Twój koszyk jest pusty
Exercises with inverse proportion
Example 1
It takes \latex{ 3 } people \latex{ 18\;hours } to harvest the grapes in a given area. How long would it take \latex{ 5 } people to harvest the grapes?
Solution
Twice as many people can harvest the grapes in half as much time, while three times as many can do it in one-third of the original amount of time. The number of people and the time needed are inversely proportional.
Solution 1
\latex{3} people
\latex{5} people
\latex{1} people
\latex{5} people
\latex{18} \latex{hours}
\latex{ ?\, hours }
\latex{18} \latex{hours} \latex{\times\;3=54} \latex{hours}
\latex{54} \latex{hours} \latex{\div\;5=10.8} \latex{hours}
\latex{\times3}
\latex{\times5}
\latex{\div5}
\latex{\div3}
reduced
to one-third
to one-third
increase by
five times
five times
increase by
three times
three times
reduced
to one-fifth
to one-fifth
\latex{5} people can finish the harvest in \latex{10.8} \latex{ hours } (\latex{10} \latex{ hours } \latex{48} \latex{ minutes }).
Solution 2
You know that, in the case of inversely proportional quantities, their product is constant. Indicate the amount of time needed with \latex{ x } and make a table including the information you have.
number of people
harvest time (\latex{hours})
product of corresponding values
\latex{3}
\latex{5}
\latex{18}
\latex{x}
\latex{5\times x}
\latex{3\times 18}
\latex{5\times x=54}, thus \latex{x=54\div 5=10.8}.
\latex{5} people can finish the harvest in \latex{10.8} \latex{ hours } (\latex{10} \latex{ hours } \latex{48} \latex{ minutes }).
Example 2
Karl's car consumes \latex{ 8\;litres } of fuel over \latex{ 100 \;kilometres }, Anna's \latex{ 4 \;litres }, Cassie's \latex{ 5\; litres }, and Steve's car consumes \latex{ 10 \;litres }. How many times can each of them complete a distance of \latex{ 100 \;km } if their vehicles have \latex{ 40 \;litres } of fuel in their tanks?
Solution
Karl’s car
consumes \latex{8} \latex{ litres }
consumes \latex{40} \latex{ litres }
over \latex{100} \latex{ km }
over \latex{5\times100=500} \latex{ km }
\latex{5\times}
\latex{\times5}
Karl can complete a distance of \latex{ 100\;km } five times. Fuel consumption and the distance travelled with one full tank are inversely proportional; therefore, the product of corresponding values is constant. See the table of corresponding values below:
total fuel \latex{(litres)}
consumption \latex{\left(\frac{litre}{100\;km} \right)}
distance travelled \latex{(100\;km)}
\latex{40}
\latex{40}
\latex{40}
\latex{40}
\latex{10}
\latex{5}
\latex{4}
\latex{8}
\latex{5}
\latex{10}
\latex{8}
\latex{4}
Thus, Anna can travel \latex{ 100\; km } with one tank \latex{ 10 } times, Cassie \latex{ 8 } times, and Steve \latex{ 4 } times.
Exercises
{{exercise_number}}. During summer break, it takes \latex{ 6 } painters \latex{ 8 } \latex{ days } to paint the walls in all the classrooms in a school. At the same pace, how long would it take
- \latex{1};
- \latex{2};
- \latex{3};
- \latex{4};
- \latex{8};
- \latex{12};
- \latex{10};
- \latex{ 5 } painters to finish the work?
{{exercise_number}}. Martha got a book for her birthday. She planned to read \latex{ 16 } pages a \latex{ day } so she could finish reading the book in \latex{ 9 } \latex{ days. } Eventually, she managed to read only \latex{ 12 } pages a \latex{ day }. How long did it take Martha to finish her book?
{{exercise_number}}. At an event, the organisers calculated that they needed \latex{ 60 } cups to serve orange juice in \latex{ 300\,ml } portions. How many cups would they need if orange juice was served in \latex{ 200\; ml } portions?
{{exercise_number}}. Gravel is transported to a construction site. If \latex{ 6 } \latex{m^{3}} of gravel is loaded onto a truck, \latex{ 12 } trucks are needed to transport it to the site. However, the trucks the company uses can transport only \latex{ 4 } \latex{m^{3}} of gravel. How many trucks do they need to transport all the gravel to the construction site?
{{exercise_number}}. A staircase is being designed for a lookout tower. If each step were \latex{ 14 } \latex{ cm } high, \latex{ 234 } steps would lead to the top of the tower. How many steps will be built if the engineers decide to make the steps \latex{ 18 } \latex{ cm } high?
{{exercise_number}}. The bathroom floor can be covered with either \latex{ 40 } \latex{cm^{2}} or \latex{ 90 } \latex{cm^{2}} tiles. None of the tiles should be cut to cover the entire floor. If \latex{ 135 } tiles of the smaller ones are needed to cover the entire bathroom floor, how many larger tiles would be needed?
{{exercise_number}}. A cyclist travels on average \latex{ 24 } \latex{ km } \latex{ in } \latex{ one } \latex{ hour }. At this speed, he arrives at the city centre in \latex{ 45 } \latex{ minutes }. How long does it take a car covering a distance of \latex{ 72 } \latex{ km } \latex{ in } \latex{ one } \latex{ hour } to arrive at the same place?
{{exercise_number}}. Standing tickets for a rock concert cost €\latex{ 75 }, while seated tickets cost €\latex{ 80 }. If all the tickets were sold, the income from the sale of both types of tickets would be equal. How many seats are in the arena if \latex{ 4,200 } standing tickets were printed?
{{exercise_number}}. Trucks carry soil for the construction of a river dam. If the soil were transported by \latex{ 12 } trucks, it would take \latex{ 9 } \latex{ days } to take all the soil to the construction site. How long would it take \latex{ 18 } trucks to transport the same amount of soil?
{{exercise_number}}. Trees are planted next to a residential area. Two types of saplings are suitable for the project: one costs €\latex{ 9 }, while the other €\latex{ 12 } each. The total budget is sufficient to buy \latex{ 600 } of the cheaper saplings. How many €\latex{ 12 } saplings can be purchased for the project?
{{exercise_number}}. Write a story and make a table based on the graph to the right.
\latex{90}
\latex{80}
\latex{70}
\latex{60}
\latex{40}
\latex{50}
\latex{30}
\latex{20}
\latex{10}
\latex{1}
\latex{2}
\latex{3}
{{exercise_number}}. The same number of candles are packed at a factory on corresponding days of the \latex{ week }. On Mondays, \latex{ 33 } candles are put in a box, on Tuesdays, \latex{ 68 } candles are put in a box, and on Wednesdays, \latex{ 132 } candles are put in a box. How many boxes are used every \latex{ day } if the number of candles packed is less than \latex{ 4,000 } each \latex{ day ?}
Quiz
If Louis III issued \latex{ 16 } decrees in \latex{ 4 } \latex{years}, then which Louis issued \latex{ 64 } decrees in \latex{ 2 } \latex{years?}