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Fractions, SM-Bank 070

Write reciprocals for the following.

  1. `4/5`  (1 mark)

  2. `3`  (1 mark)

  3. `1 3/4`  (2 marks)

Show Answers Only

a.    `5/4`

b.    `1/3`

c.    `4/7`

Show Worked Solution
a.    `5/4`
 `=> 4/5\ \ \ \ text{(Swap the numerator and denominator)}`

 

b.    `3` `= 3/1\ \ \ \ text{(Write whole numbers as fractions with a denominator of 1)}`  
  `=> 1/3\ \ \ \ text{(Then swap the numerator and denominator)}`  

 

c.    `1 3/4`
 `= 7/4 \ \ \ \ text{(Change mixed numbers to improper fractions)}`
  `=> 4/7 \ \ \ \ text{(Then swap the numerator and denominator)}`

Fractions, SM-Bank 061

Transmission Oils import engine oil.

They sell one brand of engine oil at `2 1/4` times the wholesale price.

If the wholesale price is $12 per litre, what is the selling price per litre?  (2 marks)

Show Answers Only

`$27\ text(per litre)`

Show Worked Solution

`text(Wholesale price) = $12\ text(per litre)`

`text(Sale price)` `= 2 1/4 xx 12`
  `= 9/4 xx 12`
  `= 9 xx 3`
  `=27`

 
`:.\ text(The selling price per litre)\ = $27`

Fractions, SM-Bank 059

Find:

  1. `1/4` of 20 apples.  (1 mark)

  2. `4/5` of 60 years.  (1 mark)

  3. `3/11` of 110 kilometres.  (1 mark)

Show Answers Only

a.    `5\ text(apples)`

b.    `48\ text(years)`

c.    `30\ text(kilometres)`

Show Worked Solution

a.    `1/4 xx 20 =1/4 xx 20/1= 5\ text(apples)`
 

b.    `4/5 xx 60`
`=4/5 xx 60/1`  
  `=4 xx 12`  
  `= 48\ text(years)`  

   

c.    `3/11 xx 110`
`=3/11 xx 110/1`
  `=3 xx 10`
  `= 30\ text(kilometres)`

 

Fractions, SM-Bank 058

Evaluate the following number sentences, giving your answers in simplest form.

  1. `2 3/7-1 2/7`  (1 mark)

  2. `3 1/2-1 3/5`  (2 marks)

  3. `4 1/4-3 1/2`  (2 marks)

Show Answers Only

a.    `1 1/7`

b.    `1 9/10`

c.    `3/4`

Show Worked Solution
a.    `2 3/7-1 2/7` `=1 3/7 -2/7`
  `= 1 1/7`

 

b.    `3 1/2-1 3/5` `=3 5/10-1 6/10`
  `= 2 5/10-6/10`
  `=1 9/10`

 

c.    `4 1/4-3 1/2` `=1 1/4-1/2`
  `= 1 1/4-2/4`
  `=3/4`

Fractions, SM-Bank 057 MC

Mallory has  `3/8`  cup of raisins.

She needs  `3/4`  cup of raisins for a fruit cake she is making.

How many more cups of raisins does Mallory need for the recipe?

  1. `1/8\ text(cup)`
  2. `1/4\ text(cup)`
  3. `3/8\ text(cup)`
  4. `1/2\ text(cup)`
Show Answers Only

`C`

Show Worked Solution
`text(Raisins needed)` `= 3/4-3/8`
  `=6/8-3/8`
  ` =3/8\ text(cup)`

 
`=> C`

Fractions, SM-Bank 056

Sally makes bird feeders to hang in her garden.

The table below shows the fraction of the bird feeder eaten each day by the birds. 

\begin{array} {|l|c|c|}
\hline \textbf{Day} & \textbf{Fraction Eaten} \\
\hline \text{Monday}  &  3/10\\
\hline \text{Tuesday} & 1/5  \\
\hline \text{Wednesday} &  1/4  \\
\hline \text{Thursday} & ?   \\
\hline \end{array}

  1. If the whole feeder was consumed in four days, what fraction did the birds eat on Thursday?  (2 marks)

  2. The bird feeder weighs 5 kilograms. How many kilograms were consumed on Monday?  (2 marks)

Show Answers Only

a.    `1/4`

b.    `1 1/2\ text(kilograms)`

Show Worked Solution
a.    `text(Eaten Thursday)` `= 1-(3/10+1/5+1/4)`
  `= 1-(6/20+4/20+5/20)`
  `= 1-15/20`
  `=5/20=1/4`

 

b.    `text(Monday Kilograms)` `= 3/10 xx 5`
  `= 15/10`
  `= 1 1/2\ text(kilograms)`

Fractions, SM-Bank 055

Evaluate the following number sentences, giving your answers in simplest form.

  1. `1/6-5/6`  (1 mark)

  2. `4/9-1/2`  (2 marks)

  3. `-3/4-1/2`  (2 marks)

Show Answers Only

a.    `-2/3`

b.    `-1/18`

c.    `-1 1/4`

Show Worked Solution
a.    `1/6-5/6` `=-4/6`
  `= -2/3`

 

b.    `4/9-1/2` `=(4 xx 2)/(9 xx 2)-(1 xx 9)/(2 xx 9)`
  `= 8/18-9/18`
  `=-1/18`

 

c.    `-3/4-1/2` `=-3/4-2/4`
  `= -5/4`
  `=-1 1/4`

Fractions, SM-Bank 054

Evaluate the following number sentences, giving your answers in simplest form.

  1. `7/12-1/4`  (1 mark)

  2. `3/5-2/7`  (2 marks)

  3. `1/2-4/9`  (2 marks)

Show Answers Only

a.    `1/3`

b.    `11/35`

c.    `1/18`

Show Worked Solution
a.    `7/12-1/4` `=7/12- (1 xx 3)/(4 xx 3)`
  `= 7/12-3/12`
  `=1/3`

 

b.    `3/5-2/7` `=(3 xx 7)/(5 xx 7)- (2 xx 5)/(7 xx 5)`
  `= 21/35-10/35`
  `=11/35`

 

c.    `1/2-4/9` `=(1 xx 9)/(2 xx 9)- (4 xx 2)/(9 xx 2)`
  `= 9/18-8/18`
  `=1/18`

Fractions, SM-Bank 047

Mike is downloading a complete hard drive onto his new computer.

It should take 24 minutes to download the full hard drive.

Mike loses his internet connection when  `5/6`  of the hard drive is downloaded.

How many more minutes are needed for Mike to complete the download?  (2 marks)

Show Answers Only

`text(4 minutes)`

Show Worked Solution

`5/6 xx 24 =\ text(20 minutes)`

`:.\ text(Minutes needed)` `= 24-20`
  `= 4\ text(minutes)`

Fractions, SM-Bank 044

Add the following mixed numbers giving your answer in simplest form.

  1. `2 5/8 + 1 3/4`  (2 mark)

  2. `4 5/6 + 3 4/9`  (2 mark)

  3. `1 3/5 + 5 2/3 + 4 7/10` (2 marks)

Show Answers Only

a.    `4 3/8`

b.    `8 5/18`

c.    `11 29/30`

Show Worked Solution

a.    Method: Adding whole number parts first

  `2 5/8+1 3/4` `=2 + 1 + 5/8 + 3/4\ \ \ text(LCD)\= 8`
    `= 3 + 5/8 + 6/8`
    ` = 3 + 11/8 = 4 3/8`

    
b.    Method: Adding whole number parts first

  `4 5/6 +3 4/9` `=4 + 3 + 5/6 + 4/9\ \ \ text(LCD)\= 18`
    `= 7 + (5 xx 3)/(6 xx 3) + (4 xx 2)/(9 xx 2)`
    ` = 7 + 15/18 + 8/18`
    `= 7 + 23/18 = 8 5/18`

  
c.    Method: Adding whole number parts first

  `1 3/5+5 2/3+4 7/10` `=1 + 5 + 4 +3/5 +2/3+7/10\ \ \ text(LCD)\= 30`
    `= 10 + (3 xx 6)/(5 xx 6) + (2 xx 10)/(3 xx 10) + (7 xx 3)/(10 xx 3)`
    ` = 10 + 18/30 + 20/30 + 21/30`
    `= 10 + 59/30 = 11 29/30`

Fractions, SM-Bank 043

Add the following mixed numbers giving your answer in simplest form.

  1. `1 1/3 + 1 2/3`  (1 mark)

  2. `2 4/9 + 1 2/9`  (2 mark)

  3. `1 1/11 + 2 5/11 + 4 7/11` (2 marks)

Show Answers Only

a.    `3`

b.    `3 2/3`

c.    `8 2/11`

Show Worked Solution

a.    Method 1: Adding whole number parts first

  `1 1/3+1 2/3` `=1 + 1 +1/3 + 2/3`
    `= 2 + 3/3`
    ` = 3`

Method 2: Changing to improper fractions

  `1 1/3+1 2/3` `=(3 xx 1 + 1)/3 + (3 xx 1 + 2)/3`
    `= 4/3 + 5/3`
    ` = 9/3 = 3`

 
b.    Method 1: Adding whole number parts first

  `2 4/9 +1 2/9` `=2 + 1 +4/9 + 2/9`
    `= 3 + 6/9`
    ` = 3 2/3`

Method 2: Changing to improper fractions

  `2 4/9 +1 2/9` `=(2  xx 9 + 4)/9 + (1 xx 9 + 2)/9`
    `= 22/9 + 11/9`
    ` = 33/9 `
    `= 3 6/9 = 3 2/3`

 
c.    Method 1: Adding whole number parts first

  `1 1/11+2 5/11+4 7/11` `=1 + 2 + 4 +1/11 +5/11+7/11`
    `= 7 + 13/11`
    ` = 8 2/11`

Method 2: Changing to improper fractions

  `1 1/11+2 5/11+4 7/11` `=(1 xx 11 + 1)/11 +(2 xx 11 + 5)/11 + (4 xx 11 + 7)/11 +`
    `= 12/11 + 27/11+51/11`
    ` = 90/11 = 8 2/11`

 

Fractions, SM-Bank 042

Add the following fractions giving your answer in simplest form.

  1. `1/3 + 1/4`  (1 mark)

  2. `3/10 + 1/5`  (2 mark)

  3. `1/2 + 1/3 + 1/8` (2 marks)

Show Answers Only

a.    `7/12`

b.    `1/2`

c.    `23/24`

Show Worked Solution
a.     `1/3+1/4` `= (1xx4)/(3 xx 4) + (1 xx 3)/(4 xx 3)\ \ \ text(LCD)\= 12`
    `= 4/12 + 3/12`
    ` = 7/12`

 

b.     `3/10 +1/5` `= 3/10 + (1 xx 2)/(5 xx 2)\ \ \ text(LCD)\= 10`
    `= 3/10 + 2/10`
    ` = 5/10 = 1/2`

 

c.     `1/2 +1/3 + 1/8` `= (1 xx 12)/(2 xx 12) + (1 xx 8)/(3 xx 8) + (1 xx 3)/(8 xx 3)\ \ \ text(LCD)\= 24`
    `= 12/24 + 8/24 + 3/24`
    ` = 23/24`

Fractions, SM-Bank 041

Add the following fractions giving your answer as a mixed number where necessary.

  1. `4/9 + 1/9`  (1 mark)

  2. `5/7 + 6/7`  (1 mark)

  3. `2/15 + 7/15 + 8/15` (2 marks)

Show Answers Only

a.    `5/9`

b.    `1 4/7`

c.    `1 2/15`

Show Worked Solution

a.    `4/9 + 1/9 = 5/9`

b.    `5/7 + 6/7 = 11/7 = 1 4/7`

c.    `2/15 + 7/15 + 8/15 = 17/15 = 1 2/15`

Fractions, SM-Bank 040

Three friends are baking cupcakes for the Spring Fair.

Maryanne will bake `1/4` of the cupcakes, Martin has a large oven and will bake `2/3` of the cupcakes and Monty will bake the rest.

What fraction of the cupcake baking does Monty complete?  (2 marks)

Show Answers Only

`1/12`

Show Worked Solution

`text(Monty’s Baking)`

`= 1-(1/4 + 2/3)\ \ text(LCD)\= 12`

`= (1xx12)/(1 xx 12)-((1xx3)/(4 xx 3) + (2 xx 4)/(3 xx 4))`

`= 12/12-(3/12 + 8/12)`

`= 12/12-11/12 = 1/12`

`:.\ text(Monty bakes)\ 1/12\ text(of the cupcakes)`

Fractions, SM-Bank 039

Three friends are running as a team in a fun run.

Thanh hopes to finish `3/7` of the run, Jamieson is confident he can complete `1/2` of the run and Jules will complete the rest.

What fraction of the fun run does Jules complete?  (2 marks)

Show Answers Only

`1/14`

Show Worked Solution

`text(Jules)`

`= 1-(3/7 + 1/2)\ \ text(LCD)\= 14`

`= (1xx14)/(1 xx 14)-((3xx2)/(7 xx 2) + (1 xx 7)/(2 xx 7))`

`= 14/14-(6/14 + 7/14)`

`= 14/14-13/14 = 1/14`

`:.\ text(Jules completes)\ 1/14\ text(of the race)`

Fractions, SM_Bank 038

What is the lowest common denomiator of the following sets of fractions?

  1. `1/4, 1/8`   (1 mark)

  2. `2/3, 4/5`   (1 mark)

  3. `1/2, 2/3, 1/6`   (1 mark)

  4. `1/9, 5/6, 7/12`   (1 mark)
Show Answers Only

a.    `8`

b.    `15`

c.    `6`

d.    `36`

Show Worked Solution

a.    `text(8 is the lowest number that is a multiple of both 4 and 8)`

`:.\ text(LCD) (1/4, 1/8) = 8`
 

b.    `text(15 is the lowest number that is a multiple of both 3 and 5)`

`:.\ text(LCD) (2/3, 4/5)= 15`  
  

c.   `text(6 is the lowest number that is a multiple of 2, 3 and 6)`

`:.\ text(LCD) (1/2, 2/3, 1/6) = 6` 
   

d.    `text(36 is the lowest number that is a multiple of 9, 6 and 12)`

`:.\ text(LCD) (1/9, 5/6, 7/12) = 36`  

Fractions, SM-Bank 037

What is the lowest common multiple of the following sets of numbers?

  1. `2, 5`  (1 mark)

  2. `3, 9`  (1 mark)

  3. `2, 3, 6`  (1 mark)

  4. `4, 6, 8`  (1 mark)
Show Answers Only

a.    `10`

b.    `9`

c.    `6`

d.    `24`

Show Worked Solution

a.    `text(Multiples of 2:)\ 2, 4, 6, 8, 10`

`text(Multiples of 5:)\ 5, 10`

`:.\ text(LCM (2, 5))= 10`  
 

b.    `text(Multiples of 3:)\ 3, 6, 9`

`text(Multiples of 9:)\ 9`

`:.\ text(LCM (3, 9))= 9`  
  

c.   `text(Multiples of 2:)\ 2, 4, 6`

`text(Multiples of 3:)\ 3, 6`

`text(Multiples of 6:)\ 6`

`:.\ text(LCM (2, 3, 6))= 6`  
   

d.    `text(Multiples of 4:)\  4, 8, 12, 16, 20, 24`

`text(Multiples of 6:)\ 6, 12, 18, 24`

`text(Multiples of 8:)\ 8, 16, 24`

`:.\ text(LCM (4, 6, 8))= 24`  

Fractions, SM-Bank 036

Jaime and Jordan ordered 2 extra large pizzas for dinner.

Jaime ate `2/5` of an extra large pizza.

  1. If Jordan ate 3 times as much pizza as Jaime, how much pizza did Jordan eat? Give your answer as a mixed number.  (1 mark)

  2. What fraction of pizza is left over?  (2 marks)

Show Answers Only

a.    `1 1/5\ text(pizzas)`

b.    `2/5\ text(of a pizza)`

Show Worked Solution

a.    `text(Jordan’s Pizza)`

`= 3 xx 2/5`

`= 6/5`

 `= 1 1/5\ text(pizzas)`
  

b.    `text(Leftover pizza)`

`= 2-(2/5 + 1 1/5)`

`= 2-1 3/5`

`= 2/5\ text(of a pizza)`

Fractions, SM-Bank 022

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

Show Answers Only

`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)

Show Answers Only

`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)

Show Answers Only

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.    

  
Show Answers Only

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)

Show Answers Only

`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)

Show Answers Only

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)

Show Answers Only

`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`

Show Answers Only

`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)

      

Show Answers Only

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)

Show Answers Only

`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.)`