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Understanding percentages
Example 1
What is \latex{\frac{17}{100}} of \latex{ 600? }
Solution
\latex{\frac{17}{100}} of \latex{ 600 } is \latex{\frac{17}{100} \times 600 = \frac{17}{\underset{1}{\cancel{100}}} \times \overset{6}{\cancel{600}} = 102}.
The fraction \latex{\frac{1}{100}} is also known as \latex{ 1 }%, and the fraction \latex{\frac{100}{100}} stands for \latex{ 100 }% (the whole).
\latex{\frac{1}{10}} = \latex{\frac{10}{100}} part → \latex{ 10 }% ten per cent
\latex{\frac{2}{5}} = \latex{\frac{40}{100}} part → \latex{ 40 }% forty per cent
\latex{\frac{3}{2}} = \latex{\frac{150}{100}} part → \latex{ 150 }% one hundred fifty per cent
Example 2
A music shop is selling pianos at a \latex{ 10 }% discount. Matthew thinks this is the right time to buy the piano of his dreams, which costs €\latex{ 2,100 }. How much discount will he get?
Solution
We are looking for \latex{ 10 }% of the price of the piano, which is \latex{\frac{10}{100}} in fractions.
\latex{\frac{100}{100}} (\latex{ 100 }%)
\latex{\frac{1}{100}} (\latex{ 1 }%)
\latex{\frac{10}{100}} (\latex{ 10 }%)
€\latex{2,100}
€\latex{2,100\div100=21}
€\latex{10 \times (2,100 \div 100)} = \latex{210}
\latex{100\div}
\latex{100\div}
\latex{10 \times}
\latex{10 \times}
Matthew will receive a discount of €\latex{ 210. }
€\latex{ 2,100 }
↓
Base:
↓
Base:
The total number or quantity (\latex{ 100 }%).
\latex{ 10 }%
↓
Percentage:
↓
Percentage:
The proportion per hundred.
€\latex{ 210 }
↓
Part:
↓
Part:
The calculated portion of the base.
The three data associated with percentages are: base, percentage and part. By knowing any two of these, we can calculate the third.
Exercises
{{exercise_number}}. Convert the following fractions into percentages:
a) \latex{\frac{1}{2};}
b) \latex{\frac{1}{20};}
c) \latex{\frac{1}{4};}
d) \latex{\frac{1}{5};}
e) \latex{\frac{7}{10};}
f) \latex{\frac{3}{5};}
g) \latex{\frac{17}{100};}
h) \latex{\frac{3}{2};}
i) \latex{1;}
j) \latex{\frac{2}{3}.}
{{exercise_number}}. Convert the following numbers into percentages:
a) \latex{ 2; }
b) \latex{ 1.5 ;}
c) \latex{ 0.3 ;}
d) \latex{1\frac{1}{5};}
e) \latex{ 0.26; }
f) \latex{ 0.08 ;}
g) \latex{\frac{11}{20};}
h) \latex{\frac{8}{5};}
i) \latex{\frac{7}{25};}
j) \latex{1\frac{9}{10}.}
{{exercise_number}}. Convert the following percentages into simplified fractions:
a) \latex{50}%;
b) \latex{25}%;
c) \latex{20}%;
d) \latex{10}%;
e) \latex{75}%;
f) \latex{ 60 }%;
g) \latex{ 46 }%;
h) \latex{ 7 }%;
i) \latex{ 100 }%;
j) \latex{ 300 }%.
{{exercise_number}}. Convert the following percentages into fractions. Simplify the fractions if possible.
a) \latex{ 150 }%
b) \latex{ 1 }%
c) \latex{ 98 }%
d) \latex{ 4 }%
e) \latex{ 12.5 }%
f) \latex{ 160 }%
g) \latex{ 0.1 }%
h) \latex{ 1.4 }%
i) \latex{ 1.25 }%
j) \latex{ 0.05 }%
{{exercise_number}}. What percentage of the rectangles are red?
a)
b)
c)
d)
e)
f)
g)
h)
{{exercise_number}}. What percentage of the plane figures are red?
a)
b)
c)
d)
{{exercise_number}}. What is \latex{ 50 }% of \latex{ 100 } \latex{ metres? } Answer the question with the following percentages as well.
a) \latex{ 25 }%
b) \latex{ 10 }%
c) \latex{ 80 }%
d) \latex{ 150 }%
{{exercise_number}}. What is \latex{ 20 }% of \latex{ 1 } \latex{ kilogram? } Answer the question with the following percentages as well.
a) \latex{ 5 }%
b) \latex{ 92 }%
c) \latex{ 35 }%
d) \latex{ 200 }%
{{exercise_number}}. What is \latex{ 10 }% of \latex{ 1 } \latex{ hour? } Answer the question with the following percentages as well.
a) \latex{ 75 }%
b) \latex{ 40 }%
c) \latex{ 250 }%
d) \latex{ 1 }%
{{exercise_number}}. What number is \latex{ 25 }% of \latex{ 1? } Answer the question with the following number as well.
a) \latex{ 5 }
b) \latex{ 100 }
c) \latex{ 57 }
d) \latex{ 3.2 }
{{exercise_number}}. \latex{ 80 }% of what weight is \latex{ 800 } \latex{ g ?} Answer the question with the following units as well.
a) \latex{ 8 } \latex{ kg }
b) \latex{ 4,000 } \latex{ g }
c) \latex{ 20 } \latex{ kg }
d) \latex{ 36 } \latex{ g }
{{exercise_number}}. \latex{ 100 }% of what length is \latex{ 40 } \latex{ cm ?} Answer the question with the following percentages as well.
a) \latex{ 1 }%
b) \latex{ 10 }%
c) \latex{ 20 }%
d) \latex{ 75 }%
{{exercise_number}}. What percentage of \latex{ 1 } \latex{ km } is \latex{ 500 } \latex{ m? } Answer the question with the following units as well.
a) \latex{ 200 } \latex{ m }
b) \latex{ 750 } \latex{ m }
c) \latex{ 1 } \latex{ m }
d) \latex{ 40 } \latex{ m }
{{exercise_number}}. What percentage of \latex{ 1 } \latex{ hour } is \latex{ 15 } \latex{ minutes? } Answer the question with the following \latex{ minutes } as well.
a) \latex{ 6 } \latex{ minutes }
b) \latex{ 48 } \latex{ minutes }
c) \latex{ 3 } \latex{ minutes }
d) \latex{ 120 } \latex{ minutes }
{{exercise_number}}. What percentage of a number do you end up with if you increase or decrease it by the following amounts?
a) reduced by \latex{ 12 }%
b) increased by \latex{ 15 }%
c) increased by \latex{ 100 }%
d) threefold increase
{{exercise_number}}. Determine how much the quantity has changed, expressed as a percentage, in the following scenarios:
- \latex{ 250 }% of the original amount
- \latex{ 75 }% of the original amount
- \latex{ 100 }% of the original amount
- \latex{\frac{1}{4}} of the original amount
- \latex{ 1.2 } times the original amount
- one and a half times the original amount
Quiz
From each vertex of a cube with \latex{ 10 } \latex{ cm } edges, we cut off a smaller cube with \latex{ 1 } \latex{ cm } edges. By what percentage does the surface area of the cube decrease?