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Fractions, SM-Bank 022

Simplify `896/1248`, giving your answer in simplest form.  (2 marks)

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`28/39`

Show Worked Solution

`text(Note: when there are too many factors to list, divide by known common factors.)`

`896/1248` `= (896÷4)/(1248÷4) `  
  `= 224/312`  
  `= (224÷4)/(312÷4)`  
  `= 56/78`  
  `= (56÷2)/(78÷2)`  
  `=28/39`  

 
`:.\ 896/1248 = 28/39\ text(in simplest form)`

Fractions, SM-Bank 021

Simplify `2550/5000`, giving your answer in simplest form.  (2 marks)

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`51/100`

Show Worked Solution

`text(Note: when there are too many factors to list, divide by known common factors.)`

`2550/5000` `= (2550÷10)/(5000÷10) `  
  `= 255/500`  
  `= (255÷5)/(500÷5)`  
  `= 51/100`  

 
`:.\ 2550/5000 = 51/100\ text(in simplest form)`

Fractions, SM-Bank 018

  1. List all factors of 20 and 36.  (2 marks)

  2. What is the highest common factor (HCF) of 20 and 36?  (1 mark)

  3. Use the HCF of 20 and 36 to simplify `20/36`.  (1 mark)

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a.    `text(Factors of)\ 20: 1, 2, 4, 5, 10, 20`

`text(Factors of)\ 36: 1, 2, 3, 4, 6, 9, 12, 16, 36`

b.    `text(HCF) = 4`

c.    `20/36 = (20 ÷ 4)/(36 ÷ 4) = 5/9`

Show Worked Solution

a.    `text(Factors of)\ 20: 1, 2, 4, 5, 10, 20`

`text(Factors of)\ 36: 1, 2, 3, 4, 6, 9, 12, 16, 36`
  

b.   `text(Common Factors:)\ 1, 2, 4`

`text(HCF) = 4`
  

c.    `text(To simplify, divide the numerator and denominator by HCF.)`

`20/36 = (20 ÷ 4)/(36 ÷ 4) = 5/9`

Fractions, SM-Bank 017

State whether the following pairs of fractions are equivalent by using the `=` or `≠` symbol in the box between the fractions.  (3 marks)

a.    

  
b.    

  
c.    

  
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a.    `4/5 = 36/45`

b.    `56/63 ≠ 7/9`

c.    `5/8 ≠ 3/5`

Show Worked Solution
a.    `4/5` `= (4 xx 9)/(5 xx 9) = 36/45`

 
`:.\ 4/5 = 36/45`

 

b.    `56/63` `= (56 ÷ 7)/(63 ÷ 7) = 8/9`

 
`:.\ 56/63 ≠ 7/9`
 

c.    `text(Lowest common multiple of)\ 8\ text(and)\ 5 = 40`

`:.\ 5/8` `= (5 xx 5)/(8 xx 5) = 25/40`

 
`text(and)`

`:.\ 3/5` `= (3 xx 8)/(5 xx 8) = 24/40`

 
`:.\ 5/8 ≠ 3/5`

 

Fractions, SM-Bank 016

A bowl has 16 pieces of fruit in it.

12 of the pieces of fruit are oranges.

What fraction of the pieces of fruit are oranges?  Give your answer in simplest form.  (1 mark)

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`3/4`

Show Worked Solution
`text(Fraction of oranges)` `= text(number of oranges)/ text(total number of fruit pieces)`
  `=12/16`
  `= (12 ÷ 4)/(16 ÷ 4)`
  `=3/4 `

Fractions, SM-Bank 015

  1. The shape below has some triangles shaded.
    Shade some extra triangles so that `1/3` of the the shape is shaded.  (1 mark)
      
  2. Using the shape in (a.) above, or otherwise, write two more fractions that are equivalent to `1/3`.  (2 marks)

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a.    `5\ text(additional triangles need to be shaded (8 in total))`

`text(One possible solution.`

b.    `8/24 = 4/12 = 2/6 = 1/3`

`text(Note: Many other fractions equivalent to)\ 1/3\ text(are possible.)`

Show Worked Solution

a.   `text(There are 24 triangles altogether.)`

`text(Number of triangles)\ = 1/3 xx 24 = 8`

`:.\ 5\ text(additional triangles need to be shaded (8 in total))`

`text(One possible solution.`

b.   `text(Using diagram)`

`8/24 = 4/12 = 2/6 = 1/3`

`text(Note: Many other fractions equivalent to)\ 1/3\ text(are possible.)`

Fractions, SM-Bank 012

Shade  `4/7`  of the image below. (1 mark)

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`text(8 triangles must be shaded.)`

`text(Note: Many options are possible.)`

Show Worked Solution

`text(There are 14 triangles)`

`4/7 xx 14 = 8`

`:.\ text(Any 8 triangles shaded is acceptable.)`

 

Fractions, SM-Bank 011

Write the following fractions in order from largest to smallest.  (2 marks)

`5/8, 1/4, 5/24, 1/2, 1/6`

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`5/8, 1/2, 1/4, 5/24, 1/6`

Show Worked Solution

`text(Lowest common denominator)\ = 24`

`text(Converted fractions:)\ ` `5/8 xx 3/3` `=15/24\ …1`
  `1/4 xx 6/6` `=6/24\ …3`
  `5/24` `=5/24\ …4`
  `1/2 xx 12/12` `=12/24\ …2`
  `1/6 xx 4/4` `=4/24\ … 5`

 
`:.\ text(Order from largest to smallest) =>\ 5/8, 1/2, 1/4, 5/24, 1/6`

Fractions, SM-Bank 010 MC

Which set of fractions is ordered smallest to largest?

  1. `1/6, 5/24, 1/4, 5/8, 1/2`
  2. `1/4, 7/24, 1/3, 5/12, 1/2`
  3. `1/6, 7/24, 5/12, 3/8, 2/3`
  4. `1/4, 1/3, 7/24, 3/12, 1/2`
Show Answers Only

`B`

Show Worked Solution

`text(Converting option B to the lowest common denominator of 24:)`

`1/4, 7/24, 1/3, 5/12, 1/2\ \ =>\ 6/24, 7/24, 8/24, 10/24, 12/24`

 

`=> B`

Fractions, SM-Bank 008

  1. Write the following fractions in ascending order.  (1 mark)
     
    ` -2/6, 1/3 and -1 1/5?`

     
  2. Graph the fractions in part (i) on the number line below.  (1 mark)

      

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i.   `-1 1/5 ,  -2/6 ,  1/3 `

ii.   

Show Worked Solution

i.    `text(Ascending order)\ =>\ -1 1/5 ,  -2/6 ,  1/3 `

ii.   

Fractions, SM-Bank 005

The table shows the fractions of the Australian workforce in some industries.
  

\begin{array} {|l|c|}
\hline \ & \textbf{Fraction of the} \\ \textbf{Industry} \ & \textbf{Australian workforce} \\
\hline \text{Automotive}  & \dfrac{1}{12} \\
\hline \text{Finance} & \dfrac{1}{13} \\
\hline \text{Health Care} & \dfrac{1}{7} \\
\hline \text{Telecommunications} & \dfrac{1}{10} \\
\hline \end{array}

  
Which of these industries has the least number of employees in the workforce?  (1 mark)

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`text(Finance)`

Show Worked Solution

`text(The smallest fraction is)\ \ 1/13. (The smallest fraction has the largest denominator)`

`:.\ text(The finance industry has least number of employees.)`

Fractions, SM-Bank 001 MC

Riley threw a shotput in the school athletics carnival and landed it where the arrow marker is shown below.
 

 

 
Which of these is closest to the length of his throw?

  1. `4 2/3 text(m)`
  2. `5 1/3 text(m)`
  3. `5 1/2 text(m)`
  4. `5 2/3 text(m)`
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`B`

Show Worked Solution

`text(Throw is between 5 and 6 metres and not as far as)\ 5 1/2\ text(metres)`.

`:.\ 5 1/3 text(m is closest.)`

`=> B`

Integers, SM-Bank 071

Aurora had $500 in her bank account at the beginning of September.

During the month she withdrew $415 and deposited $82?

Write and solve a directed number sentence to show the balance of Aurora's account at the end of the month?  (2 marks)

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` 500-415 + 82= +$167`

Show Worked Solution

`text(Balance)\ = 500-415 + 82 = +167`

`:.\ text(Aurora’s balance)\ = +$167`

Integers, SM-Bank 070

Jamison measured the temperature at sunrise to be `-7\^circ`C.

He measured the temperature again 3 hours later and found it had risen to `5\^circ`C.

  1. By how many degrees had the temperature risen during the 3 hours since sunrise?  (1 mark)

  2. What was the average hourly increase in temperature during this 3 hours period?  (1 mark)

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a.   `+12^circ`

b.   `4^circ text(/hour)`

Show Worked Solution
a.   `text(Temperature change)` `=5-\-7`  
  `=+12\^circ`  

 

b.   `text(Average temperature)` `=12/3`
  `=4\^circ\text(/hour)`

Integers, SM-Bank 069

Evaluate the following.  (3 marks)

  1. `32/(-8)`
  2. `-27/(9)`
  3. `(-72)/(-12)`
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a.   `-4`   b.   `-3`   c.   `6`

Show Worked Solution

`text(Remember:)\  -/- = +\ `

a.   `32/(-8)= -4`

b.   `-27/(9) = -3`

c.   `(-72)/(-12) = 6`

Integers SM-Bank 068

Evaluate the following.  (3 marks)

  1. `-20÷(-4)`
  2. `-40÷(-2)`
  3. `90÷(-2)÷(-5)`
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a.   `5`   b.   `20`   c.   `9`

Show Worked Solution

`text(Remember:)\  -÷- = +\ `

a.   `-20÷(-4)= 5`

b.   `-40÷(-2) = 20`

c.   `90÷(-2)÷(-5) = -45÷(-5) = 9`

Integers, SM-Bank 067

Evaluate the following.  (3 marks)

  1. `-12÷3`
  2. `10÷(-2)`
  3. `48÷8÷(-2)`
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a.   `-4`   b.   `-5`   c.   `-3`

Show Worked Solution

`text(Remember:)\  -÷+ = -\ text(and)\ +÷- = -`

a.   `-12÷3= -4`

b.   `10÷(-2) = -5`

c.   `48÷8÷(-2) = 6÷(-2) = -3`

Integers, SM-Bank 066

When discussing multiplication of directed numbers with his friend, Jeremy remarked

 'If the number of minus signs in the question is odd I know
 the answer is going to be negative and if the number of minus
 signs is even I know the answer is going to be positive'.

Is Jeremy correct? Justify your answer with examples. (3 marks)

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`text(Examples: odd number of minus signs)`

`1.\ -1xx-1xx-1=1xx-1=-1`

`2.\ -1xx-1xx-1xx-1xx-1=1xx(-1xx-1xx-1)=1xx1xx-1=1xx-1=-1`

`text(Examples: even number of minus signs)`

`1.\ -1xx-1=1`

`2.\ -1xx-1xx-1xx-1=1xx(-1xx-1)=1xx1=1`

`:.\ text(An odd number of minus signs = negative answer)`

`text(and an even number of minus signs = positive answer.)`

Show Worked Solution

`text(Examples: odd number of minus signs)`

`1.\ -1xx-1xx-1=1xx-1=-1`

`2.\ -1xx-1xx-1xx-1xx-1=1xx(-1xx-1xx-1)=1xx1xx-1=1xx-1=-1`

`text(Examples: odd number of minus signs)`

`1.\ -1xx-1=1`

`2.\ -1xx-1xx-1xx-1=1xx(-1xx-1)=1xx1=1`

`:.\ text(An odd number of minus signs = negative answer)`

`text(and an even number of minus signs = positive answer.)`

Integers, SM-Bank 065

Find the values of the following.  (3 marks)

  1. `-3- (-15)`
  2. `7-(-3)`
  3. `-3-(-6)-2`
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a.   `12`   b.   `10`   c.   `1`

Show Worked Solution

`text(Remember:)\  -\ – = +`

a.   `-3- (-15)= -3 + 15= 12`

b.   `7-(-3)= 7 + 3 = 10`

c.   `-3-(-6)-2 = -3 + 6-2 = 1`

Integers, SM-Bank 063

Evaluate the following products.  (3 marks)

  1. `2xx-3xx-1xx-5`
  2. `-1xx-2xx-4xx-5`
  3. `3xx-2xx-2xx-1xx-1`
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a.   `-30`   b.   `40`   c.   `12` 

Show Worked Solution
`text(Remember:)\ ` `-xx+ ` `=-`
  `+xx-` `=-`
  `- xx -` `= +`

  

a.  `2xx-3xx-1xx-5` `=-6xx-1xx-5`
  `=6xx-5`
  `=-30`

 

b.  `-1xx-2xx-4xx-5` `=2xx-4xx-5`
  `-8xx-5`
  `=40`

 

c.  `3xx-2xx-2xx-1xx-1` `=-6xx-2xx-1xx-1`  
  `=12xx-1xx-1`  
  `=-12xx-1`  
  `=12`  

Integers, SM-Bank 062

Find the following products.  (3 marks)

  1. `-3xx-2`
  2. `-5xx-9`
  3. `-3xx-2xx8`
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a.   `6`   b.   `45`   c.   `48`

Show Worked Solution
`text(Remember:)\ ` `-xx+ ` `=-`
  `+xx-` `=-`
  `-xx-` `=+`

  
a.   `-3xx-2= 6`

b.   `-5xx-9= 45`

c.   `-3xx-2xx8 = 6xx8 =48`

Integers, SM-Bank 061

Find the following products.  (3 marks)

  1. `-5xx4`
  2. `10xx-9`
  3. `4xx-1xx8`
Show Answers Only

a.   `-20`   b.   `-90`   c.   `-32`

Show Worked Solution
`text(Remember:)\ ` `-xx+ ` `=-`
  `+xx-` `=-`
  `-xx-` `=+`

  
a.   `-5xx4 = -20`

b.   `10xx-9 = -90`

c.   `4xx-1xx8 = -4xx8 = -32`

Integers, SM-Bank 060

Explain why  `(-5)^2` is not the same as `-5^2\`.  (2 marks)

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`(-5)^2\ text(means)\ -5\ text(squared.)`

`:.\ (-5)^2=-5 xx -5 = 25`

 

`-5^2\ text(means)\ -1 xx5\ text(squared.)`

`:.\ -5^2=-1 xx 5 xx 5 = -25`

Show Worked Solution

`(-5)^2\ text(means)\ -5\ text(squared.)`

`:.\ (-5)^2=-5 xx -5 = 25`

 

`-5^2\ text(means)\ -1 xx5\ text(squared.)`

`:.\ -5^2=-1 xx 5 xx 5 = -25`

Integers, SM-Bank 059 MC

Which of the following best represents  `3 xx -5`?

  1. `5 + -5 + -5 `
  2. `(-5) + (-5) + (-5)`
  3. `(-3) + (-3) + (-3) + (-3) + (-3)`
  4. `5-5-5 `
Show Answers Only

`B`

Show Worked Solution

`text(Test Options)`

`text(Option A:)\  5 + -5 + -5  = + 5 + 2 xx (-5)\ :.\ text(Incorrect)`

`text(Option B:)\ (-5) + (-5) + (-5) = 3 xx (-5)\ :.\ text(Correct)`

`text(Option C:)\ (-3) + (-3) + (-3) + (-3) + (-3) = 5 xx (-3) \:. text(Incorrect)`

`text(Option D:)\ 5-5-5 = + 5-2 xx (+5)\ :.\ text(Incorrect)`

  
`=>B`

Integers, SM-Bank 058 MC

Which of the following best represents  `4 xx -3`?

  1. `3 + -3 + -3 + -3`
  2. `(-4) + (-4) + (-4)`
  3. `(-3) + (-3) + (-3) + (-3)`
  4. `3-3-3-3`
Show Answers Only

`C`

Show Worked Solution

`text(Test Options)`

`text(Option A:)\  3 + -3 + -3 + -3 = + 3 + 3 xx (-3)\ :.\ text(Incorrect)`

`text(Option B:)\ (-4) + (-4) + (-4) = 3 xx (-4)\ :.\ text(Incorrect)`

`text(Option C:)\ (-3) + (-3) + (-3) + (-3) = 4 xx (-3) \:. text(Correct)`

`text(Option D:)\ 3-3-3-3 = + 3-3 xx (+3)\ :.\ text(Incorrect)`

  
`=>C`

Integers, SM-Bank 057

Find the following products.  (3 marks)

  1. `-2xx-3`
  2. `-12xx-2`
  3. `7 xx-6xx-2`
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a.   `6`   b.   `24`   c.   `84` 

Show Worked Solution
`text(Remember:)\ ` `-xx+ ` `=-`
  `+xx-` `=-`
  `- xx -` `= +`

  
a.   `-2xx-3 = 6`

b.   `-12xx-2 = 24`

c.   `7 xx-6xx-2=-42xx-2=84`

Integers, SM-Bank 056

Find the following products.  (3 marks)

  1. `8xx(-3)`
  2. `-4xx9`
  3. `2xx(-3)xx7`
Show Answers Only

a.   `-24`   b.   `-36`   c.   `-42`

Show Worked Solution
`text(Remember:)\ ` `-xx+ ` `=-`
  `+xx-` `=-`
  `-xx-` `=+`

  
a.   `8xx(-3) = -24`

b.   `-4xx9 = -36`

c.   `2xx(-3)xx7 = -6xx7 = -42`