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Mathematics 6.

Table of contents
Divisibility
How to solve problems?
Rational numbers I.
Integers (revision)
The addition and subtraction of integers (revision)
Combining
The multiplication of integers
The division of integers
Combining decimals
Multiplication of decimals
Division with decimals
Mixed exercises
Axial symmetry
Rational numbers II.
Proportion
Percentages
Probability, statistics
Supplementary material
The addition and subtraction of integers (revision)
The addition of integers with the same sign
Example 1
Robinson and Friday are playing a card game. Every card features a number between \latex{ (–16) } and \latex{ (+16) }. Both of them draw two cards, and the winner is the one with the greater sum. (If the sums are equal, it is a tie.)
  1. Which pair of numbers would certainly win?
  1. Which pair of numbers would certainly lose?
Solution
  1. The player who draws the two largest numbers certainly wins.
\latex{(+16) + (+15) = (+31)} or \latex{(+15) + (+16) = (+31)}
  1. The player who draws the two smallest numbers certainly loses.
\latex{(-16) + (-15) = (-31)} or \latex{(-15) + (-16) = (-31)}
The addends are interchangeable.
The sum of positive numbers is a positive number.
\latex{(+16) + (+15) = (+31)}
The sum of negative numbers is a negative number.
\latex{(-16) + (-15) = (-31)}
Numbers with the same sign are added by calculating the sum of their absolute values. The sign of the sum is the same as that of the addends.
\latex{\mid-16\mid = 16}
\latex{\mid-15\mid = 15}
\latex{16+15=31}
The addition of integers with different signs
Example 2
Robinson and Friday drew two cards each from the following four cards. Both of them got cards with different signs. Which cards did Friday draw if the sum of his cards was greater? 
\latex{+15}
\latex{-15}
\latex{+6}
\latex{-6}
Solution
If one of them had drawn \latex{ +15 } and \latex{ –15 } and the other \latex{ +6 } and \latex{ –6 }, then it would have been a tie because
 
\latex{(+15) + (-15) = 0 \text{ and } (+6) + (-6) = 0}.
 
Therefore, the absolute values of the numbers on the cards must have been different.
So, the two sums are:
 
\latex{\textcolor{199fe3}{(+15)} + (-6) = (+9), \text{ since } \textcolor{199fe3}{(+9)+(+6)} + (-6) = (+9)},
\latex{\textcolor{199fe3}{(-15)} + (+6) = (-9), \text{ since } \textcolor{199fe3}{(-9)+(-6)} + (+6) = (-9)}.
Friday must have drawn \latex{ +15 } and \latex{ –6 }. The order of the cards does not matter.
In case of adding numbers with different signs,
  • if the absolute value of the positive number is greater, the sum will be positive:
\latex{(+15) + (-6) = (+9)};
  • if the absolute value of the negative number is greater, the sum will be negative:
\latex{(-15) + (+6) = (-9)};
  • if the absolute value of the two numbers is equal, the sum is zero:
\latex{(+15) + (-15) = 0};
When adding numbers with different signs, use their absolute values and subtract the smaller absolute value from the greater one. The sign of the sum will be that of the number with the larger absolute value.
If two numbers are the additive inverses of each other, their sum is zero.
\latex{\mid+15\mid = 15}
\latex{\mid-6\mid = 6}
\latex{15\gt 6}
Addition with various addends
Several methods can be used when calculating the sum of more than \latex{ 2 } addends. You can decide which one to use depending on the given addition.
  • The sum of various addends can be calculated by simply adding the numbers from left to right.
\latex{\textcolor{199fe3}{(+9)+(+8)} + 0 + (-10) + (-4) = \textcolor{199fe3}{(+17)+0} + (-10) + (-4) =\\= \textcolor{199fe3}{(+17)+(-10)} + (-4) = (+7) + (-4) = (+3) = 3.}
  • If the addends contain additive inverses, it is worth rearranging and grouping them before performing the addition.
\latex{(-143) + (-57) + (+143) + (-43) + (+57) =}
\latex{= (-143) + (+143) + (-57) + (+57) + (-43) = (-43).}
\latex{\textcolor{199fe3}{0}}
\latex{\textcolor{199fe3}{0}}
  • The addends can also be grouped according to their signs.
\latex{(\textcolor{red}{+}2) + (\textcolor{009fe3}{-}32) + (\textcolor{red}{+}15) + (\textcolor{009fe3}{-}8) + (\textcolor{red}{+}18) =\\ = [(\textcolor{red}{+}2) + (\textcolor{red}{+}15) + (\textcolor{red}{+}18)] + [(\textcolor{009fe3}{-}32) + (\textcolor{009fe3}{-}8)] =\\ = (+35) + (-40) = (-5).}
The addends are interchangeable and can be grouped.
The subtraction of integers
Example 3
Robinson and Friday are playing another card game. Every card features a number between \latex{ (–16) } and \latex{ (+16) }, but there are two of each this time. They draw two cards and calculate the difference of the numbers on the cards in the order they were drawn. The player who has the larger difference wins.
  1. What numbers could Friday draw if their difference was \latex{ 0 }?
  2. Which pair of numbers would certainly win?
  3. Which pair of numbers would certainly lose?
Solution
  1. He can get \latex{ 0 } only if he draws the same numbers.
E.g.: \latex{(+10)-(+10) = 0;\qquad(-12)-(-12) = 0;\qquad0-0 = 0}.
  1. The player who draws \latex{ +16 } first and then \latex{ –16 } certainly wins because
\latex{(+16)-(-16) = (+16) + (+16) = (+32)}.
  1. The player who draws \latex{ –16 } first and then \latex{ +16 } certainly loses because
\latex{(-16)-(+16) = (-16) + (-16) = (-32)}.
The difference of two equal numbers is zero.
Instead of subtracting a negative number, you can perform an addition using the absolute value of the number:
\latex{(+16)-(-16) = (+16) + (+16) = (+32)}.
Instead of subtracting a positive number, you can perform an addition using its additive inverse:
\latex{(-16)-(+16) = (-16) + (-16) = (-32)}.
Any number can be subtracted from another number by adding its additive inverse to it.
Exercises
{{exercise_number}}. How does your financial situation change if
  1. you receive cash;
  1. you lose your wallet full of cash;
  1. you accumulate debt;
  1. someone pays your bills for you;
  1. you win on the lottery;
  1. a part of your debt is cancelled;
  1. someone steals your bicycle?
{{exercise_number}}. Observe the following additions.
  1. \latex{\fcolorbox{f6b900}{fef6e3}{$(-12) + (+14)$}}
  1. \latex{\fcolorbox{f6b900}{fef6e3}{$(-12) + (-14)$}}
  1. \latex{\fcolorbox{f6b900}{fef6e3}{$(+12) + (-14)$}}
  1. \latex{\fcolorbox{f6b900}{fef6e3}{$(+12) + (+14)$}}
  1. \latex{\fcolorbox{f6b900}{fef6e3}{$(-12) + (-10)$}}
  1. \latex{\fcolorbox{f6b900}{fef6e3}{$(-10) + (+12)$}}
  1. The sums of which additions are positive?
  1. The sums of which additions are smaller than the first addend?
  1. The sum of which addition is the greatest?
  1. What is the sum of the smallest and greatest result?
{{exercise_number}}. Fill in the missing numbers and operators.
a)
b)
\latex{-60}
\latex{+(-17)}
\latex{-(\qquad)}
\latex{-100}
\latex{-45}
\latex{-77}
\latex{(\qquad)}
\latex{(\qquad)}
\latex{(\qquad)}
\latex{(\qquad)}
\latex{(\qquad)}
\latex{(\qquad)}
\latex{-(-50)}
\latex{+(+19)}
\latex{+(\qquad)}
\latex{(\qquad)}
\latex{(\qquad)}
\latex{-(-49)}
{{exercise_number}}. What is the sum?
  1. \latex{-1 + 2-3 + 4-5 + 6-7 + 8-9}
  1. \latex{1-2+3 - 4+5 - 6+7 - 8+9}
  1. Write an addition, like the one above, whose sum is \latex{ (+1) }.
  1. Write an addition, like the one above, whose sum is \latex{ (–1) }.
{{exercise_number}}. Dad had \latex{ 360 \;euros } in his bank account, \latex{ 45 \;euros } in his wallet and \latex{ 15 \;euros } in his pocket. This morning, he paid \latex{ 22 \;euros } for the telephone bill and \latex{ 18\; euros } for the water bill. Moreover, he also bought a book for \latex{ 12 \;euros } as a birthday present for his daughter.
How much money does Dad have now?
{{exercise_number}}. Convert the following subtractions to additions and perform them.
  1. \latex{(+13)-(+31)}
  1. \latex{(+29)-(-13)}
  1. \latex{(-15)-(+17)}
  1. \latex{(-53)-(-11)}
  1. \latex{(+58)-(+21)}
  1. \latex{(+11)-(-43)}
  1. \latex{(-18)-(+12)}
  1. \latex{(-77)-(-99)}
{{exercise_number}}. Arrange the differences in ascending order.
  1. A:  \latex{\fcolorbox{f6b900}{fef6e3}{$(-3)-(-11)$}\qquad}  B:  \latex{\fcolorbox{f6b900}{fef6e3}{$(-3)-(+11)$}\qquad} C: \latex{\fcolorbox{f6b900}{fef6e3}{$(-11)-(-3)$}\qquad} D: \latex{\fcolorbox{f6b900}{fef6e3}{$(+11)-(-3)$}}
  1. A:  \latex{\fcolorbox{008a43}{eaf4f2}{$(-4)+(-7)$}\qquad} B:  \latex{\fcolorbox{008a43}{eaf4f2}{$(-7)-(-9)$}\qquad} C:  \latex{\fcolorbox{008a43}{eaf4f2}{$(-9)+(-2)$}\qquad} D:  \latex{\fcolorbox{008a43}{eaf4f2}{$(+8)+(-8)$}}
  1. A:  \latex{\fcolorbox{ba0217}{fbeae5}{$\mid5-7\mid-\mid-2\mid$}\qquad} B:  \latex{\fcolorbox{ba0217}{fbeae5}{$\mid5-7\mid-\mid5-7\mid$}\qquad} C:  \latex{\fcolorbox{ba0217}{fbeae5}{$\mid-2\mid-\mid7-5\mid$}}
  1. A: \latex{\fcolorbox{0062ae}{e0e8f7}{$(-1)-(7+13)$}\qquad} B: \latex{\fcolorbox{0062ae}{e0e8f7}{$\mid-1\mid-(7-13)$}\qquad} C: \latex{\fcolorbox{0062ae}{e0e8f7}{$(-1)-\mid7-13\mid$}\qquad} D: \latex{\fcolorbox{0062ae}{e0e8f7}{$\mid12-5\mid$}}
{{exercise_number}}. Calculate the sums in the easiest way possible.
  1. \latex{(+726) + (+911) + (+274) + (+1,089)}
  1. \latex{(+2,007) + (+8,893) + (11,100)}
  1. \latex{(-1,001) + (-10,100) + (-100,010)}
  1. \latex{(-11,111) + (+1,111) + (-111) + (+11) + (-1)}
  1. \latex{(-117) + (-43) + (+34) + (-34)}
{{exercise_number}}. Substitute the letters with integers to make each equation true.
  1. \latex{a + (-18) = -25}
  1. \latex{b-(-18) = -25}
  1. \latex{(-73)-(-45) = c}
  1. \latex{d-(+37) = -98}
  1. \latex{(-57) + e = -10}
  1. \latex{f -(-130) = -100}
{{exercise_number}}. Mark the vertices of an \latex{ ABCD } quadrilateral in a Cartesian coordinate system, then connect the points. Use the same coordinate system but different colours for each exercise.
\latex{A(3; 0) \quad B(1; 2) \quad C(-1; 2) \quad D(-3; 0)}
  1. Add \latex{ (+4) } to the first coordinate of every point, and mark the resulting points.
  1. Add \latex{ (–4) } to the second coordinate of the original quadrilateral's vertices and mark the resulting points.
  1. Subtract \latex{ (–3) } from every coordinate of the original quadrilateral's vertices and mark the resulting points.
Quiz
What will the sum be if there are \latex{ 1,000 } addends?
  1. \latex{(+1) + (–2) + (+3) + (–4) + …}
  2. \latex{(–1) + (+2) + (–3) + (+4) + …}

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