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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.)`

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)`
Show Answers Only

`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)

Show Answers Only

` 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)

Show Answers Only

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)`
Show Answers Only

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)`
Show Answers Only

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)`
Show Answers Only

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)

Show Answers Only

`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`
Show Answers Only

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`
Show Answers Only

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`
Show Answers Only

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)

Show Answers Only

`(-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`
Show Answers Only

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`

Integers, SM-Bank 050

Complete the following table.  (3 marks)

\begin{array} {|c|c|c|c|}
\hline × & \ \ 3\ \  & -4 & -2 \\
\hline -1 &  &  \\
\hline 5 &  &  \\
\hline \end{array}

Show Answers Only

\begin{array} {|c|c|c|c|}
\hline × & \ \ 3\ \  & -4 & -2 \\
\hline -1 & -3 & 4 & 2 \\
\hline 5 & 15 & -20 & -10 \\
\hline \end{array}

Show Worked Solution

\begin{array} {|c|c|c|c|}
\hline × & \ \ 3\ \  & -4 & -2 \\
\hline -1 & -3 & 4 & 2 \\
\hline 5 & 15 & -20 & -10 \\
\hline \end{array}

Integers, SM-Bank 049

Complete the following table.  (3 marks)

\begin{array} {|c|c|c|c|}
\hline + & \ \ 4\ \  & -5 & -3 \\
\hline -2 &  &  \\
\hline 7 &  &  \\
\hline -10 &  &  \\
\hline \end{array}

Show Answers Only

\begin{array} {|c|c|c|c|}
\hline + & \ \ 4\ \  & -5 & -3 \\
\hline -2 & 2 & -7 & -5 \\
\hline 7 & 11 & 2 & 4 \\
\hline -10 & -6 & -15 & -13 \\
\hline \end{array}

Show Worked Solution

\begin{array} {|c|c|c|c|}
\hline + & \ \ 4\ \  & -5 & -3 \\
\hline -2 & 2 & -7 & -5 \\
\hline 7 & 11 & 2 & 4 \\
\hline -10 & -6 & -15 & -13 \\
\hline \end{array}

Integers, SM-Bank 048

Find the values of the following.  (3 marks)

  1. `4 -(-7)`
  2. `-1 -(-3)`
  3. `-21-(-16)`
Show Answers Only

a.   `11`   b.   `2`   c.   `-5`

Show Worked Solution

`text(Remember:)\  − − = +`

a.   `4-(-7)= 4 + 7 = 11`

b.   `-1-(-3) = -1 + 3 = 2`

c.   `-21-(-16) = -21 + 16 = -5`

Integers, SM-Bank 046

Evaluate the following.  (3 marks)

  1. `15 + (-8)`
  2. `-7 + 4`
  3. `-7 + 5 + (-2)`
Show Answers Only

a.   `7`   b.   `-3`   c.   `-4`

Show Worked Solution

a.   `15 + (-8) = 15-8 = 7`

b.   `-7 + 4 = -3`

c.   `-7 + 5 + (-2) = -7 + 5-2 = -4`

Integers, SM-Bank 045

Evaluate the following.  (3 marks)

  1. `-1 + (-5)`
  2. `-12 + 5`
  3. `-3 + 4 + (-1)`
Show Answers Only

a.   `-6`   b.   `-7`   c.   `0`

Show Worked Solution

a.   `-1 + (-5) = -1-5 = -6`

b.   `-12 + 5 = -7`

c.   `-3 + 4 + (-1) = -3 + 4-1 = 0`

Integers, SM-Bank 039

Jennifer opened a bank account with $500. Each week for 4 weeks she withdrew $122 for rent. At the end of the month the bank deducted $15 in fees.

What was the balance of Jennifer's bank account at the end of the month? (2 marks)

Show Answers Only

`-$3`

Show Worked Solution

`text(Balance)\ = 500-122 xx 4-15 = 500-488-15 = -3`

`:.\ text(Jennifer’s balance)\ = -$3`

Integers, SM-Bank 037

Jasper is a meteorologist working in Antartica. One of his jobs each week is to upload the daily maximum and minimum temperatures to the Base Camp website.

Jasper has partially completed his log book for this week but will need his additional notes to fill in the missing entries before he can upload them to the website.

Using Jasper's notes below, complete the missing log book entries.  (4 marks)

 Jasper's Additional Notes:

  1. Tuesday's maximum is 3 less than Monday's maximum.
  2. Thursday's temperature range is 12.
  3. Saturday's minimum is 3 times smaller than Wednesday's minimum.
  4. Saturday's maximum is half of Sunday's maximum.
     
    TEMPERATURE LOG BOOK
    \begin{array} {|l|c|c|}
    \hline \textbf{Day} & \textbf{Minimum} &  \textbf{Maximum}  \\
    \hline \text{Monday} & -10°  &  -1° \\
    \hline \text{Tuesday} & -15°  &   \\
    \hline \text{Wednesday} & -7°  &  -2° \\
    \hline \text{Thursday} &   &  -8° \\
    \hline \text{Friday} & -17°  &  -6° \\
    \hline \text{Saturday} &  &   \\
    \hline \text{Sunday} & -19°  &  -4° \\
    \hline \end{array}

Show Answers Only

TEMPERATURE LOG BOOK
\begin{array} {|l|c|c|}
\hline \textbf{Day} & \textbf{Minimum} &  \textbf{Maximum}  \\
\hline \text{Monday} & -10°  &  -1° \\
\hline \text{Tuesday} & -15°  & \colorbox{lightblue}{−4°} \\
\hline \text{Wednesday} & -7°  &  -2° \\
\hline \text{Thursday} &  \colorbox{lightblue}{−20°} &  -8° \\
\hline \text{Friday} & -17°  &  -6° \\
\hline \text{Saturday} & \colorbox{lightblue}{−21°} &  \colorbox{lightblue}{−2°} \\
\hline \text{Sunday} & -19°  &  -4° \\
\hline \end{array}

Show Worked Solution

`1.\   text{Tuesday’s Maximum}\= -1-3 = -4`

`2.\   text{Thursday’s Minimum}\= -8-12 = -20`

`3.\   text{Saturday’s minimum}\= 3 xx -7 = -21`

`4.\   text{Saturday’s maximum}\= -4 ÷ 2 = -2`
  

TEMPERATURE LOG BOOK
\begin{array} {|l|c|c|}
\hline \textbf{Day} & \textbf{Minimum} &  \textbf{Maximum}  \\
\hline \text{Monday} & -10°  &  -1° \\
\hline \text{Tuesday} & -15°  & \colorbox{lightblue}{−4°} \\
\hline \text{Wednesday} & -7°  &  -2° \\
\hline \text{Thursday} &  \colorbox{lightblue}{−20°} &  -8° \\
\hline \text{Friday} & -17°  &  -6° \\
\hline \text{Saturday} & \colorbox{lightblue}{−21°} &  \colorbox{lightblue}{−2°} \\
\hline \text{Sunday} & -19°  &  -4° \\
\hline \end{array}

Integers, SM-Bank 036

James runs a tyre fitting company.

At the beginning of February he took out a `$50\ 000` bank loan to expand his workshop.

In July James had flood damage in his workshop and added `$20\ 000` to his bank loan to pay for repairs. 

By the end September he had made loan repayments totalling `$12\ 000` and had also deposited business profits of `$15\ 000` into his loan account.

Write a number sentence and use it to calculate how much James owed at the end of September. (3 marks)
  

Show Answers Only

`-$43\ 000`

Show Worked Solution
`text(Amount of debt)` `= -50\ 000-20\ 000+12\ 000+15\ 000`  
  `= -70\ 000+27\ 000 `  
  `= -43\ 000`  

`:.\ text(James’ debt at the end of September)\= -$43\ 000`

Integers, SM-Bank 032

The daily high and low temperatures are measured in four ski resorts and recorded in the table below.

\begin{array} {|l|c|c|}
\hline \textbf{Ski Resort} & \textbf{Low} &  \textbf{High}  \\
\hline \text{Thredbo} & -1°  &  6° \\
\hline \text{Charlotte's Pass} & -5°  &  1° \\
\hline \text{Mount Hotham} & -2°  &  3° \\
\hline \text{Perisher} & -4°  &  4° \\
\hline \end{array}

  1. Which resort had the lowest recorded temperature range?  (1 mark)

  2. Which resort had a recorded temperature range of 6 degrees?  (1 mark)

Show Answers Only

a.   `text(Mount Hotham)`

b.   `text(Charlotte’s Pass)`

Show Worked Solution

`text(Considering all options.)`

`text(Thredbo)` `=6-(-1)` `=7`
`text(Charlotte’s Pass)` `=1-(-5)` `=6`
`text(Mount Hotham)` `=3-(-2)` `=5`
`text(Perisher)` `=4-(-4)` `=8`

 
a.   `text(Lowest Range)\ =\text(Mount Hotham)`

b.   `text(Range of)\ 6 = text(Charlotte’s Pass)`

Integers, SM-Bank 028

Lindy is using a number line to show a pattern.
  

What are the next two numbers in Lindy’s pattern? (2 marks)

Show Answers Only

`-7\ text(and) -4`

Show Worked Solution

`text(The numbers are increasing by 3 each time.)`

`text(The first number is: )-10+3 =-7`

`text(and the second number is: )-7+3 = -4`

`:.\ -7\ text(and) -4\ text(are the next 2 numbers)`

Integers, SM-Bank 027

Trad is using a number line to show a pattern.
  

What are the next two numbers in Trad’s pattern? (2 marks)

Show Answers Only

`-6\ text(and) -11`

Show Worked Solution

`text(The numbers are decreasing by 5 each time.)`

`text(The first number is: )-1-5 = -6`

`text(and the second number is: )-6-5 = -11`

`:.\ -6\ text(and) -11\ text(are the next 2 numbers)`