Twój koszyk jest pusty
Proportional division
Example 1
Mr. Steve is returning to his house. When he is \latex{ 200 \;metres } from his home, his dog, Charlie, starts running towards him to greet him. Charlie runs three times the distance Mr. Steve walks until they meet. How many \latex{ metres } are they from the house when they meet?
Solution
Since the dog is three times faster, Charlie covers \latex{ 3 } units of distance, while Mr. Steve covers only \latex{ 1 } unit. Thus, the ratio of the distances covered is \latex{ 1:3 }.
Show the distances covered on a line segment.
\latex{200\;m} should be divided into \latex{1 + 3 = 4} equal parts.
\latex{200\;m} should be divided into \latex{1 + 3 = 4} equal parts.
Total distance: \latex{200\;m}
Mr. Steve
Charlie
\latex{3}
\latex{1}
\latex{:}
Total distance:
Mr. Steve:
Charlie:
\latex{\div\,4}
\latex{\times\,3}
\latex{4} units
\latex{1} unit
\latex{3} units
\latex{\times\,3}
\latex{\div\,4}
\latex{200\,m}
\latex{200\,m \div 4 = 50\,m}
\latex{3 \times 50\,m = 150\,m}
Check: The sum of the completed distances: \latex{50 \,m + 150 \,m = 200 \,m}.
The ratio of the distances: \latex{50 : 150 = 1 : 3}.
They meet \latex{150\, m} from the house.
Example 2
Gale and Zoe made €\latex{ 360 } together. Gale worked \latex{ 5 \;hours }, while Zoe worked \latex{ 7 }. How much money do they each get if they shared their earnings fairly based on the \latex{ hours } they worked?
Solution
A line segment should represent the total earnings.
Since they worked \latex{ 5+7=12 } \latex{ hours } together, divide the line segment into \latex{ 12 } equal parts.
wage for \latex{5\;hours }
wage for \latex{7\;hours }
\latex{\div 12}
\latex{\times 5}
\latex{\times 7}
\latex{\div 12}
\latex{\times 5}
\latex{\times 7}
\latex{360}
\latex{360 \div 12=30}
\latex{5 \times 30=150}
\latex{7 \times 30=210}
\latex{7\;hours }
\latex{12\;hours }
\latex{1\;hour }
\latex{5\;hours }
Check: The sum of their wages \latex{\qquad\qquad\qquad\qquad150+210=360}.
The ratio of their wages \latex{\qquad\qquad\qquad\qquad150 : 210 = 5 : 7}.
Gale earned €\latex{ 150 }, while Zoe made €\latex{ 210 }.
Example 3
The difference of the two positive numbers is \latex{ 18 }, and their ratio is \latex{ 3:5 }. What are the two numbers?
Solution
The two numbers should be represented with a \latex{ 3-unit } and a \latex{ 5-unit } long line segment, respectively.
\latex{ 1 }st number:
\latex{ 2 }nd number:
\latex{18}
The drawing indicates that \latex{ 2\; units } equal \latex{ 18 }.
\latex{ 2 } units
\latex{ 1 } unit
\latex{ 3 } units
\latex{ 5 } units
\latex{ \div 2 }
\latex{ \times 3 }
\latex{ \times 5 }
\latex{ \div 2 }
\latex{ \times 3 }
\latex{ \times 5 }
\latex{ 18 }
\latex{ 18\div2 = 9 }
\latex{ 3\times9 = 27 }
\latex{ 5\times9 = 45 }
Check: The difference is \latex{\qquad\qquad\qquad\qquad45-27 = 18}.
The ratio is \latex{\qquad\qquad\qquad\qquad\qquad27 : 45 = 3 : 5}.
The two numbers are \latex{ 27 } and \latex{ 45 }.
Exercises
{{exercise_number}}. At a pizzeria, the ratio of medium and large pizzas sold over the weekend was \latex{ 7:4 }. How many of each size were sold if \latex{ 121 } pizzas were delivered in total?
{{exercise_number}}. How much do Kate and Emily weigh if the ratio of their weights is \latex{ 4:5 } and the scale shows \latex{ 81\; kg } when they stand on it together?
{{exercise_number}}. Divide a straight angle into two parts, so that the ratio of the resulting angles is \latex{ 3:5 }.
{{exercise_number}}. Two men painted a fence. One worked \latex{ 3 } \latex{ hours }, while the other worked \latex{ 5 }.
They received €\latex{ 120 } for the job. If they divide their earnings based on the number of \latex{ hours } they worked, how much should each get?
They received €\latex{ 120 } for the job. If they divide their earnings based on the number of \latex{ hours } they worked, how much should each get?
{{exercise_number}}. The ratio of two numbers is \latex{ 3:4 }, and their sum is \latex{ 16.1 }. What are the two numbers?
{{exercise_number}}.Matt has €\latex{ 80 } more savings than Greg. The ratio of their savings is \latex{ 9:5 }. How much money does Matt have?
{{exercise_number}}. The difference of two positive numbers is \latex{ 126 }, and their ratio is \latex{ 7:3 }. What is the sum of the two numbers?
{{exercise_number}}. The children bend a \latex{ 30 } \latex{ cm } long wire into a triangle.
- The ratio of the sides of Arnie's triangle is \latex{ 1 : 1 : 1 };
- The ratio of the sides of Bea's triangle is \latex{ 1 : 2 : 2 };
- The ratio of the sides of Cecilia's triangle is \latex{ 3 : 5 : 7 }.
What type of triangles did the children create? How long are the sides of the triangles in \latex{ centimetres? }
{{exercise_number}}. \latex{ 3 } people earned €\latex{ 480 } for a job. One of them worked \latex{ 6\;hours }, the other \latex{ 8\;hours }, and the third \latex{ 10\;hours }. How much money does each get if they split the money based on \latex{ work\, hours? }
{{exercise_number}}. The perimeter of a rectangle is \latex{ 4.2\;dm }, and the ratio of its sides is \latex{ 3:4 }. How many \latex{ square\;centimetres } is its area?
{{exercise_number}}. Vivian had to learn \latex{ 26 } Italian words. When she started studying, she realised that the ratio of the words she remembered from class and those she did not remember was \latex{ 4:9 }. How many words did she remember from class?
{{exercise_number}}. Zoe’s longest skirt is \latex{ 60 \,cm } longer than her shortest. How long are the two skirts if their ratio is \latex{ 8 : 3 ?}
{{exercise_number}}. The ratio of lottery tickets sold on Monday, Wednesday and Friday was \latex{ 3:7:10 }. \latex{ 2,780 } tickets were sold during the three \latex{ days. } How many lottery tickets were sold on Wednesday?
{{exercise_number}}. The ratio of three numbers is \latex{ 2 : 6 : 7 }. The middle number is \latex{ 10.2 }. What is the sum of the three numbers?
*{{exercise_number}}. The ratio of the two sides of a rectangle is \latex{ 5:3 }, and its area is \latex{ 60 \,cm^2 }. How many \latex{ centimetres } is its perimeter?
Quiz
A hungry traveller is invited to share dinner with two shepherds. One of the shepherds offers \latex{ 3 } buns, while the other provides \latex{ 5 } buns. They split the buns among themselves equally and eat them. The traveller leaves early in the morning and leaves the shepherds \latex{ 320\;coins } for the buns. How should the two shepherds divide the money fairly among themselves?
