Twój koszyk jest pusty
Calculating the whole
Example 1
A rock climber ascended to a height of \latex{ 180 \;m }, \latex{\frac{3}{4}} of the total height of the cliff.
How high is the cliff?
Solution
\latex{\frac{3}{4}} part
\latex{1} whole \latex{=\frac{4}{4}} part
\latex{180} \latex{ m }
\latex{ ? } \latex{ m }
\latex{\frac{1}{4}} part
\latex{\frac{4}{4}} part
\latex{180} \latex{ m } \latex{\div 3= 60} \latex{ m }
\latex{4\times 60} \latex{ m } \latex{=240} \latex{ m }
\latex{\div 3}
\latex{\div 3}
\latex{\times 4}
\latex{\times 4}
\latex{180} m
\latex{\frac{3}{4}} part
The height of the cliff is \latex{ 240 } \latex{ m }.
What is \latex{\frac{3}{4}} of \latex{240}?
\latex{240}
\latex{180}
\latex{\times \frac{3}{4} }
\latex{\div \frac{3}{4} }
The \latex{\frac{3}{4}} of which number is \latex{180}?
What is the quotient of 180 and \latex{\frac{3}{4}?}
\latex{180\div \frac{3}{4} =\frac{4}{\underset{1}{\cancel{3}}}\times \overset{60}{\cancel{180}}=240}
If \latex{180} is \latex{\frac{3}{4}} of the whole, then the whole can also be calculated by dividing \latex{180} by \latex{\frac{3}{4}}.
Example 2
Adam pays \latex{\frac{2}{3}} of his rent, \latex{ 800 } \latex{ euros }, at the beginning of the month and the rest at the end of the month. How much rent does Adam pay in total?
Solution 1 (deduction)
Rent
\latex{\frac{2}{3}} part
\latex{\frac{1}{3}} part
\latex{\frac{3}{3}} part
\latex{1} whole \latex{=\frac{3}{3}} part
€\latex{800}
€\latex{x}
€\latex{800} \latex{\div2=}€\latex{400}
\latex{3\times} €\latex{ 400= } €\latex{1,200}
\latex{\div 2}
\latex{\div 2}
\latex{\times 3}
\latex{\times 3}
Solution 2 (division)
If \latex{\frac{2}{3}} part is \latex{\longrightarrow} €\latex{800};
then \latex{\frac{3}{3}} part is €\latex{800\div \frac{2}{3}=\overset{400}{\bcancel{800}} \times \frac{3}{\underset{1}{\bcancel{2}}}=} €\latex{ 1,200 }.
then \latex{\frac{3}{3}} part is €\latex{800\div \frac{2}{3}=\overset{400}{\bcancel{800}} \times \frac{3}{\underset{1}{\bcancel{2}}}=} €\latex{ 1,200 }.
In both cases, you get the same result.
Adam's monthly rent is \latex{ 1,200 } \latex{ euros. }
Adam's monthly rent is \latex{ 1,200 } \latex{ euros. }
If you know the value of \latex{\frac{2}{3}} of a number, you can calculate the whole by dividing the number by \latex{\frac{2}{3}}.
Example 3
On the first \latex{ day } of a bicycle trip, the cyclists completed \latex{\frac{2}{7}} of the whole route, and on the second \latex{ day }, \latex{\frac{2}{5}}. How many \latex{ kilometres } are left if, on the first \latex{ day }, they cycled 30 \latex{ km? }
Solution
The distance covered on the first \latex{ day } was \latex{ 30 } \latex{ km }, \latex{\frac{2}{7}} of the whole route.
\latex{\frac{2}{7}} part
\latex{\frac{7}{7}} part
\latex{\frac{1}{7}} part
\latex{\frac{7}{7}} part
\latex{30} \latex{ km }
\latex{ ? } \latex{ km }
\latex{30} \latex{ km } \latex{\div2=15} \latex{ km }
\latex{7\times 15} \latex{ km } \latex{=105} \latex{ km }
\latex{\div2}
\latex{\div2}
\latex{\times 7}
\latex{\times 7}
The whole route is \latex{ 105 } \latex{ km }.
On the second \latex{ day }, they covered \latex{\frac{2}{5}} of the whole route.
\latex{\frac{2}{5}} part
\latex{\frac{1}{5}} part
\latex{\frac{2}{5}} part
\latex{1} whole \latex{=\frac{5}{5}} part
\latex{105} \latex{ km }
\latex{ ? } \latex{ km }
\latex{105} \latex{ km } \latex{\div5=21} \latex{ km }
\latex{2\times 21} \latex{ km } \latex{ = } \latex{42} \latex{ km }
\latex{\div 5}
\latex{\div 5}
\latex{\times 2}
\latex{\times 2}
On the second \latex{ day }, they cycled \latex{ 42 } \latex{ km }.
The distance left is: \latex{ 105\, km - 30\, km - 42\, km = 33\, km }.
\latex{ 33 } \latex{ kilometres } are left.
The distance left is: \latex{ 105\, km - 30\, km - 42\, km = 33\, km }.
\latex{ 33 } \latex{ kilometres } are left.
Exercises
{{exercise_number}}. \latex{\frac{3}{4}} of which number is
- \latex{ 24; }
- \latex{ 150; }
- \latex{ 540; }
- \latex{ 32.4; }
- \latex{ 100 ?}
{{exercise_number}}. \latex{\frac{2}{3}} of what amount is
- \latex{ 6 } \latex{ kg; }
- \latex{ 45 } \latex{ m; }
- \latex{ 40 } \latex{ minutes };
- \latex{ 184 } \latex{cm^{2}};
- \latex{ 23.5\, km ?}
{{exercise_number}}. You have dug up \latex{\frac{7}{12}} of your garden. How many \latex{ square } \latex{ metres } is your garden if you have \latex{ 55 } \latex{m^{2}} left to dig up?
{{exercise_number}}. Connor read \latex{\frac{5}{8}} of a book and has \latex{ 147 } pages left to finish it. How many pages does the book have?
{{exercise_number}}. \latex{\frac{3}{10}} of the students have already paid for lunch at the school canteen. How many students eat at the school canteen if \latex{ 168 } have not paid yet?
{{exercise_number}}. \latex{\frac{8}{5}} of what amount is
- \latex{ 15 } \latex{ km };
- \latex{ 150 } \latex{ m };
- \latex{ 1 } \latex{ hour };
- \latex{ 0.2 } \latex{ kg };
- \latex{\frac{2}{3}\;m^{2}?}
{{exercise_number}}. \latex{\frac{3}{4}} of my savings is \latex{ 900 } \latex{ euros }. How much savings do I have?
{{exercise_number}}. On a class trip, the students completed \latex{\frac{3}{5}} of the planned route. How many \latex{ kilometres } did they walk to this point if they had \latex{ 5.6 } \latex{ km } left?
{{exercise_number}}. Patrick has \latex{ 720 } \latex{ euros }, which is \latex{\frac{2}{5}} of Zack's money and \latex{\frac{9}{4}} of the money that Sam has. How much money does Zack and Sam have? Who has the most money?
{{exercise_number}}. A barrel contains \latex{\frac{4}{5}} \latex{ hl } of water. A hole is punched on the side of the barrel, allowing \latex{\frac{3}{4}} of the water to flow out. The water remaining in the barrel is \latex{\frac{1}{6}} of the total volume of the barrel. How many \latex{ litres } is the volume of the barrel?
{{exercise_number}}. At a party, the children ate \latex{\frac{2}{5}} of the cookies. Then, after a while, they ate \latex{\frac{3}{4}} of the remaining cookies, leaving only \latex{ 6 } on the tray. How many cookies were on the tray at the beginning of the party?
{{exercise_number}}. A sixth-grade class elected a class representative. \latex{\frac{4}{5}} of the students cast their votes. Carl obtained \latex{\frac{3}{4}} of the votes. Six students did not vote for Carl. How many students are in the class?
{{exercise_number}}. Caesar ate \latex{ 4.5 } \latex{ kg } of dog food in a \latex{ week }, \latex{\frac{9}{14}} of the amount that was supposed to last two \latex{ weeks }. How many \latex{ kilograms } of dog food should be enough for two \latex{ weeks? } Which of the following series of operations describes the correct answer?
- \latex{4.5\times \frac{9}{14}}
- \latex{(4.5\div 9)\times 14}
- \latex{4.5\div \frac{9}{14}}
- \latex{(4.5\times 14)\div 9}
- \latex{4.5\times \frac{14}{9}}
- \latex{4.5\div \frac{14}{9}}
{{exercise_number}}. Erica has \latex{24} \latex{ euros }. \latex{\frac{2}{3}} of Louie's money equals \latex{\frac{3}{4}} of Erica's money. Can they buy their sister a car that costs \latex{52} \latex{ euros } as a birthday gift?
{{exercise_number}}. Write word problems that can be solved using the following series of operations.
- \latex{\left(14\times \frac{2}{3} \right)\div \frac{3}{4}}
- \latex{(14-9)\times \frac{2}{3}}
- \latex{14-9\times \frac{2}{3}}
- \latex{14\times \frac{2}{3}-9}
- \latex{14\times \frac{2}{3}-8\times \frac{3}{4}}
- \latex{14-14\times \frac{1}{3} .}
{{exercise_number}}. \latex{ 40 } thieves stole a chest of jewels from Ali Baba. Some of the thieves took \latex{ 1 } jewel and rode off. Half of the remaining thieves took \latex{ 2 } jewels each, and no jewels were left for the other half. How many jewels were in Ali Baba’s chest?
Quiz
\latex{\frac{7}{8}} of the class participated in a sports competition, while \latex{\frac{1}{2}} of the students went to a spelling bee. How many students are in the class if every student participated in at least one of the competitions, and \latex{ 9 } students participated in both?