Twój koszyk jest pusty
Calculating the fraction of a whole
Example 1
The students of a class had to bring images of colourful marine fish to school. \latex{ 15 } photos were placed on the wall, and \latex{\frac{2}{5}} of these depicted yellow fish. How many images featured yellow fish?
Solution
Think of all the photos on the wall as a whole.
\latex{\div5}
\latex{\div5}
\latex{2\times}
\latex{2\times}
\latex{1} whole = \latex{\frac{5}{5}} part
\latex{\frac{2}{5}} part
\latex{\frac{1}{5}} part
\latex{\frac{2}{5}} part
\latex{15} images
\latex{?} images
\latex{15} images \latex{\div\;5=3} images
\latex{2\times3} images \latex{=6} images
There were \latex{ 6 } images depicting yellow fishes.
What is the result of \latex{ 15 } multiplied by \latex{\frac{2}{5}}?
\latex{\frac{2}{5} \times 15 = \frac{2}{\underset{\cancel1}{5}} \times \overset{\cancel3}{15} = 6}
Based on the previous example, it can be observed that \latex{\frac{2}{5}} of \latex{ 15 } can also be calculated by multiplying \latex{ 15 } by \latex{\frac{2}{5}}.
The \latex{\frac{2}{5}} part of \latex{ 15 = } \latex{\frac{2}{5} \times 15}.
Example 2
\latex{\frac{3}{4}} of a garden with an area of \latex{ 480 } \latex{ m^{2} } was grassed over. What is the area of the grassy part of the garden?
Solution 1 (deduction)
\latex{ 480 } \latex{ m^{2} }
\latex{÷4}
\latex{÷4}
\latex{3\times}
\latex{3\times}
\latex{1} whole = \latex{\frac{4}{4}} part
\latex{\frac{3}{4}} part
\latex{\frac{1}{4}} part
\latex{\frac{3}{4}} part
\latex{480\,m^2}
? \latex{m^2}
\latex{480\,m^2\div4=120\,m^2}
\latex{3\times120\,m^2=360\,m^2}
Solution 2 (multiplication)
\latex{1} whole \latex{\longrightarrow } \latex{480\,m^2};
\latex{\frac{3}{4}} part\latex{\longrightarrow }\latex{\frac{3}{\underset{1}{\bcancel4}}\times\overset{120}{\bcancel{480}}} \latex{ m^2 }\latex{=}\latex{360} \latex{ m^2 }
The area of the grassy part measures \latex{ 360 } \latex{ m^{2} } .
Example 3
What is the \latex{\frac{1}{2}} part of the \latex{\frac{4}{3}} part of \latex{24}?
Solution 1 (deduction)
First, determine the \latex{\frac{4}{3}} part of \latex{24}.
\latex{\div3}
\latex{\div3}
\latex{4\times}
\latex{4\times}
\latex{1} whole = \latex{\frac{3}{3}} part
\latex{\frac{4}{3}} part
\latex{\frac{1}{3}} part
\latex{\frac{4}{3}} part
\latex{24}
?
\latex{24\div3=8}
\latex{4\times8=32}
Then, calculate the \latex{\frac{1}{2}} part of \latex{ 32 }.
\latex{\div2}
\latex{\div2}
\latex{1} whole = \latex{\frac{2}{2}} part
\latex{\frac{1}{2}} part
\latex{\frac{1}{2}} part
\latex{32}
?
\latex{32\div2=16}
The \latex{\frac{1}{2}} part of the \latex{\frac{4}{3}} part of \latex{ 24 } is \latex{ 16 }.
Solution 2 (multiplication)
The \latex{\frac{4}{3}} part of \latex{24} is: \latex{\frac{4}{3} \times 24 = \frac{4}{\underset{1}{\cancel3}} \times \overset{8}{\cancel{24}} = 32}.
The \latex{\frac{1}{2}} part of \latex{32} is: \latex{\frac{1}{2} \times 32 = \frac{1}{\underset{1}{\cancel2}} \times \overset{16}{\cancel{32}} = 16}.
The calculations can also be performed in one step:
\latex{\frac{1}{2} \times \left(\frac{4}{3} \times 24\right) = \frac{1}{\underset{1}{\cancel2}} \times \frac{\overset{2}{\cancel4}}{3} \times 24 = \frac{2}{\underset{1}{\cancel3}} \times \overset{8}{\cancel{24}} = 16}.
The \latex{\frac{1}{2}} part of the \latex{\frac{4}{3}} part of \latex{24} is \latex{16}.
Exercises
{{exercise_number}}. Determine the following fractions of €\latex{ 240 }.
- \latex{\frac{1}{2}}
- \latex{\frac{2}{3}}
- \latex{\frac{1}{10}}
- \latex{\frac{3}{4}}
- \latex{\frac{4}{5}}
{{exercise_number}}. How many \latex{ minutes } constitute the following fractions of an \latex{ hour? }
- \latex{\frac{1}{2}}
- \latex{\frac{5}{6}}
- \latex{\frac{7}{10}}
- \latex{\frac{5}{4}}
- \latex{\frac{5}{2}}
{{exercise_number}}. Write the following fractions of \latex{ 1,600 } \latex{ kg } as multiplications, then calculate the results.
- \latex{\frac{3}{5}}
- \latex{\frac{3}{2}}
- \latex{\frac{7}{4}}
- \latex{0.1}
- \latex{2.5}
In which case is the result greater than \latex{ 1,600 } \latex{ kg ?}
{{exercise_number}}. One of the edges of a rectangular cuboid is \latex{ 12 } \latex{ cm } long, the second edge equals the \latex{\frac{3}{4}} part of the first one and the third equals the \latex{\frac{2}{3}} part of the first one. Determine the surface area and the volume of the rectangular cuboid.
{{exercise_number}}. One of the edges of a square prism is \latex{ 20 } \latex{ cm } long, while the other edge equals the \latex{\frac{3}{5}} part of the first one. Determine the surface area and the volume of the square prism.
{{exercise_number}}. Which value is greater?
- \latex{\frac{3}{4}} part of \latex{\frac{2}{3}} or \latex{\frac{2}{3}} part of \latex{\frac{3}{4}}
- \latex{\frac{5}{8}} part of \latex{\frac{8}{5}} or \latex{\frac{8}{5}} part of \latex{\frac{5}{8}}
- \latex{3\frac{2}{5}} part of \latex{2\frac{1}{2}} or \latex{2\frac{1}{2}} part of \latex{3\frac{2}{5}}
{{exercise_number}}. George spent the \latex{\frac{3}{7}} part of €\latex{ 420 }. How much money does he have now? What fraction is this of the original amount?
{{exercise_number}}. How many faces of a white cube were painted red if the following fractions of the surface area of the cube are red?
- \latex{\frac{1}{6}}
- \latex{\frac{2}{3}}
- \latex{\frac{1}{2}}
- \latex{\frac{5}{6}}
{{exercise_number}}. Claire and Rachel are running on a \latex{ 400 }-\latex{ metre } track. Claire has completed the \latex{\frac{4}{5}} part of the race, while Rachel is at the \latex{\frac{3}{4}} part of the track. What is the distance between them?
{{exercise_number}}. Laura’s homework was to reduce the number \latex{ 600 } by its \latex{\frac{1}{4}} part. After she completed the exercise, she called her classmates to ask them how they resolved it. Kate multiplied 600 by \latex{\frac{3}{4}}, Ben multiplied \latex{ 600 } by \latex{\frac{1}{4}}, Dorothy divided \latex{ 600 } by \latex{\frac{3}{4}}, while Adam subtracted \latex{\frac{1}{4}} from \latex{ 600 }. Which operation gives the correct result?
{{exercise_number}}. In the craft club of the school, the total number of fifth and sixth graders is less than \latex{ 22 }. \latex{\frac{9}{14}} of the girls are fifth graders, the rest are sixth graders. How many of the girls are sixth graders? How many boys are there in the craft club at most?
{{exercise_number}}. What is the \latex{\frac{5}{4}} part of the \latex{\frac{4}{5}} part of \latex{ 20 }?
{{exercise_number}}.
- What fraction is the \latex{\frac{1}{2}} part of a number of the \latex{\frac{3}{4}} part of the same number?
- What fraction of \latex{ 45 } is the \latex{\frac{7}{5}} part of the \latex{\frac{2}{3}} part of \latex{ 45 }?
{{exercise_number}}. Write the following fractions of \latex{ 1,600 } \latex{ kg } as multiplications, then calculate the results.
- \latex{\frac{3}{5}}
- \latex{\frac{3}{2}}
- \latex{\frac{7}{4}}
- \latex{0.1}
- \latex{2.5}
In which case is the result greater than \latex{ 1,600 } \latex{ kg }?
{{exercise_number}}. Calculate the following fractions.
- \latex{\frac{2}{3}} part of \latex{\frac{3}{4}}
- \latex{\frac{7}{4}} part of \latex{\frac{4}{7}}
- \latex{\frac{5}{4}} part of \latex{2\frac{3}{5}}
- \latex{\frac{2}{5}} part of \latex{4\frac{1}{3}}
- \latex{\frac{7}{10}} part of \latex{1\frac{1}{2}}
- \latex{2\frac{1}{2}} part of \latex{3\frac{3}{4}}
{{exercise_number}}. The sides of a rectangular plot are \latex{ 16 } and \latex{ 56 \,m } long. \latex{\frac{2}{7}} of the plot is occupied by a building, \latex{\frac{3}{8}} of it is used as a vegetable garden, while the rest of it is grassed over. What is the area of the grassy part? What fraction is this of the entire plot?
{{exercise_number}}. Think of word problems that can be solved using the following operations.
- \latex{\frac{2}{3} \times 3}
- \latex{30 + \frac{2}{3}}
- \latex{30 \times \frac{5}{3}}
- \latex{30 \div\frac{2}{3}}
{{exercise_number}}. Robert, Florence and Emma are hiking. Robert’s backpack is \latex{ 9.6 } \latex{ kg }, Flora’s backpack weighs \latex{\frac{3}{4}} of that of Robert, while Emma's weighs \latex{\frac{2}{3}} of Robert's backpack. How much do the three backpacks weigh in total?
Quiz
What number equals \latex{ 7 } if divided by the \latex{\frac{1}{7}} part of itself?