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Inverse proportion
Example 1
\latex{ 10 }-\latex{ year }-old Nora, her \latex{ 2 }-\latex{ year }-old sister Lily, and their dad measured the same distance by placing their feet exactly in front of each other when walking. Nora's shoes are \latex{ 22 \;cm } long, Lily's are \latex{ 11 \;cm } long, and their dad's are \latex{ 33\;cm } long. If Nora took \latex{ 6 } steps to measure the distance, how many steps would Lily and their dad take?
Solution
Lily's shoes are half as long as Nora's; therefore, she has to take twice as many steps as Nora to cover the same distance.
number of steps:
\latex{11\;cm } Lily’s shoes
\latex{22\;cm } Nora’s shoes
number of steps:
\latex{6}
\latex{12}
\latex{\div2}
\latex{2\times}
Dad's shoes are \latex{ 3 } times as long as Lily's; therefore, he needs one-third as many steps as Lily.
number of steps:
\latex{33\,cm } dad’s shoes
\latex{11\,cm } Lily’s shoes
number of steps:
\latex{12}
\latex{4}
\latex{3\times}
\latex{\div3}
Thus, it takes Lily \latex{ 12 } steps and Dad \latex{ 4 } to cover the same distance.
shoe length (\latex{cm})
number of steps
\latex{11}
\latex{22}
\latex{33}
\latex{4}
\latex{6}
\latex{12}
Two quantities are inversely proportional if one increases by a certain amount while the other decreases by the same amount.
Example 2
Dora received chocolate bars for her birthday. If she ate one bar every \latex{ day }, she would have enough chocolate for six \latex{ days. } How many \latex{ days } would the chocolate bars last if she ate \latex{\frac{1}{3}}, \latex{\frac{1}{2}}, \latex{ 2 }, \latex{ 3 }, or \latex{ 6 } bars each \latex{ day? } Make a table and show the data pairs in a coordinate system.
Solution
\latex{6} \latex{ days }
\latex{\frac{1}{3} } bar a \latex{ day }
\latex{1} bar a \latex{ day }
\latex{6\times 3=18 } \latex{ days }
\latex{\div3}
\latex{3\times}
The smaller the amount of chocolate Dora eats each \latex{ day }, the more \latex{ days } it will last.
\latex{6} \latex{ days }
\latex{2 } bar a \latex{ day }
\latex{1} bar a \latex{ day }
\latex{6\div 2=3 } \latex{ days }
\latex{2\times}
\latex{\div2}
The more chocolate she eats a \latex{ day }, the fewer \latex{ days } it will last.
amount of chocolate
eaten a \latex{day}
eaten a \latex{day}
number of \latex{days}
\latex{1}
\latex{6}
\latex{18}
\latex{12}
\latex{3}
\latex{2}
\latex{1}
\latex{6}
\latex{3}
\latex{2}
\latex{\frac{1}{2} }
\latex{\frac{1}{3} }
The number of \latex{ days } and the amount of chocolate consumed daily are inversely proportional.
The \latex{ x }-axis of the coordinate system shows the amount of chocolate eaten each \latex{ day }, while the \latex{ y }-axis represents the number of \latex{ days } the chocolate lasts.
number of \latex{ days }
amount of chocolate eaten a \latex{ day }
\latex{18}
\latex{12}
\latex{6}
\latex{3}
\latex{1}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{5}
\latex{6}
\latex{\frac{1}{2} }
\latex{\frac{1}{3} }
The points represented in the coordinate system are not found on the same line.
Example 3
\latex{ }Rectangles are created using \latex{ 12 } squares with \latex{ 1\;cm } long sides. How long can the sides of the rectangles be? Make a table.
Solution
Let's start with the rectangle created by placing the squares one after the other.
\latex{1\;cm }
\latex{12\;cm }
If the length of one of the sides is doubled, the length of the other side is halved.
\latex{2\;cm }
\latex{6\;cm }
If the length of one of the sides is tripled, that of the other side is reduced to one-third.
\latex{3\;cm }
\latex{4\;cm }
If one of the sides becomes four times longer, the length of the other side becomes one-fourth.
\latex{4\;cm }
\latex{3\;cm }
The lengths of the sides of rectangles consisting of \latex{ 12 } squares are inversely proportional.
\latex{a\;(cm)}
\latex{b\;(cm)}
\latex{A=a\times b\; (cm^{2})}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{6}
\latex{12}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{6}
\latex{12}
\latex{12}
\latex{12}
\latex{12}
\latex{12}
\latex{12}
\latex{12}
\latex{12\times1}
\latex{6\times2}
\latex{4\times3}
\latex{3\times4}
\latex{2\times6}
\latex{1\times12}
The area, or, in other words, the product of the lengths of the sides, is constant:
\latex{A=12} \latex{cm^{2}}.
If two variables are inversely proportional, then the product of their corresponding values is constant.
Exercises
{{exercise_number}}. It takes
- a giant \latex{1} step;
- Gulliver \latex{100} steps;
- a Lilliputian \latex{1,200} steps to cover a distance of \latex{ 7\;miles. }
How many \latex{ miles } is the length of Gulliver's, the Liliputian's and the giant's step? What is the relationship between the number of steps and their lengths?
{{exercise_number}}. It takes Gulliver \latex{ 100 } steps to cover \latex{ 7\;miles }. How many steps does he need to cover \latex{ 14 \;miles }, \latex{ 21 \;miles } and \latex{ 49 \;miles? }
What is the relationship between the distance covered and the number of steps?
{{exercise_number}}. Zoe and her friends want to buy pastries for €\latex{10 }.
a) Apple turnover €\latex{1.5}
b) Berry basket €\latex{2}
c) Poppy ring €\latex{0.5}
How many of each can they buy if they only want to buy one kind of pastry? What is the relationship between the price of the pastries and the number that can be bought for €\latex{10?}
{{exercise_number}}. An apple turnover costs €\latex{ 1.5 }. How many do \latex{ 5 }, \latex{ 7 }, and \latex{ 12 } turnovers cost? What is the relationship between the number of turnovers and the total amount paid?
{{exercise_number}}. The following tables show the corresponding values of two quantities. Which table shows inverse proportion?
quantity A
quantity B
\latex{4}
\latex{0.5}
\latex{16}
\latex{0.25}
\latex{144}
\latex{4}
\latex{72}
\latex{9}
quantity A
quantity B
\latex{\frac{2}{3} }
\latex{1}
\latex{\frac{4}{7} }
\latex{8}
\latex{\frac{1}{16} }
\latex{\frac{7}{8} }
\latex{0.5}
\latex{\frac{3}{4} }
{{exercise_number}}. The numerator of a fraction is \latex{ 24 }.
- Determine the value of the fraction if its denominator is \latex{ 24 }; \latex{ 12 }; \latex{ 8 }; \latex{ 6 }; \latex{ 4 }; \latex{ 3 }; \latex{ 2 } and \latex{ 1 }. Make a table.
- Show the values in a coordinate system. The \latex{ x }-axis should represent the denominator, while the \latex{ y }-axis the numerator.
- What is the relationship between the value of a fraction and its denominator if the numerator is constant?
{{exercise_number}}. It takes \latex{ 3 } machines \latex{ 8 } \latex{ days } to complete the construction of a road.
- How many \latex{ days } would it take \latex{1; 2; 3; 5; 6; 8; 12} and \latex{16} machines to complete the construction? Make a table.
- What is the relationship between the number of machines and the \latex{ days } needed to finish the construction?
- Show the corresponding values in a coordinate system.
- Discuss whether the mathematical solution of the exercise is always applicable in real life.
{{exercise_number}}. You want to fill a kiddie pool with water. It takes you twelve \latex{ 10 }-\latex{ litre } vessels to fill it. How many of the following containers do you need to fill the pool? Make a table.
- \latex{2\;litres}
- \latex{5\;litres}
- \latex{6\;litres}
- \latex{8\;litres}
- \latex{15\;litres}
{{exercise_number}}. The coordinate systems below show the relationship between two quantities marked with different colours. Make a table including the coordinates of the points. Which quantities are inversely proportional?
\latex{8}
\latex{7}
\latex{6}
\latex{5}
\latex{4}
\latex{3}
\latex{2}
\latex{1}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{5}
\latex{6}
\latex{7}
\latex{8}
y
x
\latex{8}
\latex{7}
\latex{6}
\latex{5}
\latex{4}
\latex{3}
\latex{2}
\latex{1}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{5}
\latex{6}
\latex{7}
\latex{8}
y
x
Quiz
What is the relationship between the time that has passed from a \latex{day} and the remaining time?