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Parts of a fraction

Snow White made a cake for the seven dwarfs, who asked her to divide it into \latex{ 8 } equal parts with \latex{ 3 } cuts.

How could Snow White fulfil the dwarfs' wish?

Dividing a whole into equal parts

Cut a cake into \latex{2; 3; 4} equal parts.

Take one of the equal parts.

\latex{ 1 } whole

\latex{ 1 } half

\latex{ 1 } third

\latex{ 1 } quarter

\latex{1}

>

\latex{\frac{1}{2}}

>

\latex{\frac{1}{3}}

>

\latex{\frac{1}{4}}

The more equal parts a whole is divided into, the smaller the parts become.

One whole can be expressed in several ways:

	\latex{ 1 } half \latex{ + } \latex{ 1 } half \latex{ = } \latex{ 2 } halves;
	\latex{ 1 } third \latex{ + } \latex{ 1 } third \latex{ + } \latex{ 1 } third \latex{ = 3 } thirds;
	\latex{ 1 } fourth \latex{ + } \latex{ 1 } fourth \latex{ + } \latex{ 1 } fourth \latex{ + } \latex{ 1 } fourth \latex{ = 4 } fourths.

Example 1

Matt ate three slices of the pizza shown in the image.
What fraction of the pizza did he eat?
What fraction did he leave?

Solution

One slice is one-eighth of the whole pizza. Matt ate three slices, which is three-eighths of the whole pizza.

Matt left five-eighths of the whole pizza.

Matt

left

\latex{\frac{5}{8} \text{ }}

\latex{\frac{3}{8} \text{ }}

When a whole is divided into equal parts, and some of the parts are taken away, you get a fraction.

Fractions are written in the following way:

numerator

denominator

fraction bar

\latex{\huge{\frac{3}{8}}}

- The numerator shows how many of the equal parts are chosen.

- The denominator shows how many equal parts \latex{ 1 } whole is divided into.

Example 2

Kate, Hannah and Dora won two chocolate bars in the spelling bee finals. One was milk chocolate, and the other was white chocolate.

How can they divide the two chocolates equally if

everyone wants to get the same amount of both chocolates?
Dora only likes milk chocolate, while Kate only likes white chocolate, so Dora trades her white chocolate with Kate for her milk chocolate. How much chocolate does Dora get? Draw the solutions.

Solution

Everyone gets one-third of the two chocolate bars.

Kate

Hannah

Dora

b) After the trade:

Kate

Hannah

Dora

Dora gets \latex{\frac{2}{3}} of the milk chocolate, that is, \latex{\frac{2}{3}} of a bar of chocolate.

One-third of two chocolate bars \latex{(2 \div 3)} is the same as dividing a chocolate bar into three equal parts and taking two parts from it.

you get \latex{\frac{2}{3}} if	you divide one whole into \latex{ 3 } parts and take \latex{ 2 } of them.
		you take one-third of \latex{ 2 } wholes.

The quotient of two whole numbers is called a fraction.

Example 3

Arrange the dominoes in such a way that the same fractions are underneath each other.

numerator:
\latex{ 2 }
denominator:
\latex{ 7 }

\latex{\frac{4}{5}}

\latex{\frac{3}{4}}

\latex{\frac{5}{6}}

\latex{\frac{7}{2}}

\latex{\frac{2}{7}}

\latex{\frac{2}{5}}

\latex{\frac{1}{4}}

one-fifth
of
two wholes

five-sixths

Solution

five-sixths

\latex{\frac{4}{5}}

\latex{\frac{2}{7}}

\latex{\frac{2}{5}}

\latex{\frac{7}{2}}

\latex{\frac{5}{6}}

\latex{\frac{1}{4}}

\latex{\frac{3}{4}}

numerator:
\latex{ 2 }
denominator:
\latex{ 7 }

one-fifth
of
two whole

Exercises

{{exercise_number}}. In which drawing is one-fourth of the area coloured with red?
(Each figure is a whole.)

A)

B)

C)

D)

E)

F)

G)

H)

{{exercise_number}}. Ted is going to school. Is he halfway there? Estimate it first, then use the squares of the quad paper to calculate it. (→)

school

Ted

home

{{exercise_number}}. Write the following fractions using numbers:

a) three-fifths

b) two-sevenths

c) nine-ninths

d) two-sixths

e) six-sevenths

f) five-tenths

g) ten-fifths

h) fifty-fifths

i) thirteen-eighths

{{exercise_number}}. Write down the following fractions.

a) numerator: \latex{ 7 }, denominator: \latex{ 10 }

b) denominator: \latex{ 8 }, numerator: \latex{ 3 }

c) numerator: \latex{ 17 }, denominator: \latex{ 12 }

d) numerator: \latex{ 21 }, denominator: \latex{ 53 }

{{exercise_number}}. The cake was cut into equal pieces. What fraction of the cake is gone? What fraction of the cake is left? (→)

{{exercise_number}}. What fraction of the drawings is coloured with red if each figure is one whole?

e)

a)

b)

c)

d)

h)

g)

f)

{{exercise_number}}. What fraction of the squares is white, and what fraction is coloured if the large squares are considered one whole?

a)

e)

d)

c)

b)

{{exercise_number}}. A calculator displays digits, as shown in the image below. Using fractions, write down what part of the signs turn on in the case of each digit. In which case does the numerator equal the value of the digit?

{{exercise_number}}. What fraction of the chocolates is missing?

a)

b)

c)

{{exercise_number}}. What fraction of the area is occupied by the plane figures?

j)

g)

h)

c)

f)

a)

d)

b)

e)

{{exercise_number}}. Five out of the \latex{ 8 } lanes of a swimming pool are open to the public, while the rest is reserved for professional swimmers. What fraction of the pool is reserved for professional swimmers?

{{exercise_number}}. A completely rolled-out carpet reaches from point \latex{ A } to point \latex{ C }.

a) What fraction of the carpet is rolled out in the image?
b) At which mark will \latex{\frac{3}{4}} of the carpet be rolled out? (→)

{{exercise_number}}. Draw \latex{ 3 } \latex{ cm } long and \latex{ 2 } \latex{ cm } wide rectangles in your notebook. Colour the same fraction of the rectangles as shown by the circles.

d)

a)

b)

c)

{{exercise_number}}. What fraction of each flag is red if the flags are considered one whole? Which countries do these flags belong to? List countries whose flags have the same fraction of red colour as the Hungarian flag.

f)

d)

e)

a)

b)

c)

{{exercise_number}}.

a) How many \latex{ days } is one \latex{ week }? How many \latex{ weeks } is one \latex{ day ?}
b) How many \latex{ weeks } are three \latex{ days ?}
c) How many \latex{ hours } is one \latex{ day }? How many \latex{ days } is one \latex{ hour? }
d) What fraction of today do you spend at school?
e) How many \latex{ minutes } is an \latex{ hour }? How many \latex{ hours } is one \latex{ minute ?}
f) How many \latex{ seconds } is a \latex{ minute }? How many \latex{ minutes } is a \latex{ second ?}

{{exercise_number}}.

a) How many students are in your class? What fraction of the class is one student?
b) What proportion of the class do girls account for? What proportion of the class do boys account for?

{{exercise_number}}. Draw a \latex{ 5 } \latex{ cm } long line segment in your notebook. Colour its

a) \latex{\frac{1}{10}};

b) \latex{\frac{3}{10}};

c) \latex{\frac{7}{10}};

d) \latex{\frac{10}{10}};

e) \latex{\frac{5}{10}} part.

How many \latex{ centimetres } is the coloured part of the \latex{ 5 } \latex{ cm } line segment?

{{exercise_number}}. \latex{ 1 } quarter of the students brought salty cookies, half of them brought sweet cookies, three-eighths brought fruits, and seven-twelfths brought juice to a school party. How many students brought each type of food if there are \latex{ 24 } students in the class? (At most, how many students brought two types of food?)

{{exercise_number}}.

a) Draw a \latex{ 1 } \latex{ dm } long line segment in your notebook. Colour \latex{\frac{2}{5}} of the line segment red. How many \latex{ centimetres } is the red segment?
b) Draw a \latex{ 2 } \latex{ dm } long line segment in your notebook, and colour \latex{\frac{1}{5}} it blue. How many \latex{ centimetres } is the blue segment?
c) Which segment is longer? The blue or the red one?

{{exercise_number}}. Calculate, then using a protractor, construct

a) \latex{\frac{1}{3}};

b) \latex{\frac{2}{3}};

c) \latex{\frac{5}{3}} of a right angle.

What types of angles are these?

{{exercise_number}}. Calculate, then using a protractor, construct

a) \latex{\frac{1}{3}};

b) \latex{\frac{2}{3}};

c) \latex{\frac{5}{3}} of a straight angle.

What type of angles are these?

{{exercise_number}}. A mother and her son go shopping. On the way home, Mum carries \latex{ 7 } \latex{ kg } of products, while her son carries \latex{ 1,000 } \latex{ g } of groceries. \latex{\frac{1}{10}} of the items Mum carries are vegetables, while half of the groceries carried by her son are vegetables. Who carries more vegetables?

{{exercise_number}}. Rearrange the digits in the fraction \latex{\frac{39}{57}} so that you get the smallest positive fraction possible. How much greater will the sum of the digits of the denominator be than the sum of the digits of the numerator?

{{exercise_number}}. In the numerator, only single-digit positive whole numbers can be written, while the denominator can contain only two-digit positive whole numbers. What is the largest fraction that you can create this way? Which is the smallest?

{{exercise_number}}. Dan took \latex{ 1 } \latex{ litre } of tea, while Carl took \latex{ 500 } \latex{ ml } on a trip. When they stopped for a pause, Dan drank \latex{ 1 } quarter of his tea, while Carl drank half of it. Who drank more?

{{exercise_number}}. Use fractions to express the following prefixes.

a) \latex{ deci- }

b) \latex{ centi- }

c) \latex{ milli- }

d) \latex{ micro- }

Quiz

What portion of the body of zebras is striped?

Mathematics 5.

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