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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)`

Integers, SM-Bank 025 MC

A buoy is in the ocean recording the height of waves passing by.

At sea level it measures 0 metres. It moves above and below sea level.

Which list gives four sea level recordings arranged in order from lowest to highest?

  1. `0\ text(m), -2\ text(m), 4\ text(m), -4\ text(m)`
  2. `-2\ text(m), -5\ text(m), 0\ text(m), 1\ text(m)`
  3. `-5\ text(m), 2\ text(m), -3\ text(m), 0\ text(m)`
  4. `-5\ text(m), -3\ text(m), 0\ text(m), 2\ text(m)`
Show Answers Only

`D`

Show Worked Solution

`=> D`

Integers, SM-Bank 021

Rachel lives in Broome where the temperature on a particular day was 46°C.

Rachel's son lives in Winnipeg, Canada, and on the same day the temperature there was −16°C.

What was the temperature difference between the two towns on this day? (2 marks)

Show Answers Only

`62°text(C)`

Show Worked Solution
`text(Temperature difference)` `= 46-(-16)`
  `= 62°text(C)`

Integers, SM-Bank 020 MC

Which of the following changes is the smallest?

  1. `-8°text(C  to)\ -4°text(C)`
  2. `-5°text(C  to)\ -2°text(C)`
  3. `-2°text(C  to)\ 3°text(C)`
  4. `3°text(C  to)\ 7°text(C)`
Show Answers Only

`B`

Show Worked Solution

`text(Consider each option:)`

`-8\ text(to)\ -4` `=\ text(4° change)`
`-5\ text(to)\ -2` `=\ text(3° change)`
`-2\ text(to)\ 3` `=\ text(5° change)`
`3\ text(to)\ 7` `=\ text(4° change)`

 

`:.\ -5^@\ text(C to)\-2^@text(C is the smallest change.)`

`=> B`

Integers, SM-Bank 019

Death Valley is the lowest point in the United States at `-`85 metres, which is below sea level.

King's Peak, also in the United States, is 4128 metres above sea level.

What is the difference, in metres, between Death Valley and King's Peak?  (2 marks)

Show Answers Only

`4213\ text(metres)`

Show Worked Solution
`text(Difference)` `= 4128- (-85)`
  `= 4213\ text(metres)`

Integers, SM-Bank 018 MC

Dilbert and Smacka were playing a game where they both started with zero points.

Dilbert's final score was 165 points and Smacka's final score was `-90`.

How many more points did Dilbert have than Smacka?

  1. `-255`
  2. `-75`
  3. `75`
  4. `255`
Show Answers Only

`D`

Show Worked Solution
`text(Difference)` `= 165 – (-90)`
  `= 255`

`=> D`

Integers, SM-Bank 017

Murray lives in Bundaberg, QLD, where the temperature on a particular day was 39°C.

Murray's son lives in Salt Lake City, USA, and on the same day the temperature there was `-`15°C.

What was the temperature difference between the two towns on this day?  (2 marks)

Show Answers Only

`54°text(C)`

Show Worked Solution
`text(Temperature difference)` `= 39-(-15)`
  `= 54°text(C)`

Integers, SM-Bank 016 MC

Manou placed an apple pie into his oven.

The 1st measurement of the apple pie's temperature was  `-` 4°C.

Twenty minutes later, a second temperature reading was taken, which measured 21°C.

What was the change in temperature between the two measurements?

  1. decrease of  25°C
  2. decrease of  17°C
  3. increase of  25°C
  4. increase of  17°C
Show Answers Only

`C`

Show Worked Solution
`text(Change in temperature)` `=21-(-4)`
  `=25^@ text(C)`

 

`:.\ text(An increase of  25°C.)`

`=> C`

Integers, SM-Bank 014

Pen wrote the following number sequence.

`4, 2, 0, \-\2, \-\4`

What are the next four numbers in Pen's number sequence?  (2 marks)

Show Answers Only

`\-\6, \-\8, \-\10, \-\12`

Show Worked Solution

`text(The numbers are decreasing by 2 each time.)`

`:.\ text(The next four numbers are: )\-\6, \-\8, \-\10, \-\12`

Integers, SM-Bank 008

Ivan left home and turned off the heater.

Before turning off the heater, the temperature in the room was 25°C.

When Ivan arrived home from work that night, the temperature in the room was `\-\`2°C.

What was the change in temperature in the room? Write your answer as a directed number.  (2 marks)

Show Answers Only

`-27text(°C)`

Show Worked Solution

`text(Change in temperature) = (25 + 2) = 27text(° decrease)`

`:.\ text(Change in temperature) = -27text(°C)`

Integers, SM-Bank 007

Karen arrived home and turned on the heater.

Before turning on the heater, the temperature in the room was −5°C.

After fifteen minutes, the temperature in the room was 12°C.

What was the change in temperature in the room?  Write your answer as a directed number.  (2 marks)

Show Answers Only

`\+\17\text(°C)`

Show Worked Solution

`text(Change in temperature) = 5 + 12 = 17\text(°C increase)`

`:.\ text(Change in temperature) = + 17\text(°C)`

Integers, SM-Bank 006

The Nevis Bungy in Queenstown is the highest bungee jump in New Zealand.

Jumpers launch downwards from a platform and take just 8.5 seconds to freefall 134 metres before the bungee cord retracts for the first time.

Express the length of the freefall as a directed number. Consider the platform to be at zero length.  (1 mark)

Show Answers Only

`-134\ text(metres)`

Show Worked Solution

`134\ text(metres below the platform) = -134\ text(metres)`

Integers, SM-Bank 002

Pixie enters the elevator in her high rise office building on level 5.

She travels down in the lift for 8 floors.

By considering the floors above ground level to be positive and the floors below ground level to be negative, write and solve a number sentence that gives Pixie's position in the building when she gets out of the elevator.  (2 marks)

Show Answers Only

`5 \-\8 = \-\3`

Show Worked Solution

`text(Pixie enters at level 5 and descends 8 floors, therefore the number sentence is:`

`5 \-\8= \-\3`

Integers, SM-Bank 001 MC

Jerome is on holidays in Thredbo. His family is hoping for a good covering of snow as they are having snowboarding lessons.

Which of the following outside temperatures would produce the snow they need?

  1. 30º C
  2. 18º C
  3. 10º C
  4. `-`15º C
Show Answers Only

`D`

Show Worked Solution

`text(The temperature must be below 0º C for snow to fall.)`
`text(Therefore, the outside temperature would need to be −15ºC.)`

`=> D`

Data Analysis, SM-Bank 042

The following data represents the ages of people on a riverboat cruise in Europe.
 

28,  35,  39,  39,  28,  34,  40,  43,  51,  34,  35,  39,  40,  46,  60

 

  1. Organise the data into an ordered stem-and-leaf plot.  (2 marks)

  2. What is the median of the ages?  (1 mark)

  3. What is the modal age of the travellers?  (1 mark)

  4. Is this age data skewed or symmetrical?  (1 mark)

Show Answers Only

a.   

\begin{array} {r|lll} \textbf{Stem} & \textbf{Leaf}  \\ \hline  2 & 8\   8 \\  3 & 4\ 4\ 5\ 5 \ 9\ 9\ 9\  \\  4 & 0\ 0\ 3\ 6 \\ 5 & 1 \\ 6 & 0 \\ \end{array}

b.    `39`

c.    `39`

d.    `text(Skewed)`

Show Worked Solution

a.    `text(Ordered data:) \ 28,  28,  34,  34,  35,  35,  39,  39,  39,  40,  40,  43,  46,  51,  60`

\begin{array} {r|lll} \textbf{Stem} & \textbf{Leaf}  \\ \hline  2 & 8\   8 \\  3 & 4\ 4\ 5\ 5 \ 9\ 9\ 9\  \\  4 & 0\ 0\ 3\ 6 \\ 5 & 1 \\ 6 & 0 \\ \end{array}

b.    `text(Median)\ = \text(8th score)\ =\ 39`

c.    `text(Mode)\ = \ 39`

d.    `text(The data is skewed.)`

 

Data Analysis, SM-Bank 038

Jason recorded the following marks out of 100 in his last 8 class tests.
 

74,  65,  70,  72,  95,  68,  70,  64
 

  1. Which one of his marks is an outlier?  (1 mark)

  2. If the outlier is removed, by how many marks does the mean change?  (2 marks)

  3. Explain why it would be more appropriate to use the median rather than the mean when including the outlier in Jason's marks.  (2 marks)

Show Answers Only

a.   `95`

b.   `72.25 \-\69 = 3.25\ text(marks)`

c.   “

Show Worked Solution

a.   `text(The test mark of 95 is significantly different from the other marks)`

`:.\  95\ text(is an outlier)`
 

b.  `text(Initial Mean)`

`text(Mean)` `=(74 + 65 + 70 + 72 + 95 + 68 + 70 + 64)/8`  
  `= 578/8`  
  `= 72.25`  

 
`text(Mean without outlier)`

`text(New Mean)` `=(74 + 65 + 70 + 72  + 68 + 70 + 64)/7`
  `= 483/7`
  `= 69`

`:.\ text(The mean decreases by)\ 3.25\ text(marks)`

c.   `text(Ordered marks):\  64, \ 65, \ 68, \  70, \ 70, \ 72, \ 74, \ 95 `

`:.\ text(When 95 is included, the median is 70 where as the mean is 72.25.)`

`72.25\ text(lies between his 6th and 7th scores and is, therefore, not a)`

`text(good measure of centre for Jason’s marks.)`

Data Analysis, SM-Bank 036

The ages of boys competing in an inter-school futsal competition are shown in the frequency distribution table below.
 

\begin{array} {|c|c|}
\hline \textbf{Age (years)} & \textbf{Frequency} \\
\hline 13 & 4 \\
\hline 14 & 6  \\
\hline 15 & 11\\
\hline 16 & 6\\
\hline 17 & 3\\
\hline \end{array} 

  1. How many boys took part in the competition?  (1 mark)

  2. Calculate the mean age of the competitors, correct to the nearest whole number.  (2 marks)

  3. State the median age of the competitors?  (2 marks)

Show Answers Only

\begin{array} {ll} \textbf{a.} &  30 \\ \textbf{b.} & 15 \text{ years} \\ \textbf{c.} & 15 \text{ years} \end{array}

Show Worked Solution
a.   `text(  Number of boys)` `= 4 + 6 + 11 + 6 + 3`
    `= 30`

 

b.   `text(  Mean age of boys)` `= (13 xx 4 + 14 xx 6 + 15 xx 11 + 16 xx 6 + 17 xx 3)/30`
    `= (52 + 84 + 165 + 96 + 51)/30`
    `= 448/30`
    `= 14.9333…..`
    `~~ 15\ text(years (nearest whole number))`

 

c.   `text(  Median age of boys )` `=  text(average of 15th and 16th scores)`
    `= 15\ text(years, as both the 15th and 16th scores occur in 15 years)`

Data Analysis, SM-Bank 035

A data set has a range of 50 and a mean of 20.

Give an example of a dataset using 4 numbers that satisfies this condition.  (2 marks)

Show Answers Only

`0,  10,  20,  50`

`text(Note: other answers are possible.)`

Show Worked Solution

`0,  10,  20,  50`

`text(Range) = 50 \ -\ 0 = 50`

`text(Sum of the numbers) = 20 xx 4 = 80`

`text(Mean) = (0 + 10 + 20 +50)/4\ = 80/4 \= 20`

`text(Note: other answers are possible.)`