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Algebra, STD1 A3 2024 HSC 23

Carrie is organising a fundraiser.

The cost of hiring the venue and the band is $2500. The cost of providing meals is $50 per person.

  1. Complete the table of values to show the total cost of the fundraiser.   (1 mark)

\begin{array} {|l|c|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \text{Number of people} \rule[-1ex]{0pt}{0pt} & \ \ \ \ 0\ \ \ \  & 25 & 50 & 75 & 100 & 125 & 150 \\
\hline
\rule{0pt}{2.5ex} \text{Cost} \rule[-1ex]{0pt}{0pt} & & 3750 & 5000 & 6250 & 7500 & 8750 & 10\,000 \\
\hline
\end{array}

  1. Carrie decides that tickets should be sold at $70 per person. The graph shows the expected revenue at this ticket price. Using the information in part (a), plot the line that shows the cost of the fundraiser.   (2 marks)

     

  1. How many tickets need to be sold for the fundraiser to break even?   (1 mark)

  2. Carrie sold 300 tickets. How much profit did the fundraiser make?   (3 marks)

Show Answers Only

a.    \(\text{Cost when 0 people }= $2500\)

b.


c.    \(100\ \text{tickets}\)

d.    \(\text{Profit }=$3500\)

Show Worked Solution

a.    \(\text{Cost when 0 people }= $2500\)

b.


c.    \(\text{Point of intersection}\ \ \Rightarrow\ \ \text{break-even}\)

\(\therefore\ \text{Break-even when 100 tickets sold.}\)
 

d.    \(\text{Revenue}\ (R)=70n\ \ (n=\ \text{number of people)}\)

\(\text{Cost}\ (C)=2500 + \Big(\dfrac{1250}{25}\Big)n=2500+50n\)

\(\text{Find profit}\ (P)\ \text{when}\ \ n=300:\)

\(P\) \(=R-C\)
  \(=70\times 300-(2500+50\times300)\)
  \(=21\,000-17\,500\)
  \(=$3500\)

Algebra, STD1 A3 2022 HSC 25

Sam is making cupcakes to sell at a market. It costs Sam $60 to hire a stall, and each cupcake costs $1.50 to make. Sam intends to sell each cupcake for $4.00.

The equations representing Sam's cost `($ C)` and revenue `($ R)`, are

`C=1.5 x+60`  and  `R=4 x`, where `x` is the number of cupcakes sold.

The graphs of `C` and `R` are shown below.
 


 

  1. How many cupcakes must Sam sell in order to break even?  (1 mark)

  2. If Sam sells 60 cupcakes, what profit is made?
  3. You may assume that  Profit = Revenue – Cost.  (2 marks)

Show Answers Only
  1. 24
  2. $90
Show Worked Solution

a.   Break even occurs where the two graphs intersect.

→ 24 cupcakes
 

b.   If cupcakes sold (`x`) = 60:

`C = 1.5 xx 60 + 60= $150`

`R = 4 xx 60=$240`
  
`:.\ text{Profit}\ = 240-150 = $90`

Algebra, STD1 A3 2019 HSC 30

A small business makes and sells bird houses.

Technology was used to draw straight-line graphs to represent the cost of making the bird houses `(C)` and the revenue from selling bird houses `(R)`. The `x`-axis displays the number of bird houses and the `y`-axis displays the cost/revenue in dollars.
 


 

  1. How many bird houses need to sold to break even?  (1 mark)

  2. By first forming equations for cost `(C)` and revenue `(R)`, determine how many bird houses need to be sold to earn a profit of $1900.  (3 marks)

Show Answers Only
  1. `20`
  2. `96`
Show Worked Solution

a.   `20\ \ (xtext(-value at intersection))`

 

b.   `text(Find equations of both lines):`

♦ Mean mark part (b) 47%.

`(0, 500)\ text(and)\ (20, 800)\ text(lie on)\ \ C`

`m_C = (800 – 500)/(20 – 1) = 15`

`=> C = 500 + 15x`
 

`(0,0)\ text(and)\ (20, 800)\ text(lie on)\ \ R`

`m_R = (800 – 0)/(20 – 0) = 40`

`=> R = 40x`
 

`text(Profit) = R – C`

`text(Find)\ \ x\ \ text(when Profit = $1900:)`

`1900` `= 40x – (500 + 15x)`
`25x` `= 2400`
`x` `= 96`

Algebra, STD2 A4 2018 HSC 27d

The graph displays the cost (`$c`) charged by two companies for the hire of a minibus for `x` hours.
 


  

Both companies charge $360 for the hire of a minibus for 3 hours.

  1. What is the hourly rate charged by Company A?  (1 mark)

  2. Company B charges an initial booking fee of $75.

     

    Write a formula, in the form of  `c = mx + b`, for the cost of hiring a minibus from Company B for `x` hours.  (2 marks)

  3. A minibus is hired for 5 hours from Company B.

     

    Calculate how much cheaper this is than hiring from Company A.  (2 marks)

Show Answers Only
  1. `$120`
  2. `c = 95x + 75`
  3. `$50`
Show Worked Solution
i.    `text(Hourly rate)\ (A)` `= 360 ÷ 3`
    `= $120`

 

ii.   `m = text(hourly rate)`

`text(Find)\ m,\ text(given)\ c = 360,\ text(when)\ \ x = 3\ \ text(and)\ \ b = 75`

`360` `= m xx 3 + 75`
`3m` `= 285`
`m` `= 95`

 
`:. c = 95x + 75`
 

iii.    `text(C)text(ost)\ (A)` `= 120 xx 5 = $600`
  `text(C)text(ost)\ (B)` `= 95 xx 5 + 75 = $550`

 
`:.\ text(Company)\ B’text(s hiring cost is $50 cheaper.)`

Algebra, STD2 A4 SM-Bank 4

Penny is a baker and makes meat pies every day.

The cost of making `p` pies, `$C`,  can be calculated using the equation

`C = 675 + 3.5 p`

Penny sells the pies for $5.75 each, and her income is calculated using the equation

`I = 5.75 p`

  1. On the graph, draw the graphs of  `C`  and  `I`.  (2 marks)

  2. On the graph, label the breakeven point and the loss zone.  (2 marks)

Show Answers Only

(i) and (ii)

Show Worked Solution
i.   

 

ii.  `text(Loss zone occurs when)\ C > I,\ text(which is shaded)`

`text(in the diagram above.)`

Algebra, STD2 A4 2005 HSC 28b

Sue and Mikey are planning a fund-raising dance. They can hire a hall for $400 and a band for $300. Refreshments will cost them $12 per person.

  1. Write a formula for the cost ($C) of running the dance for `x` people. (1 mark)

The graph shows planned income and costs when the ticket price is $20 

2005 28b

  1. Estimate the minimum number of people needed at the dance to cover the costs.  (1 mark)

  2. How much profit will be made if 150 people attend the dance? (1 mark)

Sue and Mikey plan to sell 200 tickets. They want to make a profit of $1500.

  1. What should be the price of a ticket, assuming all 200 tickets will be sold?  (3 marks)

Show Answers Only
  1. `700 + 12x`
  2. `text(Approximately 90)`
  3. `$500`
  4. `$23`
Show Worked Solution
i.    `$C` `= 400 + 300 + (12 xx x)`
    `= 700 + 12x`

 

ii.  `text(Using the graph intersection)`

`text(Approximately 90 people are needed)`

`text(to cover the costs.)`

 

iii.  `text(If 150 people attend)`

`text(Income)` `= 150 xx $20`
  `= $3000`
`text(C)text(osts)` `= 700 + (12 xx 150)`
  `= $2500`

 

`:.\ text(Profit)` `= 3000 − 2500`
  `= $500`

 

iv.  `text(C)text(osts when)\ x = 200:`

`C` `= 700 + (12 xx 200)`
  `= $3100`

 

`text(Income required to make $1500 profit)`

`= 3100 + 1500`

`= $4600`
 

`:.\ text(Price per ticket)` `= 4600/200`
  `= $23`

Algebra, STD2 A4 2011 HSC 20 MC

A function centre hosts events for up to 500 people. The cost `C`, in dollars, for the centre
to host an event, where `x` people attend, is given by:

`C = 10\ 000 + 50x`

The centre charges $100 per person. Its income `I`, in dollars, is given by:

`I = 100x`
 

2UG 2011 20

How much greater is the income of the function centre when 500 people attend an event, than its income at the breakeven point?

  1.  `$15\ 000`
  2. `$20\ 000`
  3. `$30\ 000` 
  4. `$40\ 000`
Show Answers Only

`C`

Show Worked Solution
♦ Mean mark 50%
COMMENT: Students can read the income levels directly off the graph to save time and then check with the equations given.

`text(When)\ x=500,\ I=100xx500=$50\ 000`

`text(Breakeven when)\ \ x=200\ \ \ text{(from graph)}`

`text(When)\ \ x=200,\ I=100xx200=$20\ 000`

`text(Difference)` `=50\ 000-20\ 000`
  `=$30\ 000`

 
`=> C`

Algebra, STD2 A4 2010 HSC 24b

Ashley makes picture frames as part of her business. To calculate the cost,  `C`, in dollars, of making  `x`  frames, she uses the equation  `C=40+10x`.

She sells the frames for $20 each and determines her income,  `I`, in dollars, using the equation  `I=20x`.
 

Use the graph to solve the two equations simultaneously for  `x`  and explain the significance of this solution for Ashley's business.   (2 marks)

Show Answers Only

`x=4`

Show Worked Solution

`text(From the graph, intersection occurs at)\ x=4`

♦ Mean mark 36%.
MARKER’S COMMENT: The intersection on the graph is the same point at which the two simultaneous equations are solved for the given value of `x`.

`=>\ text(Break-even point occurs at)\ x=4`

`text(i.e. when 4 frames sold)`

`text(Income)` `=20xx4=$80\ \ \ text(is equal to)`
`text(C)text(osts)` `=40+(10xx4)=$80`

 

`text(If)\ <4\ text(frames sold)=>\ text(LOSS for business)`

`text(If)\ >4\ text(frames sold)=>\ text(PROFIT)`