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Mathematics 6.

Table of contents
Divisibility
How to solve problems?
Rational numbers I.
Axial symmetry
Rational numbers II.
Fractions (revision)
Operations with fractions (revision)
Negative fractions
Multiplying fractions by fractions
The reciprocals of numbers
Division by a fraction
Properties of operations
Rational numbers
Mixed exercises
Proportion
Percentages
Probability, statistics
Supplementary material
Negative fractions
On a winter afternoon, Zack measured the temperature every hour. According to his measurements, the temperature decreased uniformly between \latex{1\;PM } and \latex{5 \;PM } from \latex{ –1 °C } to \latex{ –5 °C }. What was the temperature at \latex{ 1:30} \latex{ PM}, \latex{ 2:30} \latex{ PM }, ...?
The temperature decreased uniformly by \latex{ 1 °C } every hour, that is, by \latex{ 0.5 °C } every half an hour. This can be expressed by defining the negative fractions between two negative numbers.
Example 1
Perform the following subtractions.
  1. \latex{\frac{4}{5}-\frac{1}{5}}
  1. \latex{\frac{1}{5}-\frac{4}{5}}
Solution
Apply what you have learned when subtracting fractions. (Calculate the difference of the numerators and leave the denominator unchanged.)
  1. \latex{\frac{4}{5}-\frac{1}{5}=\frac{4-1}{5}=\frac{3}{5}\gt 0.}
\latex{-1}
\latex{0}
\latex{1}
\latex{\frac{3}{5} }
\latex{\frac{4}{5} }
\latex{-\frac{1}{5} }
  1. \latex{\frac{1}{5} -\frac{4}{5}=\frac{1-4}{5}=\frac{-3}{5}\lt 0.}
\latex{-1}
\latex{0}
\latex{1}
\latex{-\frac{3}{5} }
\latex{\frac{1}{5} }
In case b), the numerator is negative.
\latex{-\frac{3}{5} = -\frac{3}{5}}
Negative fractions are smaller than 0.
The position of negative fractions on the number line
Negative halves: \latex{ 1 } unit is divided into two equal parts.
\latex{-2}
\latex{-1}
\latex{0}
\latex{-\frac{4}{2} }
\latex{-\frac{3}{2} }
\latex{-\frac{2}{2} }
\latex{-\frac{1}{2} }
\latex{0 }
Negative thirds: \latex{ 1 } unit is divided into three equal parts.
\latex{-2}
\latex{-1}
\latex{0}
\latex{-\frac{6}{3} }
\latex{-\frac{5}{3} }
\latex{-\frac{4}{3} }
\latex{-\frac{3}{3} }
\latex{0 }
\latex{-\frac{2}{3} }
\latex{-\frac{1}{3} }
Negative fourths: \latex{ 1 } unit is divided into four equal parts.
\latex{-2}
\latex{-1}
\latex{0}
\latex{-\frac{8}{4} }
\latex{-\frac{7}{4} }
\latex{-\frac{6}{4} }
\latex{-\frac{5}{4} }
\latex{0 }
\latex{-\frac{4}{4} }
\latex{-\frac{3}{4} }
\latex{-\frac{2}{4} }
\latex{-\frac{1}{4} }
Negative fractions with larger denominators can be represented similarly. Negative fractions can be simplified and expanded similarly to positive fractions.

\latex{-\frac{6}{4}=-\frac{3}{2}}


\latex{-\frac{2}{4}=-\frac{1}{2}}

The additive inverse and absolute value of negative fractions
\latex{-1}
\latex{0}
\latex{1}
\latex{-\frac{3}{3} }
\latex{-\frac{2}{3} }
\latex{-\frac{1}{3} }
\latex{\frac{1}{3} }
\latex{0}
\latex{\frac{2}{3} }
\latex{\frac{3}{3} }
The distance of \latex{-\frac{2}{3}} and \latex{\frac{2}{3}} from zero on the number line is equal.
\latex{-\left(-\frac{2}{3} \right)=\frac{2}{3}}
\latex{-\frac{2}{3}} is the additive inverse of \latex{\frac{2}{3}}, while \latex{\frac{2}{3}} is the additive inverse of \latex{-\frac{2}{3}}.
Two numbers are additive inverses of each other if their distance from zero on a number line is equal.
 
\latex{\frac{2}{3}+\left(-\frac{2}{3} \right)=0}, that is, \latex{\frac{2}{3}- \frac{2}{3}=0}.
 
If two numbers are the additive inverses of each other, their sum is zero.
The absolute value of a number is its distance from \latex{ 0 } on the number line.

So, the absolute values of \latex{-\frac{2}{3}}  and  \latex{\frac{2}{3}} are equal, that is, \latex{\left|-\frac{2}{3} \right| =\left|\frac{2}{3} \right|=+\frac{2}{3}} .
The absolute value of a positive fraction is the number itself, while the absolute value of a negative fraction is its additive inverse.
Comparing negative fractions
Compare the negative fractions shown on the number line below.
\latex{-1}
\latex{0}
\latex{0 }
\latex{-\frac{5}{5} }
\latex{-\frac{4}{5} }
\latex{-\frac{3}{5} }
\latex{-\frac{2}{5} }
\latex{-\frac{1}{5} }
\latex{\lt }
\latex{\lt }
\latex{\lt }
\latex{\lt }
\latex{\lt }
Any negative fraction is smaller than 0 and any positive number.
The smaller the absolute value of a negative fraction, the larger the fraction is.
\latex{-\frac{2}{5}\lt -\frac{1}{5}}
\latex{-\frac{4}{5}\lt -\frac{2}{5}}
\latex{\left|-\frac{2}{5} \right|\gt \left|-\frac{1}{5} \right|}
\latex{\left|-\frac{4}{5} \right|\gt \left|-\frac{2}{5} \right|}
Operations with negative fractions
Operations with negative fractions are similar to those with positive fractions. The sign of the result is determined according to the already known rules of dealing with negative integers.
Example 2
Perform the following operations.
  1. \latex{-\frac{4}{5}+\left(-\frac{2}{3} \right)+\frac{8}{15}}
  1. \latex{3\times \left(-\frac{5}{8} \right)}
  1. \latex{-\frac{9}{10}\div (-2)}
Solution
  1. \latex{-\frac{4}{5}+\left(-\frac{2}{3} \right)+\frac{8}{15}=-\frac{4}{5}-\frac{2}{3}+\frac{8}{15}=-\frac{12}{15}-\frac{10}{15}+\frac{8}{15}=-\frac{14}{15}} . 
Eliminate the brackets, find a common denominator and perform the operations.
  1. \latex{3\times \left(-\frac{5}{8} \right)=-\frac{15}{8}=-1\frac{7}{8}}.
The product of a negative and a positive number is negative.
 
  1. \latex{-\frac{9}{10}\div (-2)=\frac{9}{20}.}
The quotient of two negative numbers is positive.
\latex{\begin{aligned}&2\div 3=\frac{2}{3}\\[5pt]-&2\div 3=-\frac{2}{3}\\[5pt]&2\div (-3)=-\frac{2}{3}\\[5pt]-&2\div (-3)=\frac{2}{3}\end{aligned}}
Mixed numbers containing negative fractions
Fractions smaller than \latex{ –1 } can be written as mixed numbers:
 
\latex{-\frac{12}{5}=-\left(\frac{10}{5}+\frac{2}{5} \right)=-\left(2\frac{2}{5} \right)=-2\frac{2}{5}}.
 
In a different way:
 
\latex{-\frac{12}{5}=-\frac{10}{5}+\left(-\frac{2}{5} \right)=-2+\left(-\frac{2}{5} \right)=-2\frac{2}{5}.}
If a fraction is expressed as a negative mixed number, it can be converted to a fraction in the following way:
 
\latex{-5\frac{2}{7}=-\left(5+\frac{2}{7} \right)=-\left(\frac{35}{7}+\frac{2}{7} \right)=-\frac{37}{7},}
\latex{-\left(5\frac{2}{7} \right)=-\frac{35}{7}+\left(-\frac{2}{7} \right)=-\frac{37}{7}.}
Exercises
{{exercise_number}}. Which of the fractions \latex{-\frac{2}{3};\;\;\;\frac{6}{7};\;\;\;-\frac{6}{2};\;\;\;-\frac{3}{2};\;\;\;-\frac{7}{6}; \;\;\;-\frac{9}{3};\;\;\;\frac{6}{2};\;\;\;\frac{3}{2};\;\;\;-\frac{11}{2}},
  1. have equal absolute values;
  2. are additive inverses of each other;
  3. are smaller than \latex{ –1 };
  4. is the smallest;
  5. is the greatest;
  6. can be written as a mixed number?
{{exercise_number}}. Replace the symbols with positive integers to make the following equalities and inequalities true. How many solutions are there in each case?
  1. \latex{-\frac{3}{4}\lt -\frac{\square }{4} }
  1. \latex{-2\leq -\frac{\square }{5} }
  1. \latex{-1\gt -\frac{3}{\square } }
  1. \latex{-\frac{2}{\square }=-\frac{4}{10} }
{{exercise_number}}. Perform the additions and subtractions.
  1. \latex{\frac{7}{8}-\frac{8}{9} }
  1. \latex{1\frac{4}{5}-2\frac{2}{5} }
  1. \latex{\frac{9}{4}-2\frac{3}{4} }
  1. \latex{4\frac{1}{3}-7\frac{3}{7} }
  1. \latex{-\frac{4}{5}+\frac{2}{3} }
  1. \latex{-\frac{7}{8}-\frac{1}{2}-\frac{3}{4} }
  1. \latex{2\frac{2}{3}+1\frac{1}{2}-5\frac{5}{6}}
{{exercise_number}}. What number should you write in the \latex{\square} to make the equalities true?
  1. \latex{-\frac{1}{5}+\square =-1 }
  1. \latex{\frac{1}{2}-\square =-\frac{3}{4} }
  1. \latex{\square +3\frac{1}{3}=2\frac{5}{6} }
  1. \latex{-3-\square =\frac{1}{2} }
  1. \latex{3\frac{1}{5}-\left(1\frac{1}{2}+\square \right)=0 }
  1. \latex{-2\frac{1}{3}-4\frac{1}{2}-\square =-1 }
{{exercise_number}}. Perform the multiplications and divisions.
  1. \latex{5\times \left(-\frac{2}{3} \right)}
  1. \latex{(-3)\times \frac{4}{5}}
  1. \latex{-\frac{7}{9}\times 3}
  1. \latex{(-8)\times \left(-\frac{9}{16} \right)}
  1. \latex{-\frac{9}{10}\times (-12)}
  1. \latex{-\frac{4}{5}\div 2}
  1. \latex{\frac{6}{8}\div (-3)}
  1. \latex{2\frac{1}{5}\div (-11)}
  1. \latex{-4\frac{2}{5}\div (-5)}
  1. \latex{-\frac{9}{11}\div (-18)}
  1. \latex{(-8)\times \left(2\frac{1}{2}-\frac{3}{4} \right)}
  1. \latex{\left(2\frac{1}{2}-\frac{3}{4} \right)\div (-7)}
{{exercise_number}}. What number should you write in the \latex{\square} to make the equalities true?
  1. \latex{-\frac{2}{3}\div \square =-\frac{2}{9} }
  1. \latex{-\frac{2}{3}\div \square =\frac{2}{9} }
  1. \latex{\frac{2}{3}\div \square =-\frac{2}{9} }
  1. \latex{\frac{\square }{7}\div 5 =-\frac{3}{35}}
  1. \latex{-\frac{\square }{7}\div 5 =-\frac{3}{35} }
  1. \latex{\frac{\square }{7}\div (-5) =-\frac{3}{35}}
{{exercise_number}}. Continue each sequence with four additional terms.
  1. \latex{-\frac{1}{2};-\frac{1}{4};-\frac{1}{8};...}
  1. \latex{-7\frac{1}{5} ;-1\frac{1}{5};-\frac{1}{5};...}
  1. \latex{-\frac{1}{8} ;\frac{5}{8};-3\frac{1}{8};...}
{{exercise_number}}. What number is
  1. one-tenth of \latex{-\frac{5}{4};}
  1. ten times \latex{-\frac{5}{4};}
  1. two times \latex{-\frac{5}{4};}
  1. \latex{-\frac{5}{4}} when multiplied by \latex{-3;}
  1. half of \latex{-\frac{5}{4};}
  1. five times \latex{-\frac{5}{4}?}
{{exercise_number}}.  There was \latex{\frac{3}{4}} \latex{ kilogram } of bread left. Three persons ate \latex{\frac{3}{25}} \latex{ kg } of bread each for breakfast. In the afternoon, each of them made a sandwich using \latex{\frac{1}{10}} \latex{ kg } of bread. For lunch, they ate \latex{ 90 \;grams } in total. How much bread do they have left for dinner?
Quiz
What numbers can replace the symbols?     \latex{\frac{1}{\square } +\frac{1}{\triangle }+\frac{1}{\bigcirc}}=\latex{1}
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