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Rates, SM-Bank 044 MC

Fleur lives 15 kilometres from her work.

On Wednesday, she drove to work and averaged 60 kilometres per hour.

On Thursday, she took the bus which averaged 15 kilometres per hour.

What was the extra time of the bus journey, in minutes, compared to when she drove on Wednesday?

  1. \(15\)
  2. \(45\)
  3. \(60\)
  4. \(75\)
Show Answers Only

\(B\)

Show Worked Solution
\(\text{Time on Wednesday}\) \(=\dfrac{15}{60}\)
  \(= 0.25\ \text{hour}\)
  \(= 15\ \text{minutes}\)

 

\(\text{Time on Thursday}\) \(=\dfrac{15}{15}\)
  \(= 1\ \text{hour}\)
  \(= 60\ \text{minutes}\)

 

\(\therefore\ \text{The extra time taking the bus}\)

\(=60-15\)

\(= 45\ \text{minutes}\)

 
\(\Rightarrow B\)

Rates, SM-Bank 043 MC

Elvis walks from his home to the beach through a park.
 

  
Which of the following situations best fits the distance/time graph above, where distance is Elvis' distance from home?

  1. Elvis ran to the park, sat down on a bench, and then ran home.
  2. Elvis walked to the park, sat down on a bench, then ran to the beach.
  3. Elvis ran to the beach, sat down, then ran home.
  4. Elvis walked to the park, sat down on a bench, then walked to the beach.
Show Answers Only

\(D\)

Show Worked Solution

\(\text{The horizontal section of the graph indicates Elvis’ distance}\)

\(\text{from home is not changing and he is resting.}\)

\(\text{The slope before and after the rest break is the same, so Elvis’}\)

\(\text{speed of travel was the same before and after the rest.}\)

 

\(\therefore\ \text{Elvis walked, rested then walked.}\)

 
\(\Rightarrow D\)

Rates, SM-Bank 042 MC

Columbo had a full drum of water.
 

 
  

He put two holes in the bottom and the water began leaking out.

After a few minutes, Columbo closed off one of the holes in the drum and the water poured out more slowly.

Which graph below best shows the depth of water in the drum against time?

  
A. B. C. D.
Show Answers Only

\(B\)

Show Worked Solution

\(\text{The depth decreases quickly at a constant rate at the start}\)

\(\text{(steep decline) and then slows down when one hole is }\)

\(\text{closed (less steep decline).}\)

\(\Rightarrow B\)

Rates, SM-Bank 041 MC

Kelly drives her motorised scooter to the shopping centre 9 km away at an average speed of 45 km per hour.

How long does the trip take?

  1. 5 minutes
  2. 12 minutes
  3. 24 minutes
  4. 40.5 minutes
Show Answers Only

\(B\)

Show Worked Solution
\(\text{Time}\) \(=\dfrac{\text{Distance}}{\text{Speed}}\)
  \(= \dfrac{9}{45}\)
  \(= \dfrac{1}{5}\ \text{hours}\)
  \(=12\ \text{minutes}\)

 
\(\Rightarrow B\)

Rates, SM-Bank 040 MC

Kingsley drives her moped to a beach 100 km away at an average speed of 60 km.

How long does the trip take?

  1. 36 minutes
  2. 1 hour 20 minutes
  3. 1 hour 24 minutes
  4. 1 hour 40 minutes
Show Answers Only

\(D\)

Show Worked Solution
\(\text{Time}\) \(=\dfrac{\text{Distance}}{\text{Speed}}\)
  \(= \dfrac{100}{60}\)
  \(= 1\dfrac{40}{60}\ \text{hours}\)
  \(=1\ \text{hour}\ 40\ \text{minutes}\)

 
\(\Rightarrow D\)

Rates, SM-Bank 039 MC

Lachlan drives his boat to an island 100 km away at an average speed of 80 km/h.

How long does the trip take?

  1. 48 minutes
  2. 1 hour 15 minutes 
  3. 1 hour 20 minutes 
  4. 1 hour 30 minutes 
Show Answers Only

\(B\)

Show Worked Solution
\(\text{Time}\) \(=\dfrac{\text{Distance}}{\text{Speed}}\)
  \(= \dfrac{100}{80}\)
  \(= 1.25\ \text{hours}\)
  \(=1\ \text{hour}\ 15\ \text{minutes}\)

 
\(\Rightarrow B\)

Rates, SM-Bank 038

Ant is travelling at 110 km/h in his car.

If he maintains this speed, how many kilometres will he travel in 1 hour and 20 minutes?  Give your answer correct to the nearest kilometre.   (2 marks)

Show Answers Only

\(147\ \text{km}\)

Show Worked Solution
\(\text{Distance}\) \(=\text{Speed}\times \text{time}\)
  \(= 110\times \dfrac{80}{60}\)
  \(= 110\times \dfrac{4}{3}\)
  \(= 146.\dot{6}\ \text{km}\)
  \(=147\ \text{km  (nearest km)}\)

Rates, SM-Bank 037

Vicki is travelling at 90 km/h in her car.

If she maintains this speed, how many kilometres will she travel in 35 minutes?  (2 marks)

Show Answers Only

\(52.5\ \text{km}\)

Show Worked Solution
\(\text{Distance}\) \(=\text{Speed}\times \text{time}\)
  \(= 90\times \dfrac{35}{60}\)
  \(= 90\times \dfrac{7}{12}\)
  \(= 52.5\ \text{km}\)

Rates, SM-Bank 036

Joy is travelling at 42 km/h on her racing bike.

If she maintains this speed, how many kilometres will she travel in 50 minutes?  (2 marks)

Show Answers Only

\(35\ \text{km}\)

Show Worked Solution
\(\text{Distance}\) \(=\text{Speed}\times \text{time}\)
  \(= 42\times \dfrac{50}{60}\)
  \(= 42\times \dfrac{5}{6}\)
  \(= 35\ \text{km}\)

Rates, SM-Bank 035

Billy Bob is travelling at 120 km/h in his car.

If he maintains this speed, how many kilometres will he travel in 40 minutes?  (2 marks)

Show Answers Only

\(80\ \text{km}\)

Show Worked Solution
\(\text{Distance}\) \(=\text{Speed}\times \text{time}\)
  \(= 120\times \dfrac{40}{60}\)
  \(= 120\times \dfrac{2}{3}\)
  \(= 80\ \text{km}\)

Rates, SM-Bank 034

Aurora works part time in a donut shop.

On weekends, she earns 2.5 times as much per hour as she earns on weekdays.

One week, she works 14.5 hours on weekdays and 3 hours on the weekend.

Her pay for the week was $319.

How much does she earn in 1 hour on a weekday?  (2 marks)

Show Answers Only

\($12.50/\text{hour}\)

Show Worked Solution
\(\text{Pay hours}\) \(= 14.5+3\times 2.5\)
  \(=22\)

 

\(\text{Earnings for 1 hour}\) \(=\dfrac{319}{22}\)
  \(=$14.50\)

 
\(\therefore \text{Aurora earns}\ $14.50\ \text{in 1 hour.}\)

Rates, SM-Bank 033

Fleur works part time in a flower shop.

On weekends, she earns 2 times as much per hour as she earns on weekdays.

One week, she works 23 hours on weekdays and 1.5 hours on the weekend.

Her pay for the week was $351.

How much does she earn in 1 hour on a weekday?  (2 marks)

Show Answers Only

\($13.50/\text{hour}\)

Show Worked Solution
\(\text{Pay hours}\) \(= 23+2\times 1.5\)
  \(=26\)

 

\(\text{Earnings for 1 hour}\) \(=\dfrac{351}{26}\)
  \(=$13.50\)

 
\(\therefore \text{Fleur earns}\ $13.50\ \text{in 1 hour.}\)

Rates, SM-Bank 032

Fanny works part time at Guzman y Gomez.

On weekends, she earns 1.5 times as much per hour as she earns on weekdays.

One week, she works 12 hours on weekdays and 8 hours on the weekend.

Her pay for the week was $414.

How much does she earn in 1 hour on a weekday?  (2 marks)

Show Answers Only

\($17.25/\text{hour}\)

Show Worked Solution
\(\text{Pay hours}\) \(= 12+ 8 \times 1.5\)
  \(=24\)

 

\(\text{Earnings for 1 hour}\) \(=\dfrac{414}{24}\)
  \(=$17.25\)

Rates, SM-Bank 031

Clay has a casual job at the local movie cinema.

On weekends, he earns 1.5 times as much per hour as he earns on weekdays.

One week, he works 20 hours on weekdays and 6 hours on the weekend.

His pay for the week was $498.80.

How much does he earn in 1 hour on a weekday?  (2 marks)

Show Answers Only

\($17.20/\text{hour}\)

Show Worked Solution
\(\text{Pay hours}\) \(= 20+6\times 1.5\)
  \(=29\)

 

\(\text{Earnings for 1 hour}\) \(=\dfrac{498.80}{29}\)
  \(=$17.20\)

 
\(\therefore \text{Clay earns}\ $17.20\ \text{in 1 hour.}\)

Rates, SM-Bank 030

Betty works part time in a clothing shop.

On weekends, she earns 2 times as much per hour as she earns on weekdays.

One week, she works 15 hours on weekdays and 3 hours on the weekend.

Her pay for the week was $367.50.

How much does she earn in 1 hour on a weekday?  (2 marks)

Show Answers Only

\($17.50/\text{hour}\)

Show Worked Solution
\(\text{Pay hours}\) \(= 15+3\times 2\)
  \(=21\)

 

\(\text{Earnings for 1 hour}\) \(=\dfrac{367.50}{21}\)
  \(=$17.50\)

 
\(\therefore \text{Betty earns}\ $17.50\ \text{in 1 hour.}\)

Rates, SM-Bank 029

Brin works part time in a coffee shop.

On weekends, he earns 1.5 times as much per hour as he earns on weekdays.

One week, he works 9 hours on a weekday and 4 hours on the weekend.

His pay for the week was $270.

How much does he earn in 1 hour on a weekday?  (2 marks)

Show Answers Only

\($18/\text{hour}\)

Show Worked Solution
\(\text{Pay hours}\) \(= 9+4\times 1.5\)
  \(=15\)

 

\(\text{Earnings for 1 hour}\) \(=\dfrac{270}{15}\)
  \(=$18\)

 
\(\therefore \text{Brin earns}\ $18\ \text{in 1 hour.}\)

Rates, SM-Bank 028

John's old tractor used 8.3 litres of fuel per 100km.

His new tractor uses 5.9 litres of fuel per 100 km.

John pays $2.15 per litre for fuel and drives 20,000 km each year.

How much money will John save on fuel each year with his new tractor?  (2 marks)

Show Answers Only

\($1032\)

Show Worked Solution

\(\text{Fuel cost of old tractor}\)

\(=8.3\times $2.15\times \dfrac{20\ 000}{100}\)

\(=8.3\times $2.15\times 200\)

\(=$3569\)
 

\(\text{Fuel cost of new tractor}\)

\(=5.9\times $2.15\times\dfrac{20\ 000}{100}\)

\(=5.9\times $2.15\times 200\)

\(=$2537\)

 

\(\therefore\ \text{John’s fuel savings each year}\)

\(=$3569-$2537\)

\(=$1032\)

Rates, SM-Bank 027

One litre of softdrink contains 90 grams of sugar.

How many millilitres of softdrink contain 4.5 grams of sugar?  (2 marks)

Show Answers Only

\( 50\ \text{mL}\)

Show Worked Solution
\(\text{Millilitres}\) \(= \dfrac{4.5}{90}\times 1000\ \text{mL}\)
  \(= 50\ \text{mL}\)

Rates, SM-Bank 026

Kurt is travelling from Newcastle to Sydney. The journey is 165 kilometres.

His car uses 8.35 litres of fuel per 100 kilometres.

How much fuel will Kurt need to make the journey?

Round your answer to the nearest litre.  (2 marks)

Show Answers Only

\(14\ \text{litres (nearest whole number)}\)

Show Worked Solution
\(\text{Fuel needed}\) \(=\dfrac{165}{100}\times 8.35\)
  \(= 13.77\dots\)
  \(= 14\ \text{litres (nearest whole number)}\)

Rates, SM-Bank 025

A laundromat can wash 12 loads of laundry in one hour at full capacity.

A standard load of laundry weighs 7 kilograms.

Here is some information about two different washing machines.
 

Washing machine

Amount of water used per
kilogram of laundry (litres)
Top loader 10.25
Front loader 6.5

 
Working at full capacity, how many litres of water would the laundromat expect to save in one hour by using the front loader instead of the top loader?  (2 marks)

Show Answers Only

\(315\ \text{L}\)

Show Worked Solution

\(\text{Top loader water used (1 load)}\)

\(= 7\times 10.25\)

\(= 71.75\ \text{L}\)
 

\(\text{Front loader water used (1 load)}\)

\(= 7\times 6.5\)

\(= 45.5\ \text{L}\)
 

\(\therefore\ \text{Expected water saved}\)

\(= 12\times (71.75-45.5)\)

\(= 315\ \text{L}\)

Rates, SM-Bank 024

Gary used 4 litres of paint to paint a wall.

The wall was a rectangle 2 metres high and 3 metres wide.

How many litres of paint would he need to paint a rectangular wall which is 3 metres high and 5 metres wide?  (2 marks)

Show Answers Only

\(10\)

Show Worked Solution

\(\text{Area of smaller wall}\)

\(= 2\times 3\)

\(= 6\ \text{m}^2\)

 

\(\text{S}\text{ince 4 litres of paint are needed to}\)

\(\text{paint the small wall:}\)

\(\rightarrow\ \text{Paint needed for 1 m}^2\)

\(= \dfrac{4}{6}\)

\(= \dfrac{2}{3}\ \text{litre}\)

\(\rightarrow\ \text{Area of larger wall}\)

\(= 3\times 5\)

\(= 15\ \text{m}^2\)

 

\(\text{Paint needed}\) \(= 15\times \dfrac{2}{3}\)
  \(= 10\ \text{litres}\)

Rates, SM-Bank 023

Genghis and Kublai are shooting arrows at a target.

Genghis shoots an arrow every 3 seconds.

Kublai shoots an arrow every 8 seconds.

They shoot their first arrow together at 10:00 am.

How many more times will they shoot arrows at exactly the same time in the next 3 minutes?  (2 marks)

Show Answers Only

\(7\)

Show Worked Solution

\(\text{Lowest common multiple of 3 and 8}=24\)

\(\therefore\ \text{Every 24 seconds, the arrows are shot at the same time.}\)

\(\rightarrow\ \text{Arrows are shot at the same time:}\)

\(24, 48, 72, 96, 120, 144, 168\ \text{seconds}\)

\(\therefore\ 7\ \text{more times  (time ≤ 180 seconds)}\)

Rates, SM-Bank 022

Patrick uses 25 litres of water every minute when he has a shower.

Kate uses 150 litres of water when she has a bath.

How many fewer litres of water does Patrick use in his 5½ minute shower than Kate does in her bath?  (2 marks)

Show Answers Only

\(12.5\ \text{litres}\)

Show Worked Solution

\(\text{Patrick water usage}\)

\(= 5.5\times 25\)

\(= 137.5\ \text{litres}\)
 

\(\therefore\ \text{Less litres}\)

 \(= 150-137.5\)
  \(=12.5\ \text{litres}\)

Rates, SM-Bank 021

Kate buys a new computer.

The disk in its hard drive makes 120 full turns every second.

How many minutes will it take for the disk to make 324 000 full turns?  (2 marks)

Show Answers Only

\(45\ \text{minutes}\)

Show Worked Solution

\(\text{Turns in 1 minute}\)

\(= 120\times 60\)

\(= 7200\)
 

\(\therefore\ \text{Minutes needed}\)

\(= \dfrac{324\ 000}{7200}\)

\(=45\ \text{minutes}\)

Rates, SM-Bank 018

A solar panel grid on a school roof produces an average of 8.6 kWh of energy per day.

How much energy will the grid produce for the school on average over 7 days?  (2marks)

Show Answers Only

\(60.2\ \text{kWh}\)

Show Worked Solution

\(\text{One multiplication strategy:}\)

\(7\times 8=56\)

\(7\times 0.6 = 4.2\)

\(7\times 8.6 = 56+4.2=60.2\)
 

\(\therefore\ \text{It produces 60.2 kWh over 7 days.}\)

Rates, SM-Bank 017

Constance used 300 grams of sugar to make 12 brownies.

How many grams of sugar will she need to make 18 brownies?  (2 marks)

Show Answers Only

\(450\ \text{grams}\)

Show Worked Solution

\(\text{Method 1}\)

\(\text{Sugar to make 18 brownies}\)

\(=\dfrac{18}{12}\times 300\)

\(=1.5\times 300\)

\(=450\ \text{grams}\)

 
\(\text{Method 2 – Unitary Method}\)

\(300\ \text{g/}12\ \text{cookies}\) \(=\dfrac{300}{12}\ \text{g/cookie}\)
  \(=25\ \text{g/cookie}\)

 

\(\text{Sugar to make 18 brownies}\) \(=25\times 18\)
  \(=450\ \text{grams}\)

Rates, SM-Bank 016

Riley lives 3 km from the park.

He jogs at a constant speed of 10 km per hour.

How many minutes does it take for Riley to get to the park?  (2 marks)

Show Answers Only

\(18\ \text{minutes}\)

Show Worked Solution
\(\text{Time}\) \(=\dfrac{\text{Distance}}{{\text{S}\text{peed}}}\)
  \(=\dfrac{3}{10}\ \text{hr}\)
  \(=\dfrac{3}{10}\times 60\)
  \(=18\ \text{minutes}\)

Rates, SM-Bank 015

Fleur lives 15 kilometres from her work.

On Wednesday, she drove to work and averaged 60 kilometres per hour.

On Thursday, she took the bus which averaged 15 kilometres per hour.

What was the extra time of the bus journey, in minutes, compared to when she drove on Wednesday?  (2 marks)

Show Answers Only

\(45\ \text{minutes}\)

Show Worked Solution
\(\text{Time on Wednesday}\) \(=\dfrac{15}{60}\)
  \(= 0.25\ \text{hour}\)
  \(= 15\ \text{minutes}\)

 

\(\text{Time on Thursday}\) \(=\dfrac{15}{15}\)
  \(= 1\ \text{hour}\)
  \(= 60\ \text{minutes}\)

 

\(\therefore\ \text{The extra time taking the bus}\)

\(=60-15\)

\(=45\ \text{minutes}\)

Rates, SM-Bank 014

A brewery can make 1250 cans of ale and 900 cans of lager per hour.

The brewery runs non-stop and each can weighs 350 grams.

How many kilograms of ale and lager, altogether, does the brewery make in 1 full day?  (2 marks)

Show Answers Only

\(18\ 060\ \text{kilograms per day}\)

Show Worked Solution

\(\text{Total cans made per hour}\)

\(=1250+900\)

\(= 2150\)
 

\(\text{Kilograms made in 1 hour}\)

\(=2150\times 350\)

\(=752\ 500\ \text{grams}\)

\(= 752.5\ \text{kilograms}\)

\(\therefore\ \text{Kilograms made in 1 day}\) \(=752.5\times 24\)
  \(=18\ 060\)

Rates, SM-Bank 013

Johnno was standing 300 metres away from the stage at a rock concert.

If the sound travelled at 330 metres per second from the stage, how many seconds did the sound take to get to Johnno? Give your answer correct to 2 decimal places.  (2 marks)

Show Answers Only

\(0.91\ \text{seconds}\)

Show Worked Solution
\(\text{Time}\) \(=\dfrac{\text{distance}}{\text{speed}}\)
  \(= \dfrac{300}{330}\)
  \(= 0.909090…\approx 0.91\ \text{(2 d.p.)}\)

  
\(\therefore \text{The sound takes approximately}\ 0.91\ \text{seconds to reach Johnno.}\)

Rates, SM-Bank 012

Rhonda rode her hovercraft at a speed of 5 metres per second.

If she rode for 3 minutes, how far did she go?  (2 marks)

Show Answers Only

\(\text{900}\ \text{m}\)

Show Worked Solution
\(3\ \text{minutes}\) \(=60\times 3\ \text{seconds}\)
  \(=180\ \text{seconds}\)

 

\(\text{Distance}\) \(=\text{Speed}\times\text{Time}\)
  \(= 5\times 180\)
  \(= 900\ \text{m}\)

Rates, SM-Bank 011 MC

In a science experiment, Albert needs to add 60 millilitres of acid to every 2 litres of water.

If Albert only has 0.5 litres of water left, how many millilitres of acid should he add?

  1. 0.15 mL
  2. 0.6 mL
  3. 15 mL
  4. 30 mL
Show Answers Only

\(C\)

Show Worked Solution
\(\text{Given}\ \ 60\ \text{mL}\) \(\rightarrow 2\ \text{litres}\)
\(30\ \text{mL}\) \(\rightarrow 1\ \text{litre}\)
\(\therefore 15\ \text{mL}\) \(\rightarrow 0.5\ \text{litres}\) 

 
\(\therefore 15\ \text{millilitres of acid should be added}\)
 
\(\Rightarrow C\)

Rates, SM-Bank 010 MC

A tram at the zoo does a complete 5 kilometre loop in 30 minutes.

If the tram travelled at the same speed, how long did it take to complete 2 kilometres?

  1. 8 minutes
  2. 12 minutes
  3. 18 minutes
  4. 24 minutes
Show Answers Only

\(B\)

Show Worked Solution
\(30\ \text{minutes}/5\ \text{kms}\) \(=\dfrac{30}{5}\ \text{minutes}/\dfrac{5}{5}\ \text{km}\)
  \(=6\ \text{minutes}/\text{km}\)

 

\(\therefore\ \text{Time for 2 kms}\) \(=6\times 2\)
  \(=12\ \text{minutes}\)

 
\(\Rightarrow B\)

 

Rates, SM-Bank 009 MC

Sid is selling bike tyre tubes at a market stall.

He makes $54 from selling 6 bike tyre tubes.

All bike tyre tubes cost the same.

How much will Sid make if he sells 11 bike tyre tubes?

  1. $65
  2. $83
  3. $99
  4. $108
Show Answers Only

\(C\)

Show Worked Solution
\($54/6\ \text{tubes}\) \(=\dfrac{$54}{6}/\dfrac{6}{6}\text{tubes}\)
  \(=$9/\text{tube}\)

 

\(\therefore\ \text{Price of 11 tubes}\) \(=11\times 9\)
  \(=$99\)

 
\(\Rightarrow C\)

Rates, SM-Bank 008

Bryce organises parking for major events in the city.

He has open air parking spaces for 3 cars on every 27 square metres of land.

How many cars could Bryce park on 4680 square metres?  (2 marks)

Show Answers Only

\(520\)

Show Worked Solution

\(\text{3 cars/}\ 27\ \text{m}^2=1\ \text{car/}9\ \text{m}^2\)

\(\therefore\ \text{cars on }4680\ \text{m}^2\) \(=\dfrac{4680}{9}\)
  \(=520\ \text{cars}\)

Rates, SM-Bank 007

A young echidna weighed 1200 grams at the beginning of November just before leaving the burrow. 

At the end of March it weighed 1850 grams.

Calculate the average rate of growth of the echidna for the 5 month period.  (2 marks)

Show Answers Only

\(130\ \text{g}/ \text{month}\)

Show Worked Solution
\(\text{Growth}\) \(=1200-1850\)
  \(=650\ \text{g}\)

 

\(\text{Rate of growth}/\text{month}\) \(=\dfrac{650}{5}\ \text{g}/\text{month}\)
  \(=130\ \text{g}/\text{month}\)

 

Rates, SM-Bank 006

Pete travelled 646 kilometres and used 76 litres of fuel.

  1. Express Pete's average fuel consumption in:
     
    i.  litres per 100 kilometres, giving your answer rounded to 1 decimal place.  (2 marks)

    ii.  kilometres per litre.  (1 mark)

      
  2. If Pete's average fuel consumption remained the same, how many litres of fuel would he use to travel 272 kilometres?  (2 marks)

Show Answers Only

a.    i.    \(11.8\ \text{L}/100\ \text{km  (1 d.p.)}\)

ii.   \(8.5\ \text{km}/\text{L}\)

b.    \(32\ \text{litres}\)

Show Worked Solution
a.    i.    \(\text{Average litres/100 km}\) \(=\dfrac{76}{646}\times 100/100\ \text{km}\)
    \(=11.764\dots /100\ \text{km}\)
    \(\approx 11.8\ \text{L}/100\ \text{km  (1 d.p.)}\)

 

ii.   \(\text{Average km/L}\) \(=\dfrac{646}{76}\ \text{km}/\text{L}\)
  \(=8.5\ \text{km}/\text{L}\)

 

b.    \(\text{Litres}\) \(=\dfrac{272}{8.5}\)
    \(=32\)

 
\(\therefore\ \text{Pete would use}\ 32\ \text{litres of fuel to travel}\ 272\ \text{km.}\)

Rates, SM-Bank 005

A tree was 143 cm tall at the beginning of January.  At the end of May it measured 233 cm tall.

  1. Calculate the average rate of growth of the tree for the 5 month period.  (2 marks)

  2. If this rate of growth were to continue, how many more months would it take the tree to reach a height of 611 cm?  (2 marks)

Show Answers Only

a.    \(18\ \text{cm}/ \text{month}\)

b.    \(21\ \text{months}\)

Show Worked Solution
a.    \(\text{Growth}\) \(=233-143\)
    \(=90\ \text{cm}\)

 

\(\text{Rate of growth}/\text{month}\) \(=\dfrac{90}{5}\ \text{cm}/\text{month}\)
  \(=18\ \text{cm}/\text{month}\)

 

b.    \(\text{Growth}\) \(=611-233\)
    \(=378\ \text{cm}\)
     
  \(\text{Months}\) \(=\dfrac{378}{18}\)
    \(=21\)

 
\(\therefore\ \text{It would take}\ 21\ \text{months for the tree to grow to a height of }\ 611\ \text{cm.}\)

Rates, SM-Bank 004

All the water has leaked from a 10 litre bucket in 20 minutes.

  1. Calculate the leaking rate in litres/minute.  (1 mark)

  2. How long would it take to empty a 12 litre bucket at the same rate?  (2 marks)

Show Answers Only

a.    \(0.5\ \text{litres}/ \text{minute}\)

b.    \(24\ \text{minutes}\)

Show Worked Solution
a.    \(10\ \text{litres}/20\ \text{minutes}\) \(=\dfrac{10}{20}\ \text{litres}/\dfrac{20}{20}\ \text{minutes}\)
    \(=0.5\ \text{litres}/\text{minute}\)

 

b.    \(0.5\ \text{litres}/\text{hour}\) \(=1\ \text{litres}/2\ \text{minutes}\)
    \(=12\ \text{litres}/2\times 12\ \text{minutes}\)
    \(=12\ \text{litres}/24\ \text{minutes}\)

 

\(\therefore\ \text{It would take}\ 24\ \text{minutes to empty a}\ 12\ \text{litre bucket.}\)

Rates, SM-Bank 003

Write each of the following as a rate in simplified form.

  1. $136 for every 8 hours worked  (1 mark)

  2. $171 000 to purchase 3 cars  (1 mark)

  3. 112 millimetres of rain in 14 days  (1 mark)

Show Answers Only

a.    \($17/ \text{hour}\)

b.    \($57\ 000/\text{car}\)

c.    \(8\ \text{mm}/\text{day}\)

Show Worked Solution
a.    \($136/8\ \text{hours}\) \(=\dfrac{$136}{8}/\dfrac{8}{8}\ \text{hours}\)
    \(=$17/\text{hour}\)

 

b.    \($171\ 000/3\ \text{cars}\) \(=\dfrac{$171\ 000}{3}/\dfrac{3}{3}\ \text{cars}\)
    \(=$57\ 000/\text{car}\)

 

c.    \(112\ \text{mm}/14\ \text{days}\) \(=\dfrac{112}{14}\ \text{mm}/\dfrac{14}{14}\ \text{days}\)
    \(=8\ \text{mm}/\text{day}\)

Rates, SM-Bank 002

Write each of the following as a rate in simplified form.

  1. $18 for every 5 kilograms  (1 mark)

  2. 30 animals per 6 cages  (1 mark)

  3. 63 rainy days in 7 months  (1 mark)

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a.    \($3.60/ \text{kilogram}\)

b.    \(5\ \text{animals}/\text{cage}\)

c.    \(9\ \text{rainy days}/\text{month}\)

Show Worked Solution
a.    \($18/5\ \text{kg}\) \(=\dfrac{$18}{5}/\dfrac{5}{5}\ \text{kg}\)
    \(=$3.60/\text{kg}\)

 

b.    \(30\ \text{animals}/6\ \text{cages}\) \(=\dfrac{30}{6}\ \text{animals}/\dfrac{6}{6}\ \text{cages}\)
    \(=5\ \text{animals}/\text{cage}\)

 

c.    \(63\ \text{rainy days}/7\ \text{months}\) \(=\dfrac{63}{7}\ \text{rainy days}/\dfrac{7}{7}\ \text{months}\)
    \(=9\ \text{rainy days}/\text{month}\)

Ratio, SM-Bank 054

James found a fly that was 8 mm long in real life.

Calculate the scale used in this scale drawing of the fly.  (2 marks)

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\(7\ :\ 1\)

Show Worked Solution
\(\text{Scale}\) \(=\text{Scaled length}\ :\ \text{Actual length}\)
  \(=5.6\ \text{cm}\ :\ 8\ \text{mm}\)
  \(=56\ \text{mm}\ :\ 8\ \text{mm}\)
  \(=7\ :\ 1\)

Ratio, SM-Bank 053

Oliver's bike is 1.68 metres long in real life.

Calculate the scale used in this scale drawing of Oliver's bike.  (2 marks)

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\(1\ :\ 42\)

Show Worked Solution
\(\text{Scale}\) \(=\text{Scaled length}\ :\ \text{Actual length}\)
  \(=4\ \text{cm}\ :\ 1.68\ \text{m}\)
  \(=4\ \text{cm}\ :\ 168\ \text{cm}\)
  \(=1\ :\ 42\)

Ratio, SM-Bank 052

Macton and Brownville are 15 centimetres apart on a map with a scale of \(1:150\ 000\). How far apart, in kilometres, are the two cities in real life?  (2 marks)

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\(22.5\ \text{km}\)

Show Worked Solution

\(\text{Scale factor}=150\ 000\)

\(\text{Actual distance}\) \(=\text{Scaled distance}\times \text{scale factor}\)
  \(=15\times 150\ 000\ \text{cm}\)
  \(=2\ 250\ 000\ \text{cm}\)
  \(=22.5\ \text{km}\)

Ratio, SM-Bank 051

Jinderee and Exetroll are 8 centimetres apart on a map with a scale of \(1\ :\ 2000\). How far apart, in kilometres, are the two towns in real life?  (2 marks)

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\(0.16\ \text{km}\)

Show Worked Solution

\(\text{Scale factor}=2000\)

\(\text{Actual distance}\) \(=\text{Scaled distance}\times \text{scale factor}\)
  \(=8\times 2000\ \text{cm}\)
  \(=16\ 000\ \text{cm}\)
  \(=0.16\ \text{km}\)

Ratio, SM-Bank 050

Determine the scale factor if a scaled length of 1.6 centimetres represents an actual length of 0.48 kilometres.  (2 marks)

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\(30\ 000\)

Show Worked Solution

\(\text{Scale ratio}=\text{scale length}\ :\ \text{actual length}\)

\(=1.6\ \text{cm}\ :\ 0.48\ \text{km}\)

\(=1.6\ \text{cm}\ :\ 48\ 000\ \text{cm}\)

\(=1\ :30\ 000\)

\(\text{Scale factor}\)

\(=30\ 000\)

Ratio, SM-Bank 049

Determine the scale factor if an actual length of 2 centimetres represents a scaled length of 5 metres.  (2 marks)

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\(\dfrac{1}{250}\)

Show Worked Solution

\(\text{Scale ratio}=\text{scale length}\ :\ \text{actual length}\)

\(=5\ \text{m}\ :\ 2\ \text{cm}\)

\(=500\ \text{cm}\ :\ 2\ \text{cm}\)

\(=500\ :\ 2\)

\(=1\ :\ \dfrac{1}{250}\)

\(\text{Scale factor}\)

\(=\dfrac{1}{250}\)

Ratio, SM-Bank 048

Determine the scale factor if an actual length of 0.3 millimetres represents a scaled length of 15 centimetres.  (2 marks)

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\(\dfrac{1}{500}\)

Show Worked Solution

\(\text{Scale ratio}=\text{scale length}\ :\ \text{actual length}\)

\(=15\ \text{cm}\ :\ 0.3\ \text{mm}\)

\(=1500\ \text{mm}\ :\ 3\ \text{mm}\)

\(=500\ :\ 1\)

\(=1\ :\ \dfrac{1}{500}\)

\(\text{Scale factor}\)

\(=\dfrac{1}{500}\)

Ratio, SM-Bank 047

Determine the scale factor if an actual length of 2 millimetres represents a scaled length of 2 metres.  (2 marks)

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\(\dfrac{1}{1000}\)

Show Worked Solution

\(\text{Scale ratio}=\text{scale length}\ :\ \text{actual length}\)

\(=2\ \text{m}\ :\ 2\ \text{mm}\)

\(=2000\ \text{mm}\ :\ 2\ \text{mm}\)

\(=1000\ :\ 1\)

\(=1\ :\ \dfrac{1}{1000}\)

\(\text{Scale factor}\)

\(=\dfrac{1}{1000}\)

Ratio, SM-Bank 046

Determine the scale factor if a length of 3 centimetres on a map represents an actual length of 2.4 kilometres.  (2 marks)

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\(80\ 000\)

Show Worked Solution

\(\text{Scale ratio}\)

\(=3\ \text{cm}\ :\ 2.4\ \text{km}\)

\(=3\ \text{cm}\ :\ 240\ 000\ \text{cm}\)

\(=1\ :\ 80\ 000\)

\(\text{Scale factor}\)

\(=80\ 000\)

Ratio, SM-Bank 045

Determine the scale factor if a length of 5 millimetres on a map represents an actual distance of 20 metres.  (2 marks)

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\(4000\)

Show Worked Solution

\(\text{Scale ratio}\)

\(=5\ \text{mm}\ :\ 20\ \text{m}\)

\(=5\ \text{mm}\ :\ 20\ 000\ \text{mm}\)

\(=1\ :\ 4000\)

\(\text{Scale factor}\)

\(=4000\)

Ratio, SM-Bank 044

A model train is built using a scale of \(4:20\ 000\).

For each of these actual lengths, find the scaled length.

  1. \(40\ \text{m}\)  (1 mark)

  2. \(87\ 450\ \text{cm}\)  (1 mark)

  3. \(2400\ \text{m}\)  (1 mark)

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a.    \(\text{8 mm or 0.8 cm}\)

b.    \(\text{17.49 cm or 0.1749 m}\)

c.    \(\text{48 cm or 0.48 m}\)

Show Worked Solution

\(\text{Scale}\ \rightarrow\ 4:20\ 000=1:5000\)

a.    \(\text{Scaled length }\) \(\dfrac{40}{5000}\ \text{m}=\dfrac{40\ 000}{5000}\ \text{mm}\)
    \(=\text{8 mm}=\text{0.8 cm}\)

 

b.    \(\text{Scaled length}\) \(=\dfrac{87\ 450}{5000}\ \text{cm} \)
    \(=\text{17.49 cm}=\text{0.1749 m}\)

 

c.    \(\text{Scaled length}\) \(=\dfrac{2400}{5000}\ \text{m}=\dfrac{240\ 000}{5000}\ \text{cm}\)
    \(=\text{48 cm}=\text{0.48 m}\)

Ratio, SM-Bank 043

The scale on a map is \(4:20\ 000\).

For each of these scaled distances, find the actual distances in kilometres.

  1. \(2\ \text{cm}\)  (1 mark)

  2. \(5\ \text{mm}\)  (1 mark)

  3. \(18.4\ \text{cm}\)  (1 mark)

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a.    \(100\ \text{m or } 0.1\ \text{km}\)

b.    \(40\ \text{m or } 0.04\ \text{km}\)

c.    \(920\ \text{m or } 0.92\ \text{km}\)

Show Worked Solution

\(\text{Scale}=4:20\ 000=1:5000\)

a.    \(\text{Actual Distance}\) \(=2\ \text{cm}\times 5000 \)
    \(= 10\ 000\ \text{cm}\)
    \(=100\ \text{m}=0.1\ \text{km}\)

 

b.    \(\text{Actual Distance}\) \(=8\ \text{mm}\times 5000 \)
    \(=40\ 000\ \text{mm}\)
    \(=40\ \text{m}=0.04\ \text{km}\)

 

c.    \(\text{Actual Distance}\) \(=18.4\ \text{cm}\times 5000 \)
    \(= 92\ 000\ \text{cm}\)
    \(=920\ \text{m}=0.92\ \text{km}\)

Ratio, SM-Bank 042

The scale on a map is \(1:10\ 000\).

For each of these scaled distances, find the actual distances in kilometres.

  1. \(3\ \text{cm}\)  (1 mark)

  2. \(8\ \text{mm}\)  (1 mark)

  3. \(24.3\ \text{cm}\)  (1 mark)

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a.    \(300\ \text{m or } 0.3\ \text{km}\)

b.    \(80\ \text{m or } 0.08\ \text{km}\)

c.    \(2430\ \text{m or } 2.43\ \text{km}\)

Show Worked Solution
a.    \(\text{Actual Distance}\) \(=3\ \text{cm}\times 10\ 000 \)
    \(= 30\ 000\ \text{cm}\)
    \(=300\ \text{m}=0.3\ \text{km}\)

 

b.    \(\text{Actual Distance}\) \(=8\ \text{mm}\times 10\ 000 \)
    \(= 80\ 000\ \text{mm}\)
    \(=80\ \text{m}=0.08\ \text{km}\)

 

c.    \(\text{Actual Distance}\) \(=24.3\ \text{cm}\times 10\ 000 \)
    \(= 243\ 000\ \text{cm}\)
    \(=2430\ \text{m}\)
    \(=2.43\ \text{km}\)

Ratio, SM-Bank 041 MC

An ancient coin is made up of gold and silver in the ratio 1:3.

The coin weighs 21.65 grams.

How many grams of silver are in the ancient coin?

  1. \(5.4125\)
  2. \(7.217\)
  3. \(14.433\)
  4. \(16.2375\)
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\(D\)

Show Worked Solution

\(\text{Total parts}=1+3=4\) 
  

\(\text{Fraction of silver}=\dfrac{3}{4}\)

\(\text{Silver}=\dfrac{3}{4}\times 21.65=16.2375\)
  

\(\Rightarrow D\)

Ratio, SM-Bank 040

Paul stacks milk cartons into supermarket refrigerator shelves.

Each shelf is stacked with 6 full cream milk cartons, 4 lite milk cartons and 2 skim milk cartons.

Every hour Paul stacks 240 milk cartons in total.

How many lite milk cartons does he stack every hour?  (2 marks)

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\(80\)

Show Worked Solution

\(\text{Total Cartons per shelf}=6+4+2=12\)
  

\(\text{Fraction of lite}=\dfrac{4}{12}=\dfrac{1}{3}\)

\(\text{Total lite per hour}=\dfrac{1}{3}\times 240=80\)

Ratio, SM-Bank 039 MC

A Christmas pudding recipe uses 3 cups of sugar for every 4 cups of sultanas.

Select the correct combination of sugar and sultanas for this recipe.

  1. \(\dfrac{1}{3}\ \text{cup of sugar, }\dfrac{1}{4}\ \text{cup of sultanas}\)
  2. \(1\ \text{cup of sugar, }1.5\ \text{cups of sultanas}\)
  3. \(1.5\ \text{cups of sugar, }2\ \text{cups of sultanas}\)
  4. \(4\ \text{cups of sugar, }5\ \text{cups of sultanas}\)
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\(C\)

Show Worked Solution
\(\text{Ratio}\) \(= 3:4\)
  \(=1.5:2\)

 

\(\therefore\ 1.5\ \text{cups of sugar, }2\text{ cups of sultanas}\)

\(\text{is in the correct ratio.}\)

 
\(\Rightarrow C\)