SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Measurement, STD2 M7 2024 HSC 38

A cake is constructed using a cylinder and a cone-shaped top. The cylinder has diameter 30 cm and height 6 cm.
 

The ratio of the volume of the cylinder to the volume of the cone-shaped top is \(5:1\).

The cake is to be cut into equal slices with volume 212 cm\(^{3}\).

How many equal slices can be cut from the cake?   (3 marks)

Show Answers Only

\(24\ \text{slices}\)

Show Worked Solution
\(V_{\text{cylinder}}\) \(=\pi r^{2}h\)  
  \(=\pi \times 15^{2} \times 6\)  
  \(=4241.15\ \text{cm}^{3}\)  

 
\(V_{\text{cylinder}} : V_{\text{cone}} = 5:1\)

\(\Rightarrow V_{\text{cone}} = \dfrac{1}{5} \times 4241.15=848.23\ \text{cm}^{3}\)

\(\text{Volume of cake}\ = 4241.15+848.23=5089.4\ \text{cm}^{3}\)

\(\therefore\ \text{Number of slices}\ \) \(=\dfrac{5089.4}{212}\)  
  \(=24\ \text{slices}\)  
♦♦ Mean mark 36%.

Measurement, STD2 M7 2023 HSC 11 MC

A bag contains 150 jelly beans. Some of them are red and the rest are blue. The ratio of red to blue jelly beans is `2:3`.

Sophie eats 10 of each colour.

What is the new ratio of red to blue jelly beans?

  1. `2:3`
  2. `4:9`
  3. `5:8`
  4. `11:17`
Show Answers Only

`C`

Show Worked Solution

`text{Original ratio}\ = 2:3`

`text{5 parts = 150}\ \ =>\ \ text{1 part}\ = 150/5=30`

`text{Original ratio (by number)}\ = 2 xx 30:3 xx 30 = 60:90`

`text{After eating 10 of each colour:}`

`text{Ratio}\ = 50:80 = 5:8`

`=>C`

Measurement, STD2 M7 2022 HSC 38

A full container has 4.8 L of a mixture of cordial and water in the ratio 1:3.

After removing 1.2 L of the mixture, more water is added to refill the container.

What is the ratio of cordial to water in the final mixture?  (3 marks)

Show Answers Only

`3:13`

Show Worked Solution

`text{Original mixture ratio}`

`text{C : W = 1.2 litres : 3.6 litres}`

`text{Removing 1.2 L of mixture, removes}`

`text{→ 300 mL cordial and 900 mL water}`
 

`text{Cordial in mixture after refill}`

`=1200-300`

`=900\ text{mL}`
 

`text{Water in mixture after refill}`

`=3600-900+1200`

`=3900\ text{mL}`
 

`:.\ text{C : W (final mixture)}`

`=900:3900`

`=3:13`


♦♦ Mean mark 33%.

Measurement, STD2 M7 2021 HSC 15 MC

A total of 11 400 people entered a running race. The ratio of professional runners to amateurs was  `3 : 16`. All the professional runners completed the race while 600 of the amateurs did not complete the race.

For those who completed the race, what is the ratio, in its simplest form, of professional runners to amateurs?

  1. `1 : 2`
  2. `1 : 5`
  3. `1 : 8`
  4. `1 : 19`
Show Answers Only

`B`

Show Worked Solution

`3:16 =>\ text(Total parts = 19)`

♦♦ Mean mark 28%.

`text(Number of professionals) = 3/19 xx 11\ 400 = 1800`

`text(Number of amateurs) = 16/19 xx 11\ 400 = 9600`

`text(Amateurs who finished) = 9600 – 600 = 9000`
 

`:.\ text{Ratio of professionals : amateurs (who finished)}`

`= 1800 : 9000`

`=1:5`

 
`=> B`

Measurement, STD2 M7 SM-Bank 17

There are 48 people in a meeting room.

The ratio of females to males in the room is 13 to 11.

How many females need to leave the room so that the ratio of females to males in the room is 1 to 1?   (2 marks)

Show Answers Only

`4\ text(females)`

Show Worked Solution

`text(Females : Males) = 13 : 11`

`text(S)text(ince there are 48 people in the room,)`

`26\ text(females : 22 males.)`

 

`:. 4\ text(females need to leave for the ratio to become 1 : 1)`

Measurement, STD2 M7 2019 HSC 18

Andrew, Brandon and Cosmo are the first three batters in the school cricket team. In a recent match, Andrew scored 30 runs, Brandon scored 25 runs and Cosmo scored 40 runs.

  1. What is the ratio of Andrew's to Brandon's to Cosmo's runs scored, in simplest form?  (2 marks)

  2. In this match, the ratio of the total number of runs scored by Andrew, Brandon and Cosmo to the total number of runs scored by the whole team is `19:36`.
  3. How many runs were scored by the whole team?  (2 marks)

Show Answers Only
  1. `6:5:8`
  2. `180\ text(runs)`
Show Worked Solution
a.    `A:B:C` `= 30:25:40`
    `= 6:5:8`

 

b.    `text(Total runs by)\ A,B,C` `= 30 + 25 + 40`
    `= 95`

 

`text(Let)\ R` `=\ text(team runs)`
`R/95` `= 36/19`
`:. R` `= (36 xx 95)/19`
  `= 180\ text(runs)`

Measurement, STD2 M7 SM-Bank 10

A potting mix is made up of two ingredients, potting soil and manure, in the ratio 5:2.

21 kilograms of potting mix are used to fertilise the  rectangular vegetable garden pictured below.
 


 

  1. What rate must the whole 21 kilograms of potting mix be applied to be spread evenly over the garden. Give your answer in grams per square metre.  (2 marks)

  2. How much manure is needed to make the required potting mix?  (1 mark)

Show Answers Only
  1. `150\ text(g/m²)`
  2. `text(6 kg)`
Show Worked Solution

i.   `text(Potting mix: 21 kg  ⇒  21, 000 grams)`

`text(Area of garden)` `= 20 xx 7`  
  `= 140\ text(m²)`  

 

`:.\ text(Rate)` `= (21\ 000)/140`
  `= 150\ text(g/m²)`


ii.   `text(Ratio  5 : 2)\ \ =>\ \ 5/7 : 2/7`

`:.\ text(Amount of manure)` `= 2/7 xx 21`
  `= 6\ text(kg)`

Measurement, STD2 M7 SM-Bank 6

In a raffle, the total prize money is shared among the first two tickets drawn in the ratio  6 : 4.

The prize for the second ticket drawn is $360.

What is the total prize money?  (2 marks)

Show Answers Only

`$900`

Show Worked Solution

`4/10 xx text(Total prize money = $360)`

`=> 1/10 xx text(Total prize money = $90)`

`:.\ text(Total prize money)` `= 90 xx 10`
  `= $900`

Measurement, STD2 M7 SM-Bank 3 MC

There are 8 male chimpanzees in a community of 24 chimpanzees.

What is the ratio of males to females in the community?

  1.  1 : 3
  2.  1 : 2
  3.  3 : 1
  4.  2 : 1
Show Answers Only

`=>\ text(B)`

Show Worked Solution
`text(Males)` `:\ text(Females)`
`8` `: (24 – 8)`
`8` `: 16`
`1` `: 2`

 
`=>\ text(B)`

Measurement, STD2 M7 2013 HSC 30c

Joel mixes petrol and oil in the ratio  40 : 1  to make fuel for his leaf blower. 

  1. Joel pours 5 litres of petrol into an empty container to make fuel for his leaf blower.

     

    How much oil should he add to the petrol to ensure that the fuel is in the correct ratio?   (1 mark)

  2. Joel has 4.1 litres of fuel left in his container after filling his leaf blower.

     

    He wishes to use this fuel in his lawnmower. However, his lawnmower requires the petrol and oil to be mixed in the ratio  25 : 1.

     

    How much oil should he add to the container so that the fuel is in the correct ratio for his lawnmower?   (3 marks)

Show Answers Only
  1. `text(0.125 L)`
  2. `60\ text(mL)`
Show Worked Solution
♦ Mean mark 50% 
i.    `text(Petrol : Oil) = 40\ :\ 1`
  `5\ text(Litres :)\ X = 40\ :\ 1`
`:. 5/X` `= 40/1`
`40X` `=5`
`X` `= 5/40 = 0.125\ text(L)`

 

`:.\ text(Joel should add 0.125 L of oil)`

 

ii.  `text(4.1 L)\ = 4100\ text(mL)`

`text(In ratio  40:1,  there is)`

♦♦♦ Mean mark 8% 
STRATEGY: The critical information required is how much petrol is in the container. Once known, the oil required to satisfy a 25:1 ration can be easily found.

`=>\ text(4000 mL Petrol and 100 mL Oil)`

 

`text(Lawnmower ratio) = 25:1`

`text(Let Oil required for 4000 mL Petrol)\ = X\ text(mL)`

`X/4000` `=1/25`
`25X` `=4000`
`X` `=4000/25`
  `=160\ text(mL)`

 
`text(4 L of petrol requires 160 mL of oil)`

`text(Container already has 100 mL of oil)`

`:.\ text(Oil to add)\ ` `=160\ -100`
  `=60\ text(mL)`