Jūsu grozs ir tukšs
Using direct proportion to solve problems
\latex{ 5 } crates
\latex{ 125 } \latex{ kg }
\latex{ 125 } \latex{ kg }
\latex{ 7 } crates
? \latex{ kg }
? \latex{ kg }
Example 1
The mass of \latex{ 5 } crates of lemons is \latex{ 125 \;kg }. What is the mass of \latex{ 7 } crates of lemons if each crate contains the same amount of lemons?
Solution 1
The mass of lemons in fifth as many crates is the fifth of the original mass, while the mass of lemons in seven times as many crates is seven times the original mass.
\latex{5} crates
\latex{7} crates
\latex{125} \latex{ kg }
? \latex{ kg }
\latex{1} crate
\latex{7} crates
\latex{125} \latex{ kg }\latex{\div 5=25} \latex{ kg }
\latex{25} \latex{ kg }\latex{\times 7=175} \latex{ kg }
\latex{\div 5}
\latex{\div 5}
\latex{\times 7}
\latex{\times 7}
decrease to
one-fifth
one-fifth
decrease to
one-fifth
one-fifth
increase
seven times
seven times
increase
seven times
seven times
The mass of \latex{ 7 } crates of lemons is \latex{ 175 } \latex{ kg }.
Solution 2
You can use the fact that when two values are directly proportional, their quotient is constant. The mass of \latex{ 7 } crates should be marked with the letter \latex{ x }. Record the data you collected in a table.
number of crates (pcs.)
mass of lemons
in the crates (\latex{kg})
the quotient of
corresponding values
corresponding values
\latex{5}
\latex{125}
\latex{\frac{125}{5} }
\latex{7}
\latex{x}
\latex{\frac{x}{7} }
Since \latex{\frac{x}{7} =\frac{125}{5}} , thus \latex{\frac{x}{7} =25}, therefore \latex{x=7\times 25=175.}
The mass of \latex{ 7 } crates of lemons is \latex{ 175 } \latex{ kg }.
The mass of \latex{ 7 } crates of lemons is \latex{ 175 } \latex{ kg }.
Example 2
One group at a language school has \latex{ 12 } students. Nine of them have already paid the monthly fees, totalling €\latex{ 1,800 }. How many \latex{ euros } do the remaining three students have to pay in total?
Solution 1
The number of students and the amount paid is directly proportional.
\latex{9} students
\latex{3} students
\latex{3} students
€\latex{1800}
€ \latex{ x }
€\latex{1800} \latex{\div 3=} €\latex{ 600 }
\latex{\div 3}
\latex{\div 3}
decrease
to one third
to one third
decrease
to one third
to one third
The three remaining students have to pay €\latex{600}.
Solution 2
Denote the amount to be paid with x, then write the information in a table.
number of students
amount paid (€)
quotient
of the related values
of the related values
\latex{9}
\latex{3}
\latex{1800}
\latex{x}
\latex{\frac{1800}{9} }
\latex{\frac{x}{3} }
Since \latex{\frac{x}{3}=\frac{1800}{9}}, then \latex{\frac{x}{3}=200}, so \latex{x=3\times 200=600.}
€\latex{600 } is still to be paid by the three students.
€\latex{600 } is still to be paid by the three students.
Exercises
{{exercise_number}}. We paid €\latex{ 6 } for \latex{ 3 } \latex{ kg } of bananas in the shop. How much would we have paid for \latex{ 5 } \latex{ kg } of bananas? Let's solve it in several ways!
{{exercise_number}}. At the market, \latex{ 10 } eggs cost €\latex{ 6 }. How much do you pay if you buy
- \latex{ 5; }
- \latex{ 20; }
- \latex{ 25; }
- \latex{ 50; }
- \latex{ 40; }
- \latex{ 55?}
{{exercise_number}}. The wheels of a car turn \latex{ 20 } times while travelling a distance of \latex{ 38 } \latex{ m }. What distance does the car travel if its wheels turn \latex{ 150 } times?
{{exercise_number}}. The wheels of a car turn \latex{ 3 } times while travelling \latex{ 6 } \latex{ m }. How many times do the wheels of the car turn if it travels
a) \latex{ 3 \,m; }
b) \latex{ 12 \,m; }
c) \latex{ 93 \,m; }
d) \latex{ 264 \,m; }
e) \latex{ 5 \,km ?}
{{exercise_number}}. The students on a field trip walked \latex{ 7 } \latex{ km } in the first two \latex{ hours. } At a uniform pace, how many \latex{ kilometres } did they walk in \latex{ 5 } \latex{ hours? } How long did it take them to walk \latex{ 10.5 } \latex{ kilometres? }
{{exercise_number}}. A train travels \latex{ 11.2 } \latex{ km } in \latex{ 8 } \latex{ minutes. } How many \latex{ kilometres } does it travel in \latex{ 1 } \latex{ hour } at constant speed?
{{exercise_number}}. There are \latex{ 25 } congruent cubes. You want to paint their faces red. You use \latex{ 0.5 } \latex{ litres } of paint to colour eight cubes. How many \latex{ litres } of paint do you need to colour all the remaining cubes?
{{exercise_number}}. \latex{ 9 } \latex{ litres } of milk is needed to make \latex{ 250 } \latex{ g } of butter. How many \latex{ litres } of milk is needed to make \latex{ 1.2 } \latex{ kg } of butter?
{{exercise_number}}. At the bakery, we paid €\latex{ 9 } for \latex{ 6 } pieces of Caesar rolls. How much would we have paid if we had bought \latex{ 8 } pieces of Caesar rolls?
{{exercise_number}}. The \latex{ 8 }-\latex{ m }-long side of a rectangular house corresponds to \latex{ 5 } \latex{ cm } on the floor plan. How many \latex{ square } \latex{ metres } is the base area of the house if the other side is \latex{ 6 } \latex{ cm } on the floor plan?
{{exercise_number}}. A farmer cultivates wheat on a land of \latex{ 15 } \latex{ hectares. } During the first harvest, he obtains
\latex{ 40 } \latex{ kg } of wheat from an area of \latex{ 200 } \latex{ m^{2} } . How much wheat will he harvest in total?
\latex{ 40 } \latex{ kg } of wheat from an area of \latex{ 200 } \latex{ m^{2} } . How much wheat will he harvest in total?
{{exercise_number}}. \latex{ 16 } \latex{ kg } of cottage cheese can be made of \latex{ 50 } \latex{ litres } of milk. How many \latex{ litres } of milk is needed to make \latex{ 20 } \latex{ kg } of cottage cheese?
{{exercise_number}}. \latex{ 75 } pepper plants were planted in an area of \latex{ 5 } \latex{ m^{2} }. How many pepper plants can be planted over an area of \latex{ 32 } \latex{ m^{2} ?}
{{exercise_number}}. You must paint identical rectangular iron plates.
How much paint is needed for \latex{ 50 } iron plates if you used \latex{ 0.5 } \latex{ kg } for \latex{ 8 } iron plates the last time?
{{exercise_number}}. Light travels about \latex{ 3,000,000 } \latex{ km } in \latex{ 10 } \latex{ seconds. } Approximately how many \latex{ kilometres } does light travel in \latex{ 1 } \latex{ minute? } How long does it take light to cover the distance between the Sun and the Earth and the Moon-Earth distance? Find the necessary data on the Internet.
{{exercise_number}}. At the start of the trip, the fuel tank of a car was completely full. Since then, it has travelled a distance of \latex{ 135 } \latex{ km }. Here, you can see the current state of the fuel gauge. Estimate how many \latex{ kilometres } the car can travel with the remaining fuel, assuming its average consumption does not change.
Quiz
If \latex{ 2 } spiders eat \latex{ 2 } flies in two \latex{days}, then how many flies do \latex{ 6 } spiders eat in \latex{ 6 } \latex{days?}