Twój koszyk jest pusty
Calculating the base
Example 1
During a sale, every jacket is discounted \latex{ 30\% }. What is the price of a jacket if the total discount is €\latex{ 12.6? }
Solution 1 (deduction)
\latex{ 30 }%
\latex{ 100 }%
\latex{ \div30 }
\latex{ \div30 }
\latex{ 100\times }
\latex{ 100\times }
\latex{ 1 }%
\latex{ 100 }%
€?
€\latex{ 12.6 }
€\latex{ 12.6\div30= } €\latex{ 0.42 }
€\latex{ 100\times} €\latex{ 0.42 =} €\latex{ 42 }
Solution 2 (calculating the base)
The amount equal to \latex{100}% is the price of the jacket, that is, the base. The \latex{ 30}% discount corresponds to \latex{ \frac{30}{100} } of the price of the jacket.
\latex{ 30 }% \latex{ \left(=\frac{30}{100} \;part\right ) \rightarrow } €\latex{ 12.6 };
\latex{ 100 }% \latex{ \left(= 1 \;whole\right) \rightarrow } €\latex{ 12.6\div \frac{30}{100}= } €\latex{ 12.6\times \frac{100}{30}= } €\latex{ 42 }
\latex{ 100 }% \latex{ \left(= 1 \;whole\right) \rightarrow } €\latex{ 12.6\div \frac{30}{100}= } €\latex{ 12.6\times \frac{100}{30}= } €\latex{ 42 }
Solution 3 (division)
The whole part \latex{( 100 )}% can be calculated by dividing by a fraction; the divisor being one hundredth of the rate.
Original price
€(\latex{ x })
\latex{ 100 }%
€\latex{ 12.6 }
\latex{ 30 }%
\latex{ \div0.3 }
\latex{0.3 \times }
\latex{ 0.3\times x=12.6 }
\latex{ x= 12.6\div 0.3 }
\latex{ x=1,260 \div 30 }
\latex{ x=42 }
\latex{ x=1,260 \div 30 }
\latex{ x=42 }
The original price of the jacket is €\latex{ 42 }.
Example 2
\latex{ }Compared to the previous year, the number of children participating in summer camps has increased by \latex{ 12 }%. How many children went to summer camps last year if \latex{ 168 } have gone this year?
Solution 1 (deduction)
\latex{ 112 }%
\latex{ 1 }%
\latex{ 100 }%
\latex{ 100\times1.5=150 } children.
\latex{ 168\div112=1.5 };
\latex{ 168 } children;
\latex{ \div112 }
\latex{ \div112 }
\latex{ 100\times}
\latex{ 100\times}
Solution 2 (calculating the base)
The base can be calculated as \latex{ 112 }% \latex{ =\frac{112}{100} } parts of the whole.
\latex{ 112 }% \latex{ \left(=\frac{112}{100} \right)\rightarrow 168 } children;
\latex{100}% \latex{ \left(=1\;whole\right) \rightarrow 168\div \frac{112}{100}=\overset{3}{\cancel{168}}\times \frac{100}{\underset{2}{\cancel{112}} }=150 } children.
\latex{ 112 }% \latex{ \left(=\frac{112}{100} \right)\rightarrow 168 } children;
\latex{100}% \latex{ \left(=1\;whole\right) \rightarrow 168\div \frac{112}{100}=\overset{3}{\cancel{168}}\times \frac{100}{\underset{2}{\cancel{112}} }=150 } children.
Solution 3 (division)
\latex{ x } children
\latex{ 100 }%
\latex{ 168 } children
\latex{ 112 }%
\latex{ \div1.12 }
\latex{1.12 \times }
\latex{ 1.12x=168 }
\latex{ x=168\div 1.12 }
\latex{ x=16,800 \div 112 }
\latex{ x=150 }
\latex{ x=168\div 1.12 }
\latex{ x=16,800 \div 112 }
\latex{ x=150 }
\latex{ 150 } children went to summer camps last year.
Exercises
{{exercise_number}}. Determine the numbers.
a) \latex{ 1 }% of it is \latex{ 4 }
b) \latex{ 10 }% of it is \latex{ 2.5 }
c) \latex{ 20 }% of it is \latex{ 7 }
d) \latex{ 200 }% of it is \latex{ 4.8 }
e) \latex{ 18 }% of it is \latex{ 27 }
- \latex{ 25 }% of it is \latex{ 432 }
- \latex{ 92 }% of it is \latex{ 36.8 }
- \latex{ 125 }% of it is \latex{ 6 }
- \latex{ 52 }% of it is \latex{ 0.26 }
- \latex{ 47 }% of it is \latex{ 517 }
- \latex{ 7.5 }% of it is \latex{ 672 }
- \latex{ 0.1 }% of it is \latex{ 847 }
{{exercise_number}}. Determine the quantities.
a) \latex{ 50 }% of it is \latex{ 4.95 } \latex{ m }
b) \latex{ 37 }% of it is \latex{ 3.7 } \latex{ kg }
c) \latex{ 48 }% of it is \latex{ 2.88 } \latex{ km }
d) \latex{ 12.5 }% of it is \latex{ 75 } \latex{ m }
e) \latex{ 12 }% of it is \latex{ 372 } \latex{ cm }
f) \latex{ 128 }% of it is \latex{ 1664 } \latex{ m^{2} }
g) \latex{ 5 }% of it is \latex{ 43.2 } \latex{ hl }
h) \latex{ 0.5 }% of it is \latex{ 12 } \latex{ t }
{{exercise_number}}. The diagram shows the distribution of each type of tree in an orchard. How many trees are in the orchard if there are \latex{ 427 } apple trees?
\latex{ 16 }%
\latex{ 21 }%
\latex{ 35 }%
\latex{ 28 }%
apple
peach
pear
cherry
{{exercise_number}}. There are \latex{ 2,500 } literature books in a library, which account for \latex{ 8 }% of the books kept in the library. How many books are there in the library?
{{exercise_number}}. In an African savannah, a giraffe ate all the leaves of a tree up to \latex{ 64 }% of the tree's height. How tall is the tree if the giraffe can reach up to \latex{ 4 \;m } and \latex{ 80 \;cm ?}
{{exercise_number}}. \latex{ 65 }% of students at a school are girls. How many students attend the school if there are \latex{ 780 } girls in total?
{{exercise_number}}. A mushroom loses \latex{ 90 }% of its mass during drying. How many mushrooms do you need to have \latex{ 1.8 \;kg } of dried mushrooms for winter?
{{exercise_number}}. 55% of a garden has been dug up, \latex{ 405 } \latex{ m^{2} } are remaining. What is the total area of the garden?
{{exercise_number}}. The price of a dress has increased by \latex{ 15 }%, to €\latex{ 23 }. How much did it cost originally?
{{exercise_number}}. \latex{ 65 }% of a wire has been cut off, leaving a \latex{ 1.4 \;m } long segment. How long was the wire originally?
{{exercise_number}}. The price of a pair of shoes has been reduced by \latex{ 20 }%, to €\latex{ 52 }. How much did the pair of shoes cost before the discount?
{{exercise_number}}. The price of a suit increased by \latex{ 8 }%. Now it costs €\latex{ 4 } more than before. How much did the suit cost before the price increase?
{{exercise_number}}. A phone company reduced the price of calls by \latex{ 20 }% and then further decreased it by \latex{ 30 }%. What was the original price if a call now costs \latex{ 12.5 } \latex{ cents } \latex{ per } \latex{ minute? } What percentage of the original price is the new one?
{{exercise_number}}. Write word problems where the following operations can be used to calculate the base.
a) \latex{ 2,400\div \frac{3}{4} }
- \latex{67.5\div 1.25}
c) \latex{ 7.5\div \frac{5}{2} }
d) \latex{ 256\div \frac{8}{25} }
Quiz
At a university, \latex{ 60 }% of the first-year students have a language certificate in German, and \latex{ 70 }% have one in English. What percentage of the first-year students have a language certificate in at least one language?