times { { size 8{3} } over { size 8{"10"} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

(ii) 23×623×6 size 12{ { { size 8{2} } over { size 8{3} } } times 6} {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

(iii) 12×7912×79 size 12{"12" times { { size 8{7} } over { size 8{9} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

e) We would represent 6×146×14 size 12{6 times { { size 8{1} } over { size 8{4} } } } {} in another way using a number line:

Figure 1
Figure 1 (graphics1.png)

f) Represent the following on a number line: x=4×23x=4×23 size 12{x=4 times { { size 8{2} } over { size 8{3} } } } {}

18.2 Multiplying fractions with fractions

a) Look carefully at the following examples:

(i) Half ( 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}) of three quarters ( 3434 size 12{ { { size 8{3} } over { size 8{4} } } } {}) can be shown like this:

Figure 2
Figure 2 (graphics2.png)

Thus: 12×34=3812×34=38 size 12{ { { size 8{1} } over { size 8{2} } } times { { size 8{3} } over { size 8{4} } } = { { size 8{3} } over { size 8{8} } } } {}

(ii) One third ( 1313 size 12{ { { size 8{1} } over { size 8{3} } } } {})of a half ( 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {})looks like this:

Figure 3
Figure 3 (graphics3.png)

Thus 13×12=1613×12=16 size 12{ { { size 8{1} } over { size 8{3} } } times { { size 8{1} } over { size 8{2} } } = { { size 8{1} } over { size 8{6} } } } {}

b) Make your own similar sketches for:

(i) 15×1215×12 size 12{ { { size 8{1} } over { size 8{5} } } times { { size 8{1} } over { size 8{2} } } } {}

(ii) 310×12310×12 size 12{ { { size 8{3} } over { size 8{"10"} } } times { { size 8{1} } over { size 8{2} } } } {}

c) IMPORTANT!

When multiplying a fraction by a fraction, e.g. 23×3823×38 size 12{ { { size 8{2} } over { size 8{3} } } times { { size 8{3} } over { size 8{8} } } } {}

1. We first multiply the numerators together: 2 × 3 = 6

2. Then we multiply the denominators together: 3 × 8 = 24

3. We also simplify where possible: 6÷624÷6=146÷624÷6=14 size 12{ { { size 8{6~ div ~6} } over { size 8{"24"~ div ~6} } } = { { size 8{1} } over { size 8{4} } } } {}

d) Do you still remember?

To simplify you must always divide the numerator and denominator by the same number.

e) Did you know?

We can make use of cancelling to determine the product.

This entails dividing the numerator and denominator by a common factor.

Table 2
e.g.
25 × 40
30 × 48
=
1 5 × 5
15 × 6
×
5 × 8 1
6 × 8 1
Table 3
 
     
     
=
25
36

A shorter way looks like this:

Table 4
 
  25
3 30
×
40 4 1
48 12
=
25
3 × 12
Table 5
 
   
   
 
   
   
=
25
36

We can make use of cross-over and vertical cancelling.

f) Calculate the following by making use of cancelling:

(i) k=158×45k=158×45 size 12{k= { { size 8{"15"} } over { size 8{8} } } times { { size 8{4} } over { size 8{5} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

(ii) c=1825×4027c=1825×4027 size 12{c= { { size 8{"18"} } over { size 8{"25"} } } times { { size 8{"40"} } over { size 8{"27"} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

18.3 Multipilication of mixed numbers

a) TAKE NOTE!

We must first convert the mixed number to an improper fraction and then multiply.

e.g. 6 × 1 3 4 = 6 1 × 7 4 42 4 10 2 4 10 1 2 6 × 1 3 4 = 6 1 × 7 4 42 4 10 2 4 10 1 2 alignl { stack { size 12{6 times 1 { { size 8{3} } over { size 8{4} } } = { { size 8{6} } over { size 8{1} } } times { { size 8{7} } over { size 8{4} } } } {} # = { { size 8{"42"} } over { size 8{4} } } {} # ="10" { { size 8{2} } over { size 8{4} } } {} # ="10" { { size 8{1} } over { size 8{2} } } {} } } {}

Remember: Always simplify your answer!

b) Calculate the following and simplify where possible:

(i) m=834×165m=834×165 size 12{m=8 { { size 8{3} } over { size 8{4} } } times { { size 8{"16"} } over { size 8{5} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

(ii) n=523×910n=523×910 size 12{n=5 { { size 8{2} } over { size 8{3} } } times { { size 8{9} } over { size 8{"10"} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

(iii) p=335×223p=335×223 size 12{p=3 { { size 8{3} } over { size 8{5} } } times 2 { { size 8{2} } over { size 8{3} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

19.1 Can you also calculate the answers of the following?

a) h=34×43h=34×43 size 12{h= { { size 8{3} } over { size 8{4} } } times { { size 8{4} } over { size 8{3} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

b) f=79×97f=79×97 size 12{f= { { size 8{7} } over { size 8{9} } } times { { size 8{9} } over { size 8{7} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

c) e=89×98e=89×98 size 12{e= { { size 8{8} } over { size 8{9} } } times { { size 8{9} } over { size 8{8} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

d) b=1215×1512b=1215×1512 size 12{b= { { size 8{"12"} } over { size 8{"15"} } } times { { size 8{"15"} } over { size 8{"12"} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

19.2 What do you notice when you compare the answers of 19.1?

________________________________________________________________

________________________________________________________________

19.3 Did you know?

If the product of two fractions is 1, then one fraction is the RECIPROCAL of the other. Thus, to get the reciprocal of a fraction, we just swop the numerator and the denominator around!

19.4 Determine the reciprocal of each of the following:

a) 17201720 size 12{ { { size 8{"17"} } over { size 8{"20"} } } } {} = ___________________________________________________

b) 40 = ___________________________________________________

c) 615615 size 12{6 { { size 8{1} } over { size 8{5} } } } {} = ___________________________________________________

d) 918918 size 12{9 { { size 8{1} } over { size 8{8} } } } {} = ___________________________________________________

19.5 Explain to a friend how you got c and d’s answers. First write your answers down here.

________________________________________________________________

________________________________________________________________

20. BRAIN-TEASER!

Jodi is 123123 size 12{1 { { size 8{2} } over { size 8{3} } } } {} metre tall. Sandy is half her length. Grace is 1313 size 12{ { { size 8{1} } over { size 8{3} } } } {} of Sandy’s length.

a) How tall is Sandy?

___________________________________________________

___________________________________________________

b) How tall is Grace?

___________________________________________________

___________________________________________________

21. Time for self-assessment

Table 6
  • Tick the applicable block:
 YES NO
I can add fractions correctly    
I can find the lowest common multiple of two denominators.    
I can subtract fractions correctly    
I can multiply fractions with natural numbers    
I can multiply fractions with fractions    
I can multiply fractions with mixed numbers    
I know how to use cancelling when I multiply    
I know how to simplify    
I can determine the reciprocal of a fraction    

22. Let us see how quickly you can think! See if you can complete the following mental test in 2 minutes:

Table 7
   
a) 17 + 15 = ............ i) 110110 size 12{ { { size 8{1} } over { size 8{"10"} } } } {} of 70 = ............
b) 27 + ............ = 42 j) 110110 size 12{ { { size 8{1} } over { size 8{"10"} } } } {} of 1 minute is ............ seconds
c) 52 – 27 = ............ k) 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {} of a day = ............ hours
d) 72 – 47 = ............ l) 56 ÷ 7 = ............
e) 82 – 37 = ............ m) 72 ÷ 9 = ............
f) 10 × 1515 size 12{ { { size 8{1} } over { size 8{5} } } } {} = ............ n) 48 ÷ 6 = ............
g) 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {} of 32 = ............ o) 104 ÷ 102 = ............
h) 1818 size 12{ { { size 8{1} } over { size 8{8} } } } {} of 56 = ............  

(15)

  • Colour in:
Figure 4
Figure 4 (graphics4.png)

Assessment

Learning Outcome 1:The 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 Standard 1.7: We know this when the learner estimates and calculates by selecting and using operations appropriate to solving problems that involve:

1.7.3: addition, subtraction and multiplication of common fractions.

Learning Outcome 2:The learner will be able to recognise, describe and represent patterns and relationships, as well as to solve problems using algebraic language and skills.

Assessment Standard 2.1: We know this when the learner investigates and extends numeric and geometric patterns looking for a relationship or rules, including patterns:

2.1.5: represented in tables.

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