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Direct proportion
€
€
\latex{ 1.5 }
€
\latex{ 3.5 }
\latex{ 2.5 }
Example 1
At a bakery, \latex{ 4 } doughnuts cost €\latex{ 2 }.
- How much do \latex{ 8 } doughnuts cost?
- How much does \latex{ 1 } doughnut cost?
- Make a table of how much \latex{ 1; 2; 3; ... 7; } and \latex{ 8 } doughnuts cost.
Solution
- \latex{ 8 } is two times \latex{ 4 }, so \latex{ 8 } doughnuts cost twice as much as \latex{ 4 }.
\latex{ 4 } doughnuts
\latex{ 8 } doughnuts
€\latex{ 2 }
€\latex{ 4 }
\latex{\times 2}
\latex{\times 2}
\latex{ 8 } doughnuts cost €\latex{ 4 }.
- \latex{ 1 } is a quarter of \latex{ 4 }, so \latex{ 1 } doughnut costs one-fourth of \latex{ 4 } doughnuts.
\latex{ 4 } doughnuts
\latex{ 1 } doughnuts
€\latex{ 2 = } ¢\latex{ 200 }
¢\latex{ 50 }
\latex{\div 4}
\latex{\div 4}
\latex{ 1 } doughnut costs ¢\latex{ 50 }.
- Write the information in a table:
number of doughnuts | \latex{ 1 } | \latex{ 2 } | \latex{ 4 } | \latex{ 4 } | \latex{ 5 } | \latex{ 6 } | \latex{ 7 } | \latex{ 8 } |
price of doughnuts (¢) | \latex{ 50 } | \latex{ 100 } | \latex{ 150 } | \latex{ 200 } | \latex{ 250 } | \latex{ 300 } | \latex{ 350 } | \latex{ 400 } |
Two quantities are directly proportional if, as one increases, the other also increases at the same rate.
Example 2
The price of \latex{ 2 } \latex{ kg } of bread is €\latex{ 4 }.
Write the prices of \latex{ 100 } \latex{ g }, a quarter of a \latex{ kg }, half a \latex{ kg }, three-quarters of a \latex{ kg }, one and a half \latex{ kg } and \latex{ 3 } \latex{ kg } of bread in the table.
Solution
- The price of the bread will be the same proportion of €\latex{ 4 } as the mass of the bread is the proportion of \latex{ 2\; kg }.
\latex{ - }€\latex{ 4 }
\latex{ 2 } \latex{ kg }
price of \latex{ 2 \,kg } of bread
price of \latex{ 100 } \latex{ g } of bread
price of \latex{\frac{1}{4}} \latex{ kg } of bread
€\latex{ 4 = } ¢\latex{ 400 }
\latex{400 \times \frac{1}{20}}\latex{ = } ¢\latex{ 20 }
\latex{400 \times \frac{1}{8}}\latex{ = } ¢\latex{ 50 }
\latex{\times \frac{1}{20}}
\latex{\times \frac{1}{8}}
\latex{\times \frac{1}{20}}
\latex{\times \frac{1}{8}}
The other prices can be calculated similarly.
mass of bread (\latex{g})
price of bread (¢)
\latex{ 2,000 }
\latex{ 400 }
\latex{ 100 }
\latex{ 250 }
\latex{ 500 }
\latex{ 750 }
\latex{ 1,500 }
\latex{ 3,000 }
\latex{ 20 }
\latex{ 50 }
\latex{ 100 }
\latex{ 150 }
\latex{ 300 }
\latex{ 600 }
\latex{\times \frac{1}{20}}
\latex{\times \frac{1}{20}}
\latex{\times \frac{3}{8}}
\latex{\times \frac{3}{8}}
The mass of the bread and the price are directly proportional.
- The \latex{ x }-axis of the coordinate system shows the mass of the bread, while the \latex{ y }-axis represents the price.
Note that the coordinates are on a straight line.
Since \latex{ 0 } \latex{ kg } of bread costs €\latex{ 0 }, coordinate \latex{ (0;0) } is also included in the graph.
\latex{ price } (¢)
\latex{y}
\latex{ mass } (\latex{ kg })
\latex{x}
\latex{ 400 }
\latex{ 350 }
\latex{ 300 }
\latex{ 250 }
\latex{ 200 }
\latex{ 150 }
\latex{ 100 }
\latex{ 50 }
\latex{ 0 }
\latex{ \frac{1}{4} }
\latex{ \frac{1}{2} }
\latex{ \frac{3}{4} }
\latex{ 1\frac{1}{2} }
\latex{ 1 }
\latex{ 2 }
Note:
Using the graph, you can tell how much a certain amount of bread costs or how much bread can be bought for a given amount of money.
Example 3
During a quarter of an \latex{ hour }, the minute hand of a clock rotates \latex{ 90º }.
- How many \latex{ degrees } does it rotate in \latex{ 5, 10, 30, 45, } and \latex{ 60 } \latex{ minutes? } Make a table.
- Make a graph.
- Using the graph, determine how many \latex{ minutes } it takes to rotate \latex{ 120º, 150º, } and \latex{ 210º .}
- Calculate the quotient of the rotation and time in each case. Write the quotients in a table.
Solution
If the time passed is doubled, tripled, etc., the magnitude of the rotation will also be doubled, tripled, etc. The time passed and the magnitude of the rotation are directly proportional.
a)
time passed (\latex{ min }) | \latex{ 15 } | \latex{ 5 } | \latex{ 10 } | \latex{ 30 } | \latex{ 45 } | \latex{ 60 } |
rotation \latex{ (degrees) } | \latex{ 90 } | \latex{ 30 } | \latex{ 60 } | \latex{ 180 } | \latex{ 270 } | \latex{ 360 } |
\latex{ rotation\; (degrees) }
\latex{ time\; (min) }
\latex{ 360 }
\latex{ 315 }
\latex{ 270 }
\latex{ 225 }
\latex{ 180 }
\latex{ 135 }
\latex{ 90 }
\latex{ 45 }
\latex{ 0 }
\latex{ 5 }
\latex{ 10 }
\latex{ 15 }
\latex{ 20 }
\latex{ 25 }
\latex{ 30 }
\latex{ 35 }
\latex{ 40 }
\latex{ 45 }
\latex{ 50 }
\latex{ 55 }
\latex{ 60 }
- The minute hand turns
\latex{ 120° } in \latex{ 20 } \latex{ minutes },
\latex{ 150° } in \latex{ 25 } \latex{ minutes },
\latex{ 210° } in \latex{ 35 } \latex{ minutes }.
\latex{ 150° } in \latex{ 25 } \latex{ minutes },
\latex{ 210° } in \latex{ 35 } \latex{ minutes }.
\latex{ \text{time }}\latex{\text{ passed} } (\latex{ min }) | \latex{ 15 } | \latex{ 5 } | \latex{ 10 } | \latex{ 30 } | \latex{ 45 } | \latex{ 60 } |
\latex{\text{ rotation} } \latex{ (degrees) } | \latex{ 90 } | \latex{ 30 } | \latex{ 60 } | \latex{ 180 } | \latex{ 270 } | \latex{ 360 } |
\latex{\frac{\text{rotation}}{\text{time passed}} \; \left(\frac{degrees}{min}\right)} | \latex{\frac{90}{15}} | \latex{\frac{30}{5}} | \latex{\frac{60}{10}} | \latex{\frac{180}{30}} | \latex{\frac{270}{45}} | \latex{\frac{360}{60}} |
\latex{ 6 } | \latex{ 6 } | \latex{ 6 } | \latex{ 6 } | \latex{ 6 } | \latex{ 6 } |
You can notice that the quotient of corresponding values is always \latex{ 6 } \latex{\frac{degrees}{min}}.
If two variables are directly proportional, the quotient of corresponding values is constant.
Exercises
{{exercise_number}}. The following table shows the progress of two children's growth until the age of \latex{ 5 }. Is it true that the age and height of the children are directly proportional? Make a graph.
Age (\latex{ years }) | \latex{ 0 } | \latex{ 1 } | \latex{ 2 } | \latex{ 3 } | \latex{ 4 } | \latex{ 5 } |
Frank’s height (\latex{ cm }) | \latex{ 52 } | \latex{ 83 } | \latex{ 92 } | \latex{ 102 } | \latex{ 111 } | \latex{ 120 } |
Zoe’s height (\latex{ cm }) | \latex{ 48 } | \latex{ 75 } | \latex{ 83 } | \latex{ 88 } | \latex{ 91 } | \latex{ 101 } |
{{exercise_number}}. List quantities that are directly proportional.
{{exercise_number}}. List quantities where, if one of them increases, the other increases as well, but they are not directly proportional.
{{exercise_number}}. A pedestrian walking at a uniform speed covers a distance of \latex{ 4\, km} in an \latex{ hour. }
- How much does the pedestrian walk in \latex{\frac{1}{4}}; \latex{\frac{1}{2}}; \latex{ 2; 3; 4 } and \latex{ 5 } \latex{ hours? }
- Show the corresponding values in a coordinate system.
- What can you tell about the position of the coordinates?
{{exercise_number}}. Peppers at a market cost ¢\latex{ 30 } each. Show the relationship between the number of peppers bought and their price in a coordinate system. How are the coordinates positioned in the coordinate system? Can you connect the coordinates?
{{exercise_number}}. In the image, you can see the relationship between amounts. Which of the four graphs represents directly proportional amounts? Justify your answer. (→)
\latex{ y }
\latex{ x }
\latex{ 35 }
\latex{ 30 }
\latex{ 25 }
\latex{ 20 }
\latex{ 15 }
\latex{ 10 }
\latex{ 5 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{5 }
\latex{6 }
\latex{7 }
\latex{8 }
{{exercise_number}}. Is it true that,
- if you buy twice as much of a fabric, the price will be double;
- if the length of a square's sides becomes three times larger, then its area will also become three times larger;
- if you study twice as much, you will get twice as good a grade;
- if two fractions are equal and the numerator of one of the fractions is five times the other fraction's numerator, then the denominator of the first fraction is five times the other fraction's denominator?
In which case or cases are the quantities directly proportional?
Quiz
A container placed under a leaking tap gets half-filled with water in a quarter of an \latex{hour.} To what extent will the container be filled with water in one \latex{hour?}
