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Expanding and simplifying fractions
Example 1
Pete received half a pizza for dinner. He cut it in half, then cut the pieces he got in half again. Use fractions to write down the amount of pizza in all cases.
Solution
=
=
\latex{ 1 }
\latex{ 2 }
\latex{ 2 }
\latex{ 4 }
\latex{ 4 }
\latex{ 8 }
\latex{ \times2 }
\latex{ \times2 }
\latex{ \times2 }
\latex{ \times2 }
\latex{ \times4 }
\latex{ \times4 }
\latex{ \frac{1}{2} }
\latex{ \frac{4}{8} }
After the first cut, the \latex{\frac{1}{2}} pizza became \latex{\frac{2}{4}}; after the second cut, it became \latex{\frac{4}{8}} of a pizza.
The size of the pizza does not change during the cuts.
When expanding a fraction, the numerator and the denominator are multiplied by the same number (not \latex{ 0 }). This way, the value of the fraction does not change.
Example 2
Vince had swimming practice from \latex{ 12 } PM to \latex{ 4 } PM. What fraction of the day did he spend training?
Solution (\latex{ 1 } \latex{ day } = \latex{ 24 } \latex{ hours })
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ 8 }
\latex{ 9 }
\latex{ 10 }
\latex{ 11 }
\latex{ 12 }
\latex{ 13 }
\latex{ 14 }
\latex{ 15 }
\latex{ 16 }
\latex{ 17 }
\latex{ 18 }
\latex{ 19 }
\latex{ 20 }
\latex{ 21 }
\latex{ 22 }
\latex{ 23 }
\latex{ 24 }
\latex{ 0 }
\latex{ \frac{4}{24} } \latex{ day }
\latex{ 1 } \latex{ hour } \latex{ =\frac{1}{24} } \latex{ day }
\latex{ 2 }
\latex{ 4 }
\latex{ 6 }
\latex{ 8 }
\latex{ 10 }
\latex{ 12 }
\latex{ 14 }
\latex{ 16 }
\latex{ 18 }
\latex{ 20 }
\latex{ 22 }
\latex{ 24 }
\latex{ 0 }
\latex{ \frac{2}{12} } \latex{ day }
\latex{ 2 } \latex{ hour } \latex{ =\frac{1}{12} } \latex{ day }
\latex{ \div2 }
\latex{ \div2 }
\latex{ \frac{4}{24} }
\latex{ \frac{2}{12} }
\latex{ 4 }
\latex{ 8 }
\latex{ 12 }
\latex{ 16 }
\latex{ 20 }
\latex{ 24 }
\latex{ 0 }
\latex{ \frac{1}{6} } \latex{ day }
\latex{ 4 } \latex{ hour } \latex{ =\frac{1}{6} } \latex{ day }
\latex{ \div4 }
\latex{ \div4 }
\latex{ \frac{4}{24} }
\latex{ \frac{1}{6} }
Vince spent \latex{\frac{4}{24}=\frac{2}{12}=\frac{1}{6}} of the \latex{ day } at the pool.
When simplifying a fraction, the numerator and the denominator are divided by the same number (not \latex{ 0 }). This way, the value of the fraction does not change.
Example 3
a) Simplify the fraction \latex{\frac{15}{20}} by \latex{ 5 }.
b) Simplify \latex{\frac{24}{36}} by the largest factor possible.
c) Expand \latex{\frac{4}{5}}, so that the new denominator would be \latex{ 20 }.
d) Expand \latex{\frac{9}{10}}, so that the new numerator would be \latex{ 45 }.
Solution
Simplifying by \latex{ 5 } means dividing both the numerator and the denominator by \latex{ 5 },
that is,
b)
or
or
\latex{ \frac{15}{20} }
\latex{ \frac{3}{4} }.
\latex{ \div5 }
\latex{ \div5 }
\latex{ \frac{24}{36} }
\latex{ \frac{12}{18} }
\latex{ \frac{6}{9} }
\latex{ \frac{2}{3} }
\latex{ \frac{24}{36} }
\latex{ \frac{4}{6} }
\latex{ \frac{2}{3} }
\latex{ \frac{24}{36} }
\latex{ \frac{2}{3} }.
\latex{ \div2 }
\latex{ \div2 }
\latex{ \div2 }
\latex{ \div2 }
\latex{ \div3 }
\latex{ \div3 }
\latex{ \div6 }
\latex{ \div6 }
\latex{ \div2 }
\latex{ \div2 }
\latex{ \div12 }
\latex{ \div12 }
Since \latex{2 \times 2 \times 3=6 \times 2 = 12}, the fraction can be simplified by \latex{ 12 } in one step. The fraction \latex{\frac{2}{3}} cannot be further simplified.
c) When expanding a fraction to the twentieth, the denominator will be \latex{ 20 }.
The denominator is multiplied by \latex{ 4 }; therefore,
the numerator must be multiplied by \latex{ 4 } as well.
the numerator must be multiplied by \latex{ 4 } as well.
d) For the numerator to be \latex{ 45 }, \latex{ 9 } must be multiplied by \latex{ 5 }; therefore,
the denominator must be multiplied by \latex{ 5 } as well:
During expansion and simplification, the value of the fractions did not change.
\latex{ \frac{4}{5} }
\latex{ \frac{16}{20} }
\latex{ \times4 }
\latex{ \times4 }
\latex{ \frac{9}{10} }
\latex{ \frac{45}{50} }
\latex{ \times5 }
\latex{ \times5 }
Exercises
{{exercise_number}}. Write at least \latex{ 5 } fractions that are equal to
a) \latex{\frac{1}{3}};
b) \latex{\frac{2}{5}};
c) \latex{\frac{4}{7}};
d) \latex{\frac{10}{20}};
e) \latex{\frac{9}{12}}.
{{exercise_number}}. By what numbers should the \latex{\square} and \latex{\bigcirc} be replaced to make the equalities true?
a) \latex{\frac{14}{35}=\frac{\bigcirc}{5}}
b) \latex{\frac{25}{40}=\frac{5}{\bigcirc}}
c) \latex{\frac{26}{39}=\frac{\bigcirc}{3}}
d) \latex{\frac{\bigcirc}{60}=\frac{9}{12}=\frac{3}{\square}}
{{exercise_number}}. A boy is walking with \latex{ 12 } balloons in his hands. He let go \latex{ 3 } of the \latex{ 12 } balloons. What fraction of the balloons flew away? What fraction of the balloons is still in his hands? Write it in the simplest form possible.
{{exercise_number}}. Simplify the fractions.
a) \latex{\frac{4}{2}}
b) \latex{\frac{12}{3}}
c) \latex{\frac{30}{5}}
d) \latex{\frac{48}{12}}
e) \latex{\frac{66}{11}}
f) \latex{\frac{63}{9}}
{{exercise_number}}. Simplify the fractions by the largest factor possible.
a) \latex{\frac{8}{12}}
b) \latex{\frac{16}{20}}
c) \latex{\frac{15}{25}}
d) \latex{\frac{40}{60}}
e) \latex{\frac{30}{48}}
f) \latex{\frac{56}{100}}
g) \latex{\frac{75}{100}}
{{exercise_number}}. Gabe sleeps \latex{ 8 } \latex{ hours } a \latex{ day. } What fraction of the \latex{ day } does he spend sleeping?
{{exercise_number}}. By what numbers should the \latex{\square} be replaced to make the equalities true?
a) \latex{\frac{10}{7}=\frac{80}{\square}}
b) \latex{\frac{5}{5}=\frac{\square}{30}}
c) \latex{\frac{9}{17}=\frac{18}{\square}}
d) \latex{\frac{2}{9}=\frac{\square}{63}}
e) \latex{\frac{13}{12}=\frac{\square}{36}}
{{exercise_number}}. List all the fractions whose numerator contains the divisors of \latex{ 6 }, while the denominator includes the divisors of \latex{ 8 } and simplify them.
{{exercise_number}}. Simplify or expand the fractions \latex{\frac{2}{3}}, \latex{\frac{70}{150}} and \latex{\frac{3}{5}}, so their denominator becomes \latex{ 15 }. Arrange them in ascending order.
{{exercise_number}}. Expand or simplify the following fractions in such a way that
a) their denominator becomes \latex{ 12 }: \latex{\frac{3}{2};\frac{7}{3};\frac{3}{4};\frac{5}{6};\frac{40}{40};}
b) their numerator becomes \latex{ 20 }: \latex{\frac{1}{2};\frac{60}{90};\frac{5}{4};\frac{10}{7};\frac{2}{5}.}
{{exercise_number}}. Write numbers in the empty spaces to make the equalities true.
a) \latex{\frac{10}{4}=⏤\frac{}{} =\frac{}{10}}
b) \latex{\frac{6}{15}=⏤ =\frac{}{25}}
c)\latex{\frac{21}{35}=⏤ =\frac{}{10}}
d) \latex{\frac{55}{77}=⏤ =\frac{}{35}}
{{exercise_number}}. In which case are the two fractions equal?
a) \latex{\frac{4}{7};\frac{30}{35}}
b) \latex{\frac{4}{5};\frac{20}{25}}
c) \latex{\frac{9}{12};\frac{21}{28}}
d) \latex{\frac{3}{4};\frac{5}{6}}
{{exercise_number}}. Which is Pam's necklace if two-thirds of its hearts are purple?
a)
b)
c)
d)
e)
Quiz
\latex{ 6 } matchsticks were used to make \latex{\frac{1}{7}}. Move one matchstick to get \latex{\frac{1}{3}}.
