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v1 Measurement, STD2 M7 2022 HSC 38

A 5.0 L container is filled with a mixture of milk and coffee in the ratio 3:2.

After removing 1.0 L of the mixture, pure milk is added to refill the container.

What is the ratio of milk to coffee in the final mixture? (3 marks)

Show Answers Only

`17:8`

Show Worked Solution

`text{Original mixture amounts}`

`text{Milk : Coffee = 3000 mL : 2000 mL}`

`text{Removing 1.0 L (1000 mL) removes}`

`text{→ }(3/5)×1000=600\text{ mL milk and }(2/5)×1000=400\text{ mL coffee}`
 

`text{Milk after removal}`

`=3000-600=2400\text{ mL}`
 

`text{Coffee after removal}`

`=2000-400=1600\text{ mL}`
 

`text{After refilling with 1000 mL milk:}`

`text{Milk }=2400+1000=3400\text{ mL}`

`text{Coffee }=1600\text{ mL}`
 

`:.\ \text{Final ratio } \text{Milk : Coffee}`

`=3400:1600`

`=17:8`


♦♦ Mean mark 34%.

v1Measurement, STD2 M7 2021 HSC 25

A rectangular sportsground has been drawn to scale on a 1-cm grid as shown. The scale used is `1:2000`.
 

Johnny took 12 minutes to walk around the perimeter of this sportsground.

What was Johnny's average speed in kilometres per hour?  (4 marks)

Show Answers Only

`3.4 \ text{km/hr}`

Show Worked Solution
`text{Distance walked on map} \ = \ 2 times  (10 + 7) = 34 \ text{cm}`
Mean mark 54%.

 

`text{Actual distance}` ` =34 times 2000`  
  `= 68\ 000 \ text{cm}`  
  `=680 \ text{m}`  
  `= 0.68 \ text{km}`  

 
`12 \ text{minutes}\ = 12/60 = 0.2 \ text{hours}`
 

`text{Speed}` `= text{distance}/text{time}`  
  `= 0.68/0.2`  
  `= 3.4 \ text{km/hr}`  

v1 Measurement, STD1 M5 2021 HSC 26

The diagrams show two similar shapes. The dimensions of the small shape are enlarged by a scale factor of 1.5 to produce the large shape.
 

Calculate the area of the large shape.  (3 marks)

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`94.5\ text(cm)^2`

Show Worked Solution

`text(Dimension of larger shape:)`

♦♦ Mean mark 32%.

`text(Width) = 6 xx 1.5 = 9\ text(cm)`

`text(Height) = 8 xx 1.5 = 12 \ text(cm)`

`text(Triangle height) = 2 xx 1.5 = 3\ text(cm)`

`:.\ text(Area)` `= 9 xx (12-3) + 1/2 xx 9 xx 3`
  `= 94.5\ text(cm)^2`

v1 Measurement, STD2 M7 2012 HSC 27c

A topographic map has a scale of 1 : 250 000.

  1. Two lookouts are 3.6 cm apart on the map.

     

    What is the actual distance between the two lookouts, in kilometres?   (1 mark)

  2. Two towns are 42.5 km apart. How far apart are the two towns on the map, in centimetres?   (1 mark)

Show Answers Only
  1. `text(9 km)`
  2. `text(17 cm)`
Show Worked Solution
♦ Mean mark 41%
MARKER’S COMMENT: Better responses converted between centimetres, metres and kilometres systematically using the map scale (1 unit on the map represents 250 000 of the same unit in reality).
i.    `text{Actual distance (3.6 cm)}` `= 3.6 xx 250\ 000`
  `= 900\ 000\ text(cm)`
  `= 9\ 000\ text(m)`
  `= 9\ text(km)`

 

`:.\ text(The 2 lookouts are 9 km apart.)`

♦ Mean mark 45%

ii.   `text(Towns are 42.5 km apart.)`

`text{From the scale, } 1\ \text{cm} = 250\ 000\ \text{cm} = 2\ 500\ \text{m} = 2.5\ \text{km}`

`=>\ \text{On the map, } 42.5\ \text{km} = 42.5/2.5 = 17\ \text{cm}`

`:.\ \text{Distance on the map is 17 cm.}`

v1 Measurement, STD2 M7 2012 HSC 26f

The capture-recapture technique was used to estimate a population of turtles in 2015.

• 80 turtles were caught, tagged and released.

• Later, 200 turtles were caught at random.

• 40 of these 200 turtles had been tagged.

The estimated population of turtles in 2015 was 25% greater than the estimated population for 2010.

What was the estimated population for 2010? (2 marks)

Show Answers Only

`text{320 turtles}`

Show Worked Solution

`text(Let population in 2015 =)\ P(2015)`

`text(Capture)`

`⇒ 80/{P(2015)}`

`text(Recapture)`

`=> 40/200 = 1/5`

`80/{P(2015)}` `=1/5`
`:. P(2015)` `= 80 xx 5 =400`

 

`text(We know)\ P(2015)\ text(is 25% greater than P(2010))`

`text{(100% + 25%)} xx P(2010)` `= 400`
`125% xxP(2010)` `=400`
`:. P(2010)` `=400/1.25`
  `= 320`

 

`:.\ text{2010 population estimate = 320 turtles}`

v1 Measurement, STD2 M7 2020 HSC 23

In a tropical punch, the ratio of passionfruit juice to guava juice to lime juice is 12 : 10 : 6 .

  1. How much lime juice is needed if the punch is to contain 2.4 litres of passionfruit juice? (2 marks)

  2. The internal dimensions of a drink container, in the shape of a rectangular prism, are shown.
     

    To completely fill the container with the punch, how many litres of guava juice are required? (3 marks)

Show Answers Only
  1. `1.2 text{L}`
  2. `10  text{L}`
Show Worked Solution
a.    `text(12 parts)` `= 2.4\ text(L)`
  `text(1 part)` `= 2.4/12`
    `= 0.2\ text(L)`

 

`:.\ text(6 parts)` `= 6 xx 0.2`  
  `= 1.2\ text(L)`  
b.     `text{Volume of container}` `= 40 xx 20 xx 35`
    `= 28\ 000 \ text{cm}^3`

`1 \ text{mL} \ to \ 1 \ text{cm}^3`

`⇒ \ 28\ 000 \ text{mL of punch}`

`therefore \ text{Guava juice required}` `= text{Guava parts}/text{Total parts} \ xx \ 28\ 000`
  `= \frac{10}{28} \ xx \ 28\ 000`
  `= 10\ 000 \ text{mL}`
  `= 10 \ text{L}`

v1 Measurement, STD2 M7 SM-Bank 17

There are 72 students in a school hall.

The ratio of Year 12 to Year 11 students is 7 to 5.

How many Year 12 students need to leave the hall so that the ratio of Year 12 to Year 11 becomes 1 to 1?   (2 marks)

Show Answers Only

`12\ text(Year 12  students)`

Show Worked Solution

`text(Year 12 : Year  11) = 7 : 5`

`text(Since there are 72 students in the hall,)`

`42\ text(Year 12 : 30 Year 11.)`

`:. 12\ text(Year 12 students need to leave for the ratio to become 1 : 1)`

v1 Measurement, STD2 M7 2019 HSC 18

Dana, Eden and Flynn are the first three players in the school basketball team. In a recent game, Dana scored 24 points, Eden scored 18 points and Flynn scored 30 points.

  1. What is the ratio of Dana's to Eden's to Flynn's points scored, in simplest form?   (2 marks)

  2. In this game, the ratio of the total number of points scored by Dana, Eden and Flynn to the total number of points scored by the whole team is `9:20`.
  3. How many points were scored by the whole team?   (2 marks)

Show Answers Only
  1. `4:3:5`
  2. `160\ text(points)`
Show Worked Solution
a.     `D:E:F` `= 24:18:30`
    `= 4:3:5`

 

b.     `text(Total points by)\ D,E,F` `= 24 + 18 + 30`
    `= 72`

 

`text(Let)\ P` `=\ text(team points)`
`P/72` `= 20/9`
`:. P` `= (20 xx 72)/9`
  `= 160\ text(points)`

v1 Measurement, STD2 M7 2008 HSC 20 MC

A point `P` lies between a lamp post, 1.5 metres high, and a building, 7 metres high. `P` is 2.5 metres away from the base of the post.

From `P`, the angles of elevation to the top of the lamp post and to the top of the building are equal.
 

What is the distance, `x`, from `P` to the top of the tower?

  1. 10.60
  2. 12.50
  3. 13.60
  4. 14.55
Show Answers Only

`C`

Show Worked Solution

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

`text(In smaller triangle:)`

`h^2` `= 1.5^2 + 2.5^2`
  `= 8.5`
`h` `= sqrt 8.5`
   
`x/sqrt8.5` `= 7/1.5 \ \ \ text{(sides of similar Δs in same ratio)}`
`x` `= (7 sqrt 8.5)/1.5`
  `= 13.60…`

 
`=>  C`

v1 Measurement, STD2 M7 2016 HSC 16 MC

The width (`W`) of a road can be calculated using two similar triangles, as shown in the diagram.
  
  

What is the approximate width of the road?

  1. `12.8\ text(m)`
  2. `13.3\ text(m)`
  3. `14.6\ text(m)`
  4. `17.8\ text(m)`
Show Answers Only

`=> C`

Show Worked Solution

`text{Triangles are similar (equiangular)}`

`text(Using similar ratios:)`

`W/(6.5)` `= 18/8`
`:. W` `= (18 xx 6.5)/8`
  `= 14.62…`

 
`=> C`

v1 Measurement, STD2 M7 2022 HSC 4 MC

A wildlife researcher wanted to estimate the number of turtles in a swamp.

She initially caught and tagged 25 turtles before releasing them.

Two weeks later, she caught 50 turtles and found that 10 of them had tags.

What is the best estimate for the total number of turtles in the swamp?

  1. 100
  2. 110
  3. 120
  4. 125
Show Answers Only

`D`

Show Worked Solution

`text{Let}\ \ T=\ text{population of turtles in swamp}`

`text{Initial tag ratio}\ = 25/T`

`text{Recapture ratio}\ = 10/50`

`25/T` `=10/50`
`10T` `=25 xx 50`
`T` `=1250/10`
  `=125`

`=> D`


♦♦ Mean mark 35%.

v1 Measurement, STD2 M7 2023 HSC 11 MC

A garden has 180 flowers. Some are tulips and the rest are roses. The ratio of tulips to roses is `4:5`.

A gardener picks 20 tulips and 25 roses for a bouquet.

What is the new ratio of tulips to roses in the garden?

  1. `2:3`
  2. `3:4`
  3. `4:5`
  4. `5:6`
Show Answers Only

`C`

Show Worked Solution

`text{Original ratio} = 4:5`

`text{Total parts} = 4 + 5 = 9`

`text{1 part} = 180 ÷ 9 = 20`

`text{Tulips} = 4 × 20 = 80`

`text{Roses} = 5 × 20 = 100`

`text{After picking:}`

`text{Tulips} = 80 − 20 = 60`

`text{Roses} = 100 − 25 = 75`

`text{New ratio} = 60:75`

`= 4:5`

`⇒ C`