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Proportionalities (revision)
The ratio of two numbers (quantities) is the quotient of the two numbers (quantities). For example, the ratio of \latex{ 12 } and \latex{ 16 } is \latex{\frac{12}{16}=\frac{3}{4}} that is, \latex{12\div 16=3\div 4.}
You can encounter ratios in many areas of life. For example, they play an important role in cartography, physics, and arts, among others.
Example 1
The scale of a map is \latex{ 1 : 25,000 }.
- What is the distance between two villages in \latex{ kilometres } if they are found \latex{ 10 \;cm } away from each other on the map?
- How far are two points on the map from each other if the actual distance between them is \latex{ 10 } \latex{ km ?}
Solution
- \latex{ 1 } \latex{ cm } on the map corresponds to \latex{25,000} \latex{ cm } \latex{= 250} \latex{ m } \latex{= 0.25} \latex{ km }. \latex{10} \latex{ cm } equals a \latex{10} times larger distance, so \latex{10\times 0.25=2.5} \latex{ km } distance.
Map
Actual distance
\latex{1} \latex{ cm }
\latex{10} \latex{ cm }
\latex{25,000} \latex{ cm } \latex{=0.25} \latex{ km }
\latex{10\times 25,000=2.5} \latex{ km }
\latex{\times 10}
\latex{\times 10}
- A distance of \latex{10} \latex{ km } is \latex{\frac{10}{0,25}=40, } that is, \latex{40} times \latex{0.25} \latex{ km }; therefore, on the map, it corresponds to a distance of \latex{40} \latex{ cm }.
Actual distance
Map
\latex{0.25} \latex{ km } \latex{= 25,000} \latex{ cm }
\latex{1} \latex{ cm }
\latex{10} \latex{ km }
\latex{40\times 1} \latex{ cm } \latex{=40} \latex{ cm }
\latex{\times 40}
\latex{\times 40}
There are two types of proportions: direct and inverse proportion.
Direct proportion
Example 2
Cut the angle in the image out of a piece of paper and measure it with the hands of a clock.
Solution
- Estimate the angle. It is approximately \latex{ 60° }.
- Set the hands of the clock, so they are at the same angle as the paper cutout.
- What time is it according to the clock? It is \latex{ 12:12 }.
- Calculate the angles of the clock hands at \latex{ 12 } \latex{ hours } and \latex{ 12 } \latex{ minutes. }
At \latex{ 12’ }o clock the hands overlap.
Calculate by how many \latex{ degrees } the hands of the clock turn in \latex{ 12 } \latex{ minutes } if you know that the time passed and the angle are directly proportional.
Minute hand
time passed
angle
\latex{60} \latex{ minutes }
\latex{1} \latex{ minutes }
\latex{12} \latex{ minutes }
\latex{360°}
\latex{\,\frac{360°}{60} =6°}
\latex{12\times 6°=72°}
Hour hand
time passed
angle
\latex{12} \latex{ hours }
\latex{1} \latex{ hours } \latex{=60} \latex{ minutes }
\latex{1} \latex{ minute }
\latex{12} \latex{ minute }
\latex{360°}
\latex{\frac{360°}{12} =30°}
\latex{\frac{30°}{60} =0.5°}
\latex{12\times 0.5°=6°}
In \latex{ 12 } \latex{ minutes }, the minute hand turns \latex{ 72º }, while the hour hand turns \latex{ 6º }; thus, the angle formed by them is \latex{72°-6°=66°.} This is perfectly in sync with your estimate.
So, the angle shown in the image is \latex{ 66º }.
Inverse proportion
Example 3
Zack searched the Internet for data on the speed of animals: cockchafer: \latex{3\;\frac{m}{s};} salmon: \latex{5\;\frac{m}{s};} bee: \latex{6.5\;\frac{m}{s};} cheetah: \latex{30\;\frac{m}{s}.}
How long would it take these animals to cover the distance a cockchafer covers in
\latex{ 1 } \latex{ minute? }
Solution
The distance covered is known. If the distance covered within a given time is doubled (speed), then the time needed to cover the given distance is halved. This means that speed is inversely proportional to the time. Make a table.
Animal
Speed
Time
Distance
cockchafer
salmon
bee
cheetah
\latex{3\frac{m}{s}}
\latex{5\frac{m}{s}}
\latex{6.5\frac{m}{s}}
\latex{30\frac{m}{s}}
\latex{\times \frac{6.5}{3} }
\latex{\times \frac{5}{3} }
\latex{1} minute \latex{=60\;s}
\latex{36\;s}
\latex{\frac{360}{13}\approx 27.7 } \latex{ s }
\latex{60\div 10=6 \;s}
\latex{3\frac{m}{s}\times 60 } \latex{ s } \latex{=180} \latex{ m }
\latex{5\frac{m}{s}\times 36 } \latex{ s } \latex{=180} \latex{ m }
\latex{ 6.5\frac{m}{s} \times \frac{360}{13} } \latex{ s } \latex{=180} \latex{ m }
\latex{30\frac{m}{s}\times 6 } \latex{ s } \latex{=180} \latex{ m }
×\latex{10}
\latex{\div \frac{6.5}{3} }
\latex{\div \frac{5}{3} }
÷\latex{10}
It takes the salmon \latex{ 36 } \latex{ s }, the bee \latex{ 27.7 } \latex{ s } and the cheetah \latex{ 6 } \latex{ s } to cover the same distance a cockchafer flies in \latex{ 1 } \latex{ minute. }
In complex problems, you can use the nature of the quantities to determine whether they are proportional, and if so, whether they are directly or inversely proportional to each other.
Correlations between proportional quantities
Example 4
If a family of \latex{ 4 } uses \latex{ 120 } \latex{m^{3}} of water in \latex{ 4 } \latex{ months }
- how much water do \latex{ 2 } people use in \latex{ 2 } \latex{ months };
- how long does it take \latex{ 8 } people to consume \latex{ 60 } \latex{m^{3}} of water;
- how many people consumes \latex{ 120 } \latex{m^{3}} of water in \latex{ 1 } \latex{ month? }
(Let's assume that each person uses the same amount of water in \latex{ 1 } \latex{ month }.)
Solution
- Water consumption is directly proportional to the number of people and the number of months.
\latex{4} people
\latex{2} people
\latex{2} people
in \latex{4} \latex{ months }
in \latex{4} \latex{ months }
in \latex{2} \latex{ months }
consume \latex{120\;m^{3} } of water;
consume \latex{60\;m^{3} } of water;
consume \latex{30\;m^{3} } of water.
\latex{\div 2}
\latex{\div 2}
\latex{\div 2}
\latex{\div 2}
So \latex{ 2 } people consume \latex{ 30 } \latex{m^{3}} of water in \latex{ 2 } \latex{ months. }
Half as many people in the half amount of time consume a quarter of the original water amount.
- Water consumption is directly proportional to the number of people and the number of months.
\latex{4} people
\latex{8} people
\latex{8} people
use \latex{120\,m^{3} } of water
use \latex{240\,m^{3} } of water
use \latex{60\,m^{3} } of water
in \latex{4 } \latex{ months };
in \latex{4 } \latex{ months };
in \latex{1 } \latex{ months }.
\latex{\times 2}
\latex{\times 2}
\latex{\div 4}
\latex{\div 4}
So, \latex{ 8 } people consume \latex{60\;m^{3}} of water in \latex{ 1 } \latex{ month. }
- If the amount of water is constant, the number of people and \latex{ months } are inversely proportional.
\latex{120\;m^{3}} water
\latex{120\;m^{3}} water
in \latex{4} \latex{ months }
in \latex{1} \latex{ months }
is consumed by \latex{4} people;
is consumed by \latex{16} people.
\latex{\div 4}
\latex{\times 4}
So, it takes \latex{ 16 } people to use \latex{120\;m^{3}} of water in \latex{ 1 } \latex{ month. }
\latex{ x }
\latex{ y }
If two quantities are directly proportional, the quotient of their corresponding values is constant.
If two quantities are inversely proportional, the product of their corresponding values is constant.
Example 5
At a petrol station, \latex{ 36 } \latex{ litres } of fuel can be put in the tank of your car in \latex{ 1 } \latex{ minute }. During a pit stop in Formula 1 \latex{ 12 } \latex{ litres } of fuel is pumped into the tank of the race car \latex{ per } \latex{ second. }
- How long would it take to fill your car at the racetrack if it took you a \latex{ minute } and a half at the petrol station?
- How long would it take Hamilton at a petrol station to fill the same amount of fuel as the fuel pumped into his car in \latex{ 8 } \latex{ seconds } at the racetrack?
\latex{ y }
\latex{ x }
\latex{ y } and \latex{ x } are inversely proportional
\latex{x\times y=} constant
\latex{x\times y=} constant
Solution
- For easier comparison, calculate how many \latex{ litres } of fuel can be put in vehicles at the petrol station.
At the petrol station, the time is directly proportional to the volume of the fuel.
in \latex{1 } \latex{ minute } \latex{36} \latex{ litres }
in \latex{1 } \latex{ second } \latex{0.6} \latex{ litres }
\latex{\div 60}
\latex{\div 60}
So, at the petrol station, \latex{ 0.6 } \latex{ litres } of fuel can be filled into vehicles in \latex{ 1 } \latex{ second. }
If the amount of fuel is constant, the speed of the fuel flow is inversely proportional to the time.
Speed of the fuel flow \latex{\left(\frac{l}{s} \right)}
Time (\latex{ s })
\latex{ 0.6 }
\latex{ 12 }
\latex{ 90 }
\latex{ 4.5 }
\latex{\times 20}
\latex{\times \frac{1}{20} }
So, it would take you \latex{ 4.5 } \latex{ seconds } to fill your car with the same amount of fuel at the racetrack.
Speed of the fuel flow \latex{\left(\frac{l}{s} \right)}
Time (\latex{ s })
\latex{ 12 }
\latex{ 0.6 }
\latex{ 8 }
\latex{ 160 }
\latex{\times 20}
\latex{\times \frac{1}{20} }
So, Hamilton would need \latex{ 160 } \latex{ s = 2 } \latex{ minutes } and \latex{ 40 } \latex{ seconds } to fill his race car at the petrol station.
Exercises
{{exercise_number}}. The scale of a map is \latex{ 1 : 1,500,000 }.
- To what distance does \latex{ 5 } \latex{ cm } on the map correspond in reality?
- If two cities are situated \latex{ 225 } \latex{ km } from each other, what is the distance between them on the map?
{{exercise_number}}. What is the scale of the map if the distance between London and Oxford in a beeline is \latex{ 83 } \latex{ km ?}
{{exercise_number}}. Which of the following are directly, indirectly or non-proportional?
- The mass and volume of a liquid in a container.
- The time and distance covered by a vehicle performing uniform linear motion.
- The height and age of a child.
- The number of pages and the thickness of a book.
- The average fuel consumption and the distance the car can travel with a full tank.
{{exercise_number}}. A car covers \latex{ 600 } \latex{ km } in \latex{ 5 } \latex{ hours. } What distance does the car cover in \latex{ 8 } and a half \latex{ hours } at the same speed?
{{exercise_number}}. At the school canteen, \latex{ 10.5 } \latex{ kg } of meat was used to make \latex{ 70 } servings of stew. How much meat is needed to make \latex{ 100 } servings?
{{exercise_number}}. \latex{ 30 } \latex{ kg } of grapes are packed in \latex{ 20 } boxes. How many of these boxes are needed to pack \latex{ 150 } \latex{ kg } of grapes?
{{exercise_number}}. You made fruit juice at home. You filled twenty-five \latex{ 7 }-\latex{ dl } bottles. How many \latex{ 12.5 }-\latex{ dl } bottles can be filled with fruit juice?
{{exercise_number}}. The speed of an aeroplane is \latex{ 1,000 } \latex{\frac{km}{h}}, the speed of another aeroplane is \latex{\frac{3}{5}} of the previous value. If it takes the first aeroplane \latex{ 3 } \latex{ hours } to arrive at its destination, how long is the duration of the other aeroplane's flight to the same destination?
{{exercise_number}}. \latex{ 10 } \latex{ litres } of water flow out of a tap into a bath \latex{ per } \latex{ minute } and, at this rate, it takes \latex{ 20 } \latex{ minutes } to fill the bath completely. How long would it take to fill the bath if \latex{ 25 } \latex{ litres } of water flowed out of the tap every \latex{ minute? }
{{exercise_number}}. \latex{ 10 } \latex{ kg } of corn is needed to fatten a duck. How many more \latex{ kilograms } of corn are needed to fatten \latex{ 20 } ducks instead of \latex{ 15 } ducks?
{{exercise_number}}. \latex{ 5 } kids made \latex{ 80 } sandwiches in \latex{ 2 } \latex{ hours } for a birthday party. How many sandwiches would \latex{ 12 } kids make in
\latex{ 3 } \latex{ hours? }
{{exercise_number}}. If \latex{ 3 } mice eat \latex{ 5 } pieces of cheese in \latex{ 4 } \latex{ days }, then how long does it take \latex{ 5 } mice to eat \latex{ 10 } pieces of cheese?
{{exercise_number}}. Andrew, Ben, and Carl went hiking. Andrew took 4, while Ben had 5 sandwiches. Carl forgot to bring his lunch on the trip. In the end, they split the sandwiches equally among them, but Carl insisted on paying for the sandwiches he ate. Eventually, Carl paid €\latex{ 9 }. How should the other two boys divide the money fairly?
{{exercise_number}}. A solar thermal collector increases the temperature of the water stored in its boiler by \latex{ 1 }\latex{ °C } in \latex{ 30 } \latex{ minutes }. Another solar thermal collector needs \latex{ 40 } \latex{ minutes } to achieve the same rise in temperature. How long would the heating process take if both solar thermal collectors were turned on?
{{exercise_number}}. In a faraway land, a unicorn costs \latex{ 100 } gold coins and a mule. The price of a mule is \latex{ 20 } gold coins and half a pig. A pig can be bought for \latex{ 10 } gold coins and a quarter of a duck. How much does the unicorn cost if the price of a duck is \latex{ 8 } gold coins?
Quiz
On the first day of the week, the speed of the wind blowing was \latex{ 10 } \latex{\frac{km}{h}} at noon. On Tuesday, it was \latex{ 20 } \latex{\frac{km}{h}} at noon. On Wednesday, it was \latex{ 30 } \latex{\frac{km}{h}} at noon. What will the speed of the wind will be on Thursday at noon?