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Fractions (revision)
The definition of fractions
Example 1
There are four chunks of cheese of the same size in the pantry. A mouse steals one-fourth of three chunks of cheese, while another steals three-fourths of one chunk of cheese. Which mouse has stolen more cheese?
Solution
Based on the drawing above, you can see that one-fourth of 3 chunks of cheese is
\latex{\frac{1}{4} +\frac{1}{4}+\frac{1}{4}=\frac{3}{4}} of one cheese.
Three-fourths of one cheese is also \latex{\frac{3}{4}}. So, the two mice stole the same amount of cheese.
\latex{ 0.75 }
\latex{ 0 }
=
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 6 }
\latex{ 5 }
\latex{ 4 }
\latex{ 7 }
\latex{ 8 }
\latex{ 9 }
\latex{ \div }
\latex{ RCM }
\latex{ - }
%
.
\latex{ + }
\latex{ + }
\latex{ - }
\latex{ MU }
\latex{ CCE }
\latex{ CA }
\latex{ M+ }
\latex{ M- }
\latex{ \times }
\latex{ \div }
numerator
fraction bar
denominator
fraction
division sign
dividend
divisor
quotient
\latex{ 3 } wholes are divided into \latex{ 4 } equal parts
one whole is divided into \latex{ 4 } equal parts, and \latex{ 3 } of them are taken
\latex{\frac{3}{4}}
=
\latex{3\div 4}
The quotient of two integers is a fraction. (The denominator cannot be \latex{ 0 }.)
Expanding fractions
Expanding a fraction means multiplying the numerator and the denominator by the same whole number other than zero.
The expanded forms of a fraction are equal.
\latex{0}
\latex{0}
\latex{0}
\latex{1}
\latex{1}
\latex{1}
\latex{\frac{2}{3} }
\latex{\frac{4}{6} }
\latex{\frac{6}{9} }
\latex{\frac{2}{3}\;=\;\frac{4}{6}\;=\;\frac{6}{9}\;=\;\frac{8}{12}\;=\;\frac{10}{15}\;=\;\frac{12}{18}\;=\;... }
\latex{\times 3 }
\latex{\times 3 }
\latex{\times 2 }
\latex{\times 2 }
The process can be
continued endlessly.
continued endlessly.
You can find the common denominator of two fractions by expanding them until they have the same denominator.
For example, the least common multiple of the denominators of \latex{\frac{1}{6}} and \latex{\frac{3}{4}} is:
\latex{\left[6;4\right] =12.}
Expanding the fractions to the common denominator: \latex{\frac{1}{6}= \frac{2}{12};\; \frac{3}{4} =\frac{9}{12} .}
Simplifying fractions
Simplifying a fraction means dividing the numerator and the denominator by the same whole number other than zero.
The simplified forms of a fraction are equal.
\latex{\frac{24}{18}\;=\;\frac{12}{9}\;=\;\frac{8}{6}=\;\frac{4}{3}}
\latex{\div2}
\latex{\div2}
\latex{\div3}
\latex{\div3}
\latex{\div6}
\latex{\div6}
The process can be continued until the numerator and the denominator have a common factor greater than \latex{ 1 } (until the numerator and the denominator become coprimes).
\latex{\frac{1}{18}}
\latex{\frac{6}{18}}
\latex{\frac{12}{18}}
\latex{\frac{24}{18}}
\latex{\frac{12}{9}}
\latex{\frac{12}{9}}
\latex{\frac{8}{6}}
\latex{\frac{8}{6}}
\latex{\frac{4}{3}}
\latex{\frac{1}{9}}
\latex{\frac{1}{6}}
\latex{\frac{1}{3}}
\latex{\frac{2}{6}}
\latex{\frac{3}{9}}
\latex{\frac{6}{9}}
\latex{\frac{4}{6}}
\latex{\frac{2}{3}}
\latex{1}
\latex{1}
\latex{1}
\latex{1}
\latex{0}
\latex{0}
\latex{0}
\latex{0}
If a fraction is simplified by the greatest common factor, it cannot be simplified further. The greatest common factor can be found by dividing the numerator and the denominator into prime factors.
Example:
\latex{\frac{24}{18} }
\latex{\frac{168}{96} }
\latex{24 }
\latex{12 }
\latex{6 }
\latex{3 }
\latex{1 }
\latex{18 }
\latex{9 }
\latex{3 }
\latex{1 }
\latex{96 }
\latex{48}
\latex{24}
\latex{12}
\latex{6}
\latex{3}
\latex{1}
\latex{168 }
\latex{84 }
\latex{42 }
\latex{21 }
\latex{7 }
\latex{1 }
\latex{2 }
\latex{2 }
\latex{3 }
\latex{2 }
\latex{2 }
\latex{7 }
\latex{3 }
\latex{2 }
\latex{2 }
\latex{2 }
\latex{2 }
\latex{2 }
\latex{2 }
\latex{2 }
\latex{2 }
\latex{3 }
\latex{3 }
\latex{3 }
\latex{(24;18)=2\times 3=6,}
\latex{(168;96)=2\times 2\times 2\times 3=24,}
simplifying by 6
simplifying by 24
\latex{\frac{24}{18}=\frac{4}{3} .}
\latex{\frac{168}{96}=\frac{7}{4} .}
Exercises
{{exercise_number}}. Which fractions are whole numbers?
- \latex{\frac{5}{15} }
- \latex{\frac{15}{5} }
- \latex{\frac{15}{3} }
- \latex{\frac{0}{12}}
- \latex{\frac{7}{5}}
- \latex{\frac{7}{7}}
{{exercise_number}}. Use fractions to describe how much of each circle is coloured if a circle is considered one whole. Arrange the fractions in ascending order.
a)
b)
c)
d)
e)
f)
g)
{{exercise_number}}. Write three expanded forms for every fraction.
- \latex{\frac{4}{9}}
- \latex{\frac{2}{5}}
- \latex{\frac{11}{10}}
- \latex{2\frac{1}{8}}
{{exercise_number}}. Find the common denominator of the following fractions by expanding or simplifying them, then compare them.
- \latex{\frac{6}{12}} and \latex{\frac{9}{6}}
- \latex{\frac{15}{45}} and \latex{\frac{72}{90}}
- \latex{\frac{13}{18}} and \latex{\frac{60}{90}}
- \latex{\frac{7}{8}} and \latex{\frac{9}{16}}
- \latex{\frac{5}{11}} and \latex{\frac{7}{5}}
- \latex{\frac{4}{27}} and \latex{\frac{7}{18}}
{{exercise_number}}. Simplify the following fractions with the help of prime factors.
- \latex{\frac{135}{54}}
- \latex{\frac{210}{294}}
- \latex{\frac{625}{3750}}
- \latex{\frac{7440}{4960}}
{{exercise_number}}. Which fraction is the odd one out? Why?
a)
b)
\latex{\frac{14}{10}}
\latex{\frac{77}{11}}
\latex{\frac{30}{6}}
\latex{\frac{8}{4}}
\latex{\frac{8}{6}}
\latex{\frac{72}{18}}
\latex{\frac{42}{48}}
\latex{\frac{35}{40}}
\latex{\frac{11}{15}}
\latex{\frac{21}{14}}
\latex{\frac{28}{32}}
\latex{\frac{14}{16}}
{{exercise_number}}. Arrange the following fractions in descending order.
- \latex{\frac{5}{12};\frac{0}{7};\frac{1}{3};\frac{9}{24};\frac{15}{12};\frac{3}{8};\frac{5}{6}}
- \latex{\frac{7}{5};\frac{5}{6};\frac{19}{15};\frac{11}{30};\frac{1}{2};\frac{2}{3}}
{{exercise_number}}. Complete the fractions \latex{\frac{2}{3};\frac{3}{5};\frac{7}{8};\frac{9}{10};\frac{7}{24}} and \latex{\frac{17}{60}} to a) \latex{ 1 } whole; b) \latex{ 2 } wholes.
{{exercise_number}}. Draw a \latex{ 7.5\, cm } long line segment and colour
- \latex{\frac{1}{5}} of it red;
- \latex{\frac{2}{3}} of it blue;
- \latex{\frac{6}{5}} of it yellow;
- \latex{\frac{4}{3}} of it green.
How many \latex{ centimetres } is the red, blue, yellow and green part? How many \latex{ millimetres } is that?
{{exercise_number}}. How many packages of \latex{ 0.25\, kg } can be made from
- \latex{ 1,500\,g; }
- \latex{ 2,000\,g; }
- \latex{ 4,250\,g;}
- \latex{6,750\,g} of coconut flakes?
{{exercise_number}}. A cube consists of smaller cubes with \latex{ 1\,centimetre } long edges.
What portion of the cubes with A) \latex{ 3\, cm }; B) \latex{ 4\, cm } and C) \latex{ 5\, cm } long edges is coloured if
- all the small cubes found at the vertices are coloured, but the rest are not;
- all the small cubes along the edges are coloured, but the rest are not?
Express the fractions in a form that cannot be simplified any further.
{{exercise_number}}. It took Agnes \latex{3\frac{2}{3}} \latex{ hours }; Andrew \latex{ 200 } \latex{ minutes; } Gale \latex{ 2 } \latex{ hours } \latex{ 45 } \latex{ minutes }; Julie \latex{\frac{12}{5}} \latex{ hours } to do their homework. Who was the fastest to complete the homework, and who spent the most time doing their homework? Write down the amount of time each child spent completing the homework in \latex{ hours } and \latex{ minutes }.
{{exercise_number}}. In total, \latex{ 35 } students from two classes went to the theatre. Half of the number of students from class \latex{ A } equals one-third of the students from class \latex{ B }. How many students went to the theatre from each class? (Illustrate it using line segments.)
{{exercise_number}}. How old is Grandpa if half of his age is \latex{ 12 } \latex{ years } more than its third? (Illustrate it using line segments.)
{{exercise_number}}.
The windows of Aunt Lizzy are decorated with flowers of three different colours. \latex{\frac{2}{3}} of the flowers are red, \latex{\frac{2}{3}} of the rest are white, and 1 flower is pink. How many flowers does Aunt Lizzy have? (Illustrate it using line segments.)
{{exercise_number}}. In art class, the students created a poster consisting of \latex{ 3 × 5 } rectangles. Inside the rectangles, they drew digits bordered by only straight lines. What part of the rectangles should be coloured in the following cases? Design the other digits as well, and determine how much of the rectangles should be coloured in each case.
