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Simple Interest, SMB-018

Wilbur invests a sum of money into an account paying simple interest at a rate of 4.25% per annum.

The total interest earned on Wilbur's investment over 4 years is $6800.

Calculate the sum of money Wilbur originally invested.   (2 marks)

Show Answers Only

`$40\ 000`

Show Worked Solution

`I = 6800, \ \ r = 4.25/100, \ \ n = 4`

`I` `= Prn`
`6800` `= P xx 4.25/100 xx 4`
`P`  `= 6800/0.17`
  `= $40\ 000`

Simple Interest, SMB-017 MC

A sum of money is invested in an account paying simple interest at a rate of 8% per annum.

The total interest earned on this investment over 6 years is $27 000.

The sum of money invested is

  1. `$12\ 960`
  2. `$45\ 000`
  3. `$56\ 250`
  4. `$202\ 500`
Show Answers Only

`C`

Show Worked Solution

`I = 27 \ 000, \ \ r = 8/100, \ \ n = 6`

`I` `= Prn`
`27 \ 000` `= (P xx 8 xx 6)/100`
`P`  `= (2\ 700\ 000)/48`
  `= $56\ 250`

 
`=>  C`

Simple Interest, SMB-016

Kate and Amberley are Swifties and want to travel to Melbourne to go to a Tay-Tay concert.

It will cost them a total of $13 000 for tickets, accommodation and transport.

They currently have $12 500 and can invest this amount for 9 months.

What is the minimum simple interest rate, correct to 1 decimal place, at which they can invest their money to reach their goal of $13 000?   (3 marks)

Show Answers Only

`5.4\text{%}`

Show Worked Solution

`n = \frac {9}{12} = 0.75`

`text{Interest required to reach goal} =13\ 000-12\ 500=$500`

`I` `=Prn`
`500` `=12\ 500 xx r xx 0.75`
`r` `=500/9375`
  `=0.05333…`
  `=5.333…\text{%}`

 
`:.\ text{Minimum interest rate = 5.4%}`

Simple Interest, SMB-015

Carissa borrowed $4000 to buy new surfboards.

She did not make any monthly repayments and repaid the loan in full 2 years later with a lump sum of $4400.

Determine the annual simple interest rate that Carissa was charged.   (2 marks)

Show Answers Only

`5.0\text{%}`

Show Worked Solution

`text(Total interest paid)=4400-4000=$400`

`I` `=Prn`
`400` `=4000 xx r xx 2`
`r` `=400/8000`
  `=0.05`
  `=5.0\text{%}`

Simple Interest, SMB-014

Steffi borrowed $8600 to buy a new car.

She did not make any monthly repayments but instead repaid the loan in full 2 years later with a lump sum of $10 062.

Determine the annual simple interest rate that Steffi was charged.   (2 marks)

Show Answers Only

`8.50\text{%}`

Show Worked Solution

`text(Total interest paid)=10\ 062-8600=$1462`

`I` `=Prn`
`1462` `=8600 xx r xx 2`
`r` `=1462/(17\ 200)`
  `=0.0850`
  `=8.50\text{%}`

Simple Interest, SMB-013

$2800 is invested in an account that earns simple interest at the rate of 4.0% per annum.

Calculate the total interest earned in the first six months.   (2 marks)

Show Answers Only

`$56`

Show Worked Solution

`P = 2800,\ \ r = 4.0text(%),\ \ n = \frac{6}{12}=0.5`

`I` `= Prn`
  `= 2800 xx 4.0/100 xx 0.5`
  `= $56`

Simple Interest, SMB-012

$5000 is invested in an account that earns simple interest at the rate of 3.0% per annum.

Calculate the total interest earned in the first two years.   (2 marks)

Show Answers Only

`$300`

Show Worked Solution

`P = 5000,\ \ r = 3.0text(%),\ \ n = 2`

`I` `= Prn`
  `= 5000 xx 3.0/100 xx 2`
  `= $300`

Simple Interest, SMB-011

Find an expression for the total value of an investment, in dollars, if $24 000 is invested for `x` months at a simple interest rate of 4.75% per annum?   (2 marks)

Show Answers Only

`$(24\ 000 + 95x)`

Show Worked Solution
`text(Interest)` `= 24\ 000 xx x/12 xx 4.75/100`
  `= 95x`

 
`:.\ \text{Investment value}\ =$(24\ 000 + 95x)`

Simple Interest, SMB-009 MC

Albert borrowed $2000 off his brother Sid so he could attend the rugby league Magic Round in Brisbane.

He did not make any monthly repayments but instead repaid the loan in full 12 months later with a lump sum of $2350.

What was the annual simple interest rate Sid charged Albert?

  1. 3.5%
  2. 17.5%
  3. 23.5%
  4. 35.0%
Show Answers Only

`B`

Show Worked Solution

`text(Total interest paid)=2350-2000=$350`

`I` `=Prn`
`350` `=2000 xx r xx 1`
`r` `=350/2000`
  `=0.1750`
  `=17.50\text{%}`

 
`=> B`

Simple Interest, SMB-008

Sandy borrowed $7 000 for an overseas trip.

She did not make any monthly repayments but instead repaid the loan in full at the end of three years with a lump sum of $8800.

Calculate the annual simple interest rate was she charged?   (2 marks)

Show Answers Only

`8.57\text{%}`

Show Worked Solution

`text(Total interest paid)=8800-7000=$1800`

`I` `=Prn`
`1800` `=7000 xx r xx 3`
`r` `=1800/(21\ 000)`
  `=0.08571…`
  `=8.57\text{%}`

Simple Interest, SMB-007 MC

Jackie invests $4350 for 3 years. The simple interest is calculated at a rate of 4.5% per annum.

What is the total value of the investment at the end of this period, to the nearest dollar?

  1. $4395
  2. $4546
  3. $4722
  4. $4937
Show Answers Only

`D`

Show Worked Solution
`text(Interest)` `=Prn`
  `=4350 xx \frac{4.5}{100} xx 3`
  `=$587.25`

 
`:.\ text(Value of Investment)=4350 + 587 =$4937`

`=>  D`

Simple Interest, SMB-006 MC

Gutho invests $10 000 for 1 year and 8 months. The simple interest is calculated at a rate of 4.25% per annum.

What is the total value of the investment at the end of this period, to the nearest dollar?

  1. $10 425
  2. $10 571
  3. $10 708
  4. $14 562
Show Answers Only

`C`

Show Worked Solution

`text{1 year 8 months = 12 + 8 = 20 months}`

`text(Interest)` `=Prn`
  `=10\ 000 xx \frac{4.25}{100} xx \frac{20}{12}`
  `=$708`

 
`:.\ text(Value of Investment)=10\ 000+708=$10\ 708`

`=>  C`

Simple Interest, SMB-005

Johnno lives overseas and needs $6000 for his planned trip to Parramatta Stadium to watch Junior Paolo play.

If he invests $5600 for 15 months at a simple interest rate of 6.5% per annum, determine if will get to see the big fella play. Show your working.  (2 marks)

Show Answers Only
`text(Interest)` `=Prn`
  `=5600 xx \frac{6.5}{100} xx \frac{15}{12}`
  `=$455`

 

`text(Value of Investment)` `=5600+455`
  `=$6055>$6000`

 
`:.\ \text{Johnno will get to see Junior play}.`

Show Worked Solution
`text(Interest)` `=Prn`
  `=5600 xx \frac{6.5}{100} xx \frac{15}{12}`
  `=$455`

 

`text(Value of Investment)` `=5600+455`
  `=$6055`

 
`:.\ \text{Johnno will get to see Junior play}.`

Simple Interest, SMB-004

Clancy estimates that he will need $7800 for his planned trip to visit the The Stockman's Hall of Fame in Longreach.

If he invests $7000 for 18 months at a simple interest rate of 6% per annum, will he reach his goal of $7800? Show your working.   (2 marks)

Show Answers Only

`A`

Show Worked Solution
`text(Interest)` `=Prn`
  `=7000 xx \frac{6}{100} xx \frac{18}{12}`
  `=$630`

 

`text(Value of Investment)` `=7000+630`
  `=$7630`

 
`:.\ \text{Clancy won’t reach his goal of $7800}.`

Simple Interest, SMB-003

Min Woo invests $8000 for 3 years. The simple interest is calculated at a rate of 4% per annum.

Calculate the total value of the investment at the end of this period.   (2 marks)

Show Answers Only

`$8960`

Show Worked Solution
`text(Interest)` `=Prn`
  `=8000 xx \frac{4}{100} xx 3`
  `=$960`

 

`:.\ text(Value of Investment)` `=8000+960`
  `=$8960`

Simple Interest, SMB-001

$14 000 is invested in an account that earns simple interest at the rate of 2.5% per annum.

Calculate the total interest earned in the first three years.   (2 marks)

Show Answers Only

`$1050`

Show Worked Solution

`P = 14\ 000,\ \ r = 2.5text(%),\ \ n = 3`

`I` `= Prn`
  `= 14\ 000 xx 2.5/100 xx 3`
  `= $1050`

Financial Maths, SMB-027

The cash price of a large refrigerator is $2000.

A customer buys the refrigerator under a hire-purchase agreement.

She does not pay a deposit and will pay $55 per month for four years.

  1. Calculate the total amount, in dollars, the customer will pay.  (1 mark)

  2. Find the total interest the customer will pay over four years.  (1 mark)

Show Answers Only
  1. `$2640`
  2. `$640`
Show Worked Solution

i.    `text{Total amount paid} = 55 xx 4 xx 12 = $2640`

ii.   `text{Total interest} = 2640-2000= $640`

Financial Maths, SMB-026

Shaun decides to buy a new sound system on a hire-purchase plan.

The sound system is priced at $3500.

Shaun pays a 15% deposit and monthly repayments for five years.

If Shaun's total purchase price after paying the last instalment is $4425, calculate his monthly repayment.   (3 marks)

Show Answers Only

`$65`

Show Worked Solution

`text{Deposit} = \frac{15}{100} xx 3500 = $525`

`text{Balance due after deposit} = 4425-525= $3900`

`text{Number of repayments} = 5 xx 12 = 60`

`text{Monthly repayment} = \frac{3900}{60} = $65`

Financial Maths, SMB-025

Brad investigated the cost of buying a $720 washing machine under a hire purchase agreement.

A 25% deposit is required and the balance will be paid in 24 equal monthly repayments of $27.90.

Calculate the amount of interest Brad pays on this purchase.   (3 marks)

Show Answers Only

`$129.60`

Show Worked Solution

`text{Deposit} = \dfrac{25}{100} xx 720 = $180`

`text{Total instalments} = 24 xx 27.90 = $669.60`

`text{Total purchase price} = 180+669.60 = $849.60`

`text{Interest paid} = 849.60-720= $129.60`

Financial Maths, SMB-024

A $2000 lounge suite was sold under a hire-purchase agreement.

A deposit of $200 was paid.

The balance was to be paid in 36 equal monthly instalments of $68.

Calculate the total interest paid in the purchase.   (3 marks)

Show Answers Only

`$648`

Show Worked Solution

`text{Deposit} = $200`

`text(Total instalments) = 36 xx 68 = $2448`

`text{Total price paid} = 200 + 2448 = $2648`

`text{Interest paid} = 2648-2000 = $648` 

Financial Maths, SMB-023

Sandra has purchased a $4200 plasma television under a hire-purchase agreement. She paid $600 deposit and will pay the balance in equal monthly instalments of $318 over one year.

Determine the total amount of interest Sandra pays on this purchase.   (3 marks)

Show Answers Only

`$216`

Show Worked Solution

`text(Deposit = $600)`

`text(Total monthly instalments) = 12 xx 318 = $3816`

`text{Total payment} = 600 + 3816 = $4416`

`text{Interest paid} = 4416-4200 = $216`

Probability, SMB-013

Beib owns three white and five blue T-shirts. He chooses a T-shirt at random for himself and puts it on. He then chooses another T-shirt at random, from the remaining T-shirts, and gives it to his brother.

  1. What is the probability that Beib chooses a blue T-shirt for himself?  (1 mark)

  2. Complete the tree diagram by writing the correct probability on each branch.  (2 marks)
     
         

  3. Calculate the probability that both of the T-shirts are the same colour.  (2 marks)

Show Answers Only
  1. `5/8`
  2.  
  3. `13/28`
Show Worked Solution

i.    `P(B)= (text(number of blue T-shirts))/(text(total number of T-shirts)) = 5/8`

  
ii.    

       

 
iii. `Ptext((same colour))`

`= P(text(WW)) + P(text(BB))`

`= 3/8 xx 2/7 + 5/8 xx 4/7`

`= 6/56 + 20/56`

`= 13/28`

Probability, SMB-012

Two dice are rolled. What is the probability that only one of the dice shows a three?   (2 marks)

Show Answers Only

`5/18`

Show Worked Solution

`text{Each dice roll is an independent event.}`

`P(3) = 1/6, \ P\text{(not 3)} = 1-1/6=5/6`

`text{P (Only one 3)}`

`= P text{(3, not 3)} + P text{(not 3, 3)}`

`= 1/6 xx 5/6 + 5/6 xx 1/6`

`= 10/36`

`= 5/18`

Probability, SMB-011

Tay-Tay has 2 bags of apples.

Bag A contains 4 red apples and 3 green apples. 

Bag B contains 3 red apples and 1 green apple.

Tay-Tay chooses one of the bags randomly and with her eyes closed, takes one of the apples. 

  1. Complete the tree diagram below.   (2 marks)
     

  2. Determine the probability that Tay-Tay chooses a red apple?   (2 marks) 

Show Answers Only

i.    
       

ii.    `37/56`

Show Worked Solution

i.    
       
 

ii.     `Ptext{(red)}` `=1/2 xx 4/7 + 1/2 xx 3/4`
    `=37/56`

Probability, SMB-010

Each time she throws a dart, the probability that Gaga hits the dartboard is  `4/7`.

She throws two darts, one after the other.

What is the probability that she misses the dartboard with both darts?   (2 marks)

Show Answers Only

`9/49`

Show Worked Solution

`P text{(hits)} = 4/7\ \ =>\ \ P\text{(misses)} = 1-4/7=3/7`

`P text{(misses twice)}` `= 3/7 xx 3/7`
  `= 9/49`

Probability, SMB-009

Jon spins each pointer 50 times.

 

Each time he added the numbers that the pointers landed on.

His results are shown below.

\begin{array} {|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\textbf{Sum of numbers}\rule[-1ex]{0pt}{0pt} & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & \textbf{Total}\\
\hline
\rule{0pt}{2.5ex}\textbf{Number of spins}\rule[-1ex]{0pt}{0pt} & 1 & 2 & 3 & 6 & 8 & 9 & 7 & 5 & 4 & 4 & 1 & \textbf{50} \\
\hline
\end{array}

What percentage of the spins resulted in a sum of 9?   (2 marks)

Show Answers Only

`text(10%)`

Show Worked Solution
`text(Percentage)` `= text(number of 9’s)/text(number of spins) xx 100`
  `=5/50 xx 100`
  `=10 text(%)`

Probability, SMB-007 MC

Brandi spins the arrow on two identical spinners.

The arrow on each spinner is equally likely to land on 1, 2 or 3.
 

If Brandi adds up the two results, which total is she least likely to get?

  1. `2`
  2. `3`
  3. `4`
  4. `5`
Show Answers Only

`A`

Show Worked Solution

`text{2 can only result from (1, 1).}`

`text(All other totals can have more than 1 combination producing them.)`

`=>A`

Probability, SMB-006

Bromley rolls two standard dice at the same time and adds up the total.

 An incomplete table of the possible outcomes is below.

\begin{array} {|c|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\textbf{Face}  \rule[-1ex]{0pt}{0pt} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6}\\
\hline
\rule{0pt}{2.5ex}\textbf{1}\rule[-1ex]{0pt}{0pt} & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
\rule{0pt}{2.5ex}\textbf{2}\rule[-1ex]{0pt}{0pt} & 3 & 4 & 5 & & & 8 \\
\hline
\rule{0pt}{2.5ex}\textbf{3}\rule[-1ex]{0pt}{0pt} & 4 & 5 & 6 &  &  & 9 \\
\hline
\rule{0pt}{2.5ex}\textbf{4}\rule[-1ex]{0pt}{0pt} & 5 &  & 7 & 8 & 9 & 10 \\
\hline
\rule{0pt}{2.5ex}\textbf{5}\rule[-1ex]{0pt}{0pt} & 6 &  & 8 &  & 10 & 11 \\
\hline
\rule{0pt}{2.5ex}\textbf{6}\rule[-1ex]{0pt}{0pt} & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
\end{array}

  1. Complete the table.   (2 marks)

  2. What is the most likely total that Bromley will roll?   (1 mark)

Show Answers Only

i.

\begin{array} {|c|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\textbf{Face}  \rule[-1ex]{0pt}{0pt} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6}\\
\hline
\rule{0pt}{2.5ex}\textbf{1}\rule[-1ex]{0pt}{0pt} & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
\rule{0pt}{2.5ex}\textbf{2}\rule[-1ex]{0pt}{0pt} & 3 & 4 & 5 & \colorbox{lightblue}{6} & \colorbox{lightblue}{7} & 8 \\
\hline
\rule{0pt}{2.5ex}\textbf{3}\rule[-1ex]{0pt}{0pt} & 4 & 5 & 6 & \colorbox{lightblue}{7} & \colorbox{lightblue}{8} & 9 \\
\hline
\rule{0pt}{2.5ex}\textbf{4}\rule[-1ex]{0pt}{0pt} & 5 & \colorbox{lightblue}{6}  & 7 & 8 & 9 & 10 \\
\hline
\rule{0pt}{2.5ex}\textbf{5}\rule[-1ex]{0pt}{0pt} & 6 & \colorbox{lightblue}{7} & 8 & \colorbox{lightblue}{9} & 10 & 11 \\
\hline
\rule{0pt}{2.5ex}\textbf{6}\rule[-1ex]{0pt}{0pt} & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
\end{array}

ii.    `7`

Show Worked Solution

i.

\begin{array} {|c|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\textbf{Face}  \rule[-1ex]{0pt}{0pt} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6}\\
\hline
\rule{0pt}{2.5ex}\textbf{1}\rule[-1ex]{0pt}{0pt} & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
\rule{0pt}{2.5ex}\textbf{2}\rule[-1ex]{0pt}{0pt} & 3 & 4 & 5 & \colorbox{lightblue}{6} & \colorbox{lightblue}{7} & 8 \\
\hline
\rule{0pt}{2.5ex}\textbf{3}\rule[-1ex]{0pt}{0pt} & 4 & 5 & 6 & \colorbox{lightblue}{7} & \colorbox{lightblue}{8} & 9 \\
\hline
\rule{0pt}{2.5ex}\textbf{4}\rule[-1ex]{0pt}{0pt} & 5 & \colorbox{lightblue}{6}  & 7 & 8 & 9 & 10 \\
\hline
\rule{0pt}{2.5ex}\textbf{5}\rule[-1ex]{0pt}{0pt} & 6 & \colorbox{lightblue}{7} & 8 & \colorbox{lightblue}{9} & 10 & 11 \\
\hline
\rule{0pt}{2.5ex}\textbf{6}\rule[-1ex]{0pt}{0pt} & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
\end{array}

 
ii.    `text{The most frequent total = 7 (6 times)}`

`:.\ text{7 is the most likely to be rolled.}`

Probability, SMB-005 MC

Two standard dice are rolled at the same time and the two numbers are added up.
 

nap-159-nc03 

Which total is most likely?

  1. `2`
  2. `3`
  3. `4`
  4. `6`
Show Answers Only

`D`

Show Worked Solution

`text(A total of 6 is most likely.)`

`text{Note that a 6 can occur in the following ways:}`

`(5,1), (1,5), (4,2), (2,4) and (3,3)`

`text(No other option given has as many combinations.)`

Probability, SMB-004 MC

Peter has a bag of marbles. 75% of his marbles are blue.

Peter takes a green marble from his bag and loses it in a game.

If he takes another marble from the bag without looking, what are the chances it is blue?

  1. less than 75%
  2. equal to 75%
  3. greater than 75%
Show Answers Only

`C`

Show Worked Solution

`text{Taking marbles two times without replacement → dependent events.}`

`text(When taking the second marble, there will be greater than 75% chance)`

`text(of choosing blue because there are the same amount of  blue marbles)`

`text(to be chosen but 1 less marble of another colour.)`

`=>C`

Probability, SMB-003 MC

Arun flips an unbiased coin 200 times.

Which result is most likely?

  1. `20\ text(tails)`
  2. `98\ text(tails)`
  3. `108\ text(tails)`
  4. `196\ text(tails)`
Show Answers Only

`B`

Show Worked Solution

`text{Each toss is an independent event with 50% chance for both heads and tails}.`

`text{The expected result after 200 tosses is 100 tails, 100 heads.}`

`:.\ text{The most likely result = 98 tails (closest to 100)}`

`=> B`

Probability, SMB-002

A representative soccer team is chosen from 30 players who play for two clubs, Portland and Lithgow.

10 players from Portland and 20 players from Lithgow are playing in the trials, and 7 players from Portland and 8 from Lithgow are selected in the representative team.
 

 
One player at the trial is randomly selected.

What is the probability that the player is from Lithgow and is selected in the representative team?

Give your answer to two decimal places.   (2 marks)

Show Answers Only

`0.27`

Show Worked Solution

`Ptext{(player is from Lithgow and selected)}`

`= P(A and B)`

`= P(A) xx P(B)\ \ \ \text{(independent events)}`

`= 20/30 xx 8/20`

`= 8/30`

`= 0.27\ \text{(2 d.p.)}`

Probability, SMB-001

A coin is tossed 3 times. There are 8 possible outcomes.

What is the probability of getting 2 heads and 1 tail in any order?   (2 marks)

Show Answers Only

`3/8`

Show Worked Solution

`text(Strategy 1)`

`text{The table (array) below lists the possible outcomes:}`
 

  1st toss H H H H T T T T
  2nd toss H H T T H H T T
  3rd toss H T H T H T H T
       ✓  ✓    ✓      

`text(From table,)\ Ptext{(2H, 1T)} = 3/8`
  

`text(Strategy 2)`

`text(Using a probability tree:)`
 

Similarity, SMB-028

The floor plan of a home unit has been drawn to scale.
 


 

What is the total floor area of the home unit in square metres?  (2 marks)

Show Answers Only

`text{55.5 m}^2`

Show Worked Solution

`text{Conversion: 1000 mm = 1 metre}`

`text{Calculate area by splitting into 2 rectangles}`

`text{Area}`  `=  5.5 × 5 + (4.5 +2.5) xx 4`  
   `=  55.5\ text{m}^2`  

Similarity, SMB-015

Towns `A, B` and `C` are marked on the scale diagram below.
  

The distance from Town `A` to Town `B` is 9 km.

What is the distance, in kilometres, from Town `B` to Town `C`?   (2 marks)

Show Answers Only

`3.6\ text(km)`

Show Worked Solution

`text(Let)\ \ d = text(distance from Town)\ B\ text(to Town)\ \ C`

`text(S)text(ince the diagram is to scale,)`

`d/9` `= 2/5`
`:. d` `= (9 xx 2)/5`
  `= 3.6\ text(km)`

Similarity, SMB-014 MC

Spiro is making a scale drawing of his house.

  • The height of the garage in the scale model is 6 centimetres.
  • The height of Spiro's actual garage is 3 metres.

What does 1 centimetre in Spiro's scale drawing represent in his real house?

  1. 1800 cm
  2. 200 cm
  3. 100 cm
  4. 50 cm
Show Answers Only

`D`

Show Worked Solution
`text(6 cm)` `= 3\ text(m)`
  `= 300\ text(cm)`
`:. 1\ text(cm)` `= 300 -: 6`
  `= 50\ text(cm)`

 
`=>D`

Similarity, SMB-013

Libby plays on a hockey field that is 120 metres long.

She makes a scale diagram of the field using a ratio of `1: 400`.
 

How long, in centimetres, should Libby make the scale diagram of the field?   (2 marks)

Show Answers Only

`text(30 cm)`

Show Worked Solution

`text(Ratio is)\ \ 1:400`

`text{Converting to centimetres:}`

`text{120 m = 120 × 100 = 12 000 cm}`

`:.\ text(Length of the scale diagram)`

`=12\ 000 ÷ 400`

`= 30\ text(cm)`

Similarity, SMB-011

Kransky has a photo which is 25 cm wide and 10 cm high

He wants to enlarge it to make a poster with a width of 50 cm.

What will be the height of the poster?   (2 marks)

Show Answers Only

`text(20 cm)`

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`text(Scale factor)` `= text(new width)/text(old width)`
  `= 50/25`
  `= 2`

 

`:.\ text(New height)` `= 2 xx 10`
  `= 20\ text(cm)`

Similarity, SMB-010 MC

The actual body length of a beetle Brad has caught is 24 mm.

A scale drawing of the beetle is shown below.

What scale is used in the drawing?

  1. \(1\ \text{cm represents } 5\ \text{mm}\)
  2. \(1\ \text{cm represents } 2\ \text{mm}\)
  3. \(2\ \text{cm represents } 1\ \text{mm}\)
  4. \(5\ \text{cm represents } 1\ \text{mm}\)
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\(B\)

Show Worked Solution
\(12\ \text{cm}\) \(: 24\ \text{mm}\)
\(120\ \text{mm}\) \(: 24\ \text{mm}\)
\(5\) \(: 1\)
\(1\ \text{cm}\) \(: 2\ \text{mm  (possible options)}\)

 
\(\Rightarrow B\)

Similarity, SMB-009

Juanita wanted to print a larger copy of her photo of a tree.

She increased the scale by a factor of 3.

She named the first picture Photo A and the second picture Photo B.
 

 If Photo B is 3.2 cm high, how high is Photo A in centimetres, correct to 1 decimal place?    (2 marks)

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`1.1\ text(cm)`

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`text{Since Photo B is increased by a factor of 3,}`

`text(Height of Photo B)/text(Height of Photo A)` `=3`
`:.\ text(Height of Photo A)` `=text(Height of Photo B)/3`
  `=3.2/3`
  `=1.0666…`
  `=1.1\ text{cm  (1 d.p.)}`

Similarity, SMB-008 MC

The picture below shows a flower.
 

 
The picture is 2 cm wide. The actual flower is 40 cm wide.

What scale is used in the picture?

  1. 1 cm represents 80 cm
  2. 4 cm represents 20 cm
  3. 1 cm represents 20 cm
  4. 1 cm represents 2 cm
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`C`

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`2:40\ \ =>\ \ 1:20`

`text(1 cm represents 20 cm)`

`=>C`

Similarity, SMB-007

Two circles have the same centre, `O`, as shown in the diagram below.
 

The radius of the small circle is  `2/3`  the radius of the large circle.

Arc `CD` is 21 cm and the angle between the lines `AC` and `BD` is 45°.

What is the length of the arc `RS` in centimetres?   (3 marks)

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`text(42 cm)`

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`text{Circles are similar}`

`text(Arc)\ ST` `= 2/3 xx text(arc)\ CD`
  `= 2/3 xx 21`
  `= 14\ text(cm)`

 

`angleSOR` `= 180-45`
  `= 135^@`
  `= 3 xx angleSOT`

 

`:. text(Arc)\ RS` `= 3 xx text(arc)\ ST`
  `= 3 xx 14`
  `= 42\ text(cm)`

Similarity, SMB-006

Select the pair of similar triangles.

The triangles are not drawn to scale.   (2 marks)

Triangle A Triangle B  
 
Triangle C Triangle D  
 
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`text(Triangles A and D)`

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`text(Similar triangles are equiangular.)`

`text(Calculate triangle angles of each option.)`

`text(Triangle A: 55, 55, 70)`

`text(Triangle B: 50, 65, 65)`

`text(Triangle C: 50, 55, 65)`

`text(Triangle D: 55, 55, 70)`

`:.\ text{Triangles A and D are similar.}`

Similarity, SMB-005

Trapeziums `ABCD` and `AEFG` are similar.
 

What is the distance from `E` to `F`?   (2 marks)

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`2\ text(m)`

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`text{Since trapeziums are similar:}`

`(EF)/(BC)` `= (AG)/(AD)`
`(EF)/8` `= 3/12`
`EF` `= (3 xx 8)/12`
  `= 2\ text(m)`

Similarity, SMB-004

A traffic light is 2.4 m tall. Its shadow from a nearby floodlight is 3 m long.
 

What is the height of the floodlight?   (3 marks)

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`text(4.0 m)`

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`text{Triangles share common angle and both have right angles}`

`=>\ text{Triangles are similar (equiangular)}`

`text{Since corresponding sides are in similar ratios:}`

`h/5` `= 2.4/3`
`:. h` `= (5 xx 2.4)/3`
  `= 4\ text(metres)`

Similarity, SMB-003

On a map, the distance between two towns is measured at 54 millimetres.

  1. The actual distance between the two towns is 16.2 kilometres. What is the scale of the map?  (1 mark)

  2. If two other towns on the same map are 9.2 centimetres apart, what is the actual distance between the two towns in kilometres?  (1 mark)

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  1. `1: 300\ 000`
  2. `27.6\ text(km)`
Show Worked Solution

i.   `text{Convert both distance to the same unit (cm).}`

`text(54 mm)` `= 5.4\ text(cm)`
`text(16.2 km)` `= 16\ 200\ text(m)`
  `= 1\ 620\ 000\ text(cm)`

 

`:.\ text(Scale) \ \ 5.4\ ` `: 1\ 620\ 000`
`1\ ` `: 300\ 000`

 

ii.    `text(Actual distance)` `= 9.2 xx 300\ 000`
    `= 2\ 760\ 000\ text(cm)`
    `= 27\ 600\ text(m)`
    `= 27.6\ text(km)`

Similarity, SMB-002

A map is drawn to scale, on 1-cm paper, showing the position of a supermarket and a cinema. A reservoir is also shown.
 


 

It takes 10 minutes to walk in a straight line from the cinema to the supermarket at a constant speed of 3 km/h. Show that the scale of the map is 1 cm = 100 m.  (3 marks)

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`text(See Worked Solutions)`

Show Worked Solution

`text(3 km/h = 3000 metres per 60 minutes)`

♦ Mean mark part 49%.

`text(In 10 minutes:)`

`text(Actual distance) = 3000 xx 10/60 = 500\ text(metres)`

`text(Distance on map = 5 cm)`

`:.\ text(Scale   5 cm)` `: 500\ text(metres)`
`text(1 cm)` `: 100\ text(metres)`

Similarity, SMB-001

The scale on a given map is `1:80\ 000`.

If the actual distance between two points is 3.4 kilometres, how far apart on the map would be the two points be, in centimetres?  (2 marks)

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`4.25\ text(cm)`

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`text(Distance on map)`

`= text(real distance)/text(scale)`

`= text(3.4 km)/(80\ 000)`

`= text(3400 m)/(80\ 000)`

`= 0.0425\ text(m)`

`= 4.25\ text(cm)`

Volume, SMB-012

The solid below is made from joining a rectangular prism to a triangular prism, with all measurements in centimetres.
 

Find the volume of the solid, giving your answer to the nearest cubic centimetre.   (3 marks)

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`11\ 451\ text(cm)^3`

Show Worked Solution
`text{Area (face)}` `= (9 xx 13) + (1/2 xx 13 xx 7)`  
  `=162.5\ text{cm}^2`  

 

`text(Volume)` `= A h`
  `= 162.5 xx 6`
  `= 975\ text(cm)^3`

Volume, SMB-011

The cylinder shown below has a diameter of 18 centimetres and a length of 45 centimetres.
 

Find the volume of the cylinder, giving your answer to the nearest cubic centimetre.   (2 marks)

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`11\ 451\ text(cm)^3`

Show Worked Solution
`text(Volume)` `= pi r^2 h`
  `= pi xx 9^2 xx 45`
  `= 11\ 451.10…`
  `= 11\ 451\ text(cm)^3\ \ text{(nearest cm}^3 text{)}`

Volume, SMB-010

The solid below is a triangular prism. All measurements are in centimetres.
 

Find the volume of the solid, in cubic centimetres.  (2 marks)

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`378\ \text{cm}^3`

Show Worked Solution
`text{Area (triangle)}` `=1/2 xx b xx h`  
  `=1/2 xx 9 xx 12`  
  `=54\ text{cm}^2`  

 

`text(Volume)` `= A xx h`
  `= 54 xx 7`
  `= 378\ \text{cm}^3`

Volume, SMB-009

An ancient temple, modelled below, was constructed as a trapezoidal prism.
 

Find the volume of the temple in cubic metres.  (2 marks)

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`432\ \text{m}^3`

Show Worked Solution
`text{Area (trapezium)}` `=1/2h(a+b)`  
  `=1/2 xx 4 (10+14)`  
  `=48\ text{m}^2`  

 

`text(Volume)` `= A xx h`
  `= 48 xx 9`
  `= 432\ \text{m}^3`

Volume, SMB-008

A jewellery box in the shape of a rectangular prism is pictured below with all measurements in centimetres.
 

Find the volume of the jewellery box, giving your answer correct to the nearest cubic centimetre.  (2 marks)

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`271\ text{cm}^3`

Show Worked Solution
`text(Volume)` `= l xx b xx h`
  `= 9.5 xx 4.2 xx 6.8`
  `= 271.32`
  `= 271\ text{cm}^3\ text{(nearest cm}^3 text{)}`

Volume, SMB-004

Calculate the volume of earth that must be excavated so that the pool shell below fits exactly into the hole. All measurements are in metres.  (3 marks)
 

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\(136.8\ \text{m}^3\)

Show Worked Solution

\(\text{The face of the pool is a rectangle and trapezium (composite).}\)

\(\text{Area (face)}\) \(= (2.3 \times 4) + \dfrac{1}{2} h (a+b)\)
  \(= 9.2 + \dfrac{1}{2} \times 8 (2.3+1.1) \)
  \(=22.8\ \text{m}^2\)

 

\(\text{Volume}\) \(=Ah\)
  \(=22.8 \times 6\)
  \(=136.8\ \text{m}^3\)

Volume, SMB-003

Steel rods are manufactured in the shape of equilateral triangular prisms.

 

Find the volume of the prism (answer correct to 1 decimal place).  (3 marks)

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`3464.1\ text(cm)^3`

Show Worked Solution

`text{Area of triangular face (using sine rule)}`

`=1/2 xx 10 xx 10 xx sin60°`

`=43.301…`

 

`text(Volume)` `=Ah`
  `=43.301… xx 80`
  `=3464.10…`
  `=3464.1\ text{cm³  (1 d.p.)}`

Volume, SMB-002

A net is made using four rectangles and two trapeziums. It is folded to form a solid.

 What is the volume of the solid, in cm3 ?   (3 marks)

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`576\ text(cm)^3`

Show Worked Solution

`text(Volume)=Ah\ \ text(where)\ A\ text(is the area of a trapezium)`

COMMENT: Note that `h` in the “area” formula is different to the `h` used in the “volume” formula.
`A` `=1/2 h(a+b)`
  `=1/2xx8(11+5)`
  `=64\ text(cm²)`

 

`:.V=Ah=64xx9=576\ text(cm)^3`