Twój koszyk jest pusty
Combining
Perform the following additions and show them on a number line.
\latex{(+5) + (+3) = (+8)}
\latex{(-2) + (+5) = (+3)}
\latex{(+5) + (-3) = (+2)}
\latex{(-2) + (-5) = (-7)}
\latex{+3}
\latex{-4}
\latex{-3}
\latex{-2}
\latex{-1}
\latex{0}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{5}
\latex{6}
\latex{7}
\latex{8}
\latex{8}
\latex{7}
\latex{6}
\latex{5}
\latex{4}
\latex{3}
\latex{2}
\latex{1}
\latex{0}
\latex{-1}
\latex{-2}
\latex{-3}
\latex{-4}
\latex{+5}
\latex{-3}
\latex{-7}
\latex{-6}
\latex{-5}
\latex{-4}
\latex{-3}
\latex{-2}
\latex{-1}
\latex{0}
\latex{1}
\latex{2}
\latex{3}
\latex{4}
\latex{5}
\latex{4}
\latex{5}
\latex{3}
\latex{2}
\latex{1}
\latex{0}
\latex{-1}
\latex{-2}
\latex{-3}
\latex{-4}
\latex{-5}
\latex{-6}
\latex{-7}
\latex{-5}
If
- a positive number is added to a number, the sum will be greater than the given number;
- a negative number is added to a number, the sum will be smaller than the given number.
The previous additions can be expressed in a shorter form:
\latex{(+5) + (\textcolor{red}{+}3) = (+8) \quad\to\quad\;\;\; 5 \textcolor{red}{+} 3 = 8;}
\latex{(-2) + (\textcolor{red}{+}5) = (+3) \quad\to\quad -2 \textcolor{red}{+} 5 = 3;}
\latex{(+5) + (\textcolor{red}{-}3) = (+2) \quad\to\quad\;\;\; 5 \textcolor{red}{-} 3 = 2;}
\latex{(-2) + (\textcolor{red}{-}5) = (-7) \quad\to\quad -2 \textcolor{red}{-} 5 = -7.}
\latex{(-2) + (\textcolor{red}{+}5) = (+3) \quad\to\quad -2 \textcolor{red}{+} 5 = 3;}
\latex{(+5) + (\textcolor{red}{-}3) = (+2) \quad\to\quad\;\;\; 5 \textcolor{red}{-} 3 = 2;}
\latex{(-2) + (\textcolor{red}{-}5) = (-7) \quad\to\quad -2 \textcolor{red}{-} 5 = -7.}
Subtractions can always be expressed as additions; therefore, subtractions can also be expressed in a shorter form:
\latex{(+8)\textcolor{009fe3}{-}(\textcolor{red}{+}5) = (+8)\textcolor{009fe3}{+}(\textcolor{red}{-}5)=(+3)\quad\to\quad\;\;\; 8 \textcolor{red}{-} 5=\;\;\; 3;}
\latex{(+8)\textcolor{009fe3}{-}(\textcolor{red}{+}9) = (+8)\textcolor{009fe3}{+}(\textcolor{red}{-}9)=(-1)\quad\to\quad\;\;\; 8 \textcolor{red}{-}9= -1;}
\latex{(-5)\textcolor{009fe3}{-}(\textcolor{red}{+}4) = (-5)\textcolor{009fe3}{+}(\textcolor{red}{-}4)=(-9)\quad\to\quad -5 \textcolor{red}{-}4= -9;}
\latex{(-5)\textcolor{009fe3}{-}(\textcolor{red}{-}4) = (-5)\textcolor{009fe3}{+}(\textcolor{red}{+}4)=(-1)\quad\to\quad -5 \textcolor{red}{+}4= -1.}
\latex{(+8)\textcolor{009fe3}{-}(\textcolor{red}{+}9) = (+8)\textcolor{009fe3}{+}(\textcolor{red}{-}9)=(-1)\quad\to\quad\;\;\; 8 \textcolor{red}{-}9= -1;}
\latex{(-5)\textcolor{009fe3}{-}(\textcolor{red}{+}4) = (-5)\textcolor{009fe3}{+}(\textcolor{red}{-}4)=(-9)\quad\to\quad -5 \textcolor{red}{-}4= -9;}
\latex{(-5)\textcolor{009fe3}{-}(\textcolor{red}{-}4) = (-5)\textcolor{009fe3}{+}(\textcolor{red}{+}4)=(-1)\quad\to\quad -5 \textcolor{red}{+}4= -1.}
Adding and subtracting numbers with signs is called combining. When combining,
- a subtraction is expressed as an addition;
- plus signs and brackets are eliminated;
- operations are performed.
\latex{7+(+3)}
\latex{7-(-3)}
\latex{7+(-3)}
\latex{7-(+3)}
\latex{7+3}
\latex{7-3}
Combining various terms
Example 1
Use combining to perform the following operations.
- \latex{(-6)-(+15)-(-13) + (-9)}
- \latex{(+9) + (+12) + (-11)-(+7)-(-11)}
Solution
In case of several terms, combining can be done from left to right.
\latex{(-6)-(+15)-(-13) + (-9) =\\= (-6) + (-15) + (+13) + (-9) =\\=-6 -15 + 13- 9 = -21 + 13 -9 = -8- 9 =-17.}
a)
The terms can be grouped according to their signs.
\latex{(-6) -(+15) - (-13) + (-9) =\\= (-6) + (-15) + (+13) + (-9) =\\= -6 - 15 + 13 - 9 = 13 - 6 - 15 - 9 = 13 - 30 = -17.}
Calculating can be made easier by grouping the terms.
\latex{(+9) + (+12) + (-11) -(+7) - (-11) =\\= (+9) + (+12) + (-11) + (-7) + (+11) =\\= 9 + 12- 11 - 7 + 11 = 9 + 12 - 11 + 11- 7 =}
\latex{= 9 + 12- 7 = 21- 7 = 14.}
b)
\latex{\textcolor{199fe3}{0}}
\latex{\textcolor{199fe3}{-30}}
Exercises
Before performing the operations, estimate whether the results will be negative or positive.
{{exercise_number}}. Remove the brackets and perform the following operations.
- \latex{(-3) + (+3)}
- \latex{(-3) - (+3)}
- \latex{(-3) + (-3)}
- \latex{(-3) - (-3)}
- \latex{(-58) + (-24)}
- \latex{(+24) - (+77)}
{{exercise_number}}. Perform combining and select the odd one out.
- A: \latex{\fcolorbox{f6b900}{fef6e3}{$(+7)-(+3)$};\qquad} B: \latex{\fcolorbox{f6b900}{fef6e3}{$7-3$};\qquad} C: \latex{\fcolorbox{f6b900}{fef6e3}{$(+7)-(-3)$};\qquad} D: \latex{\fcolorbox{f6b900}{fef6e3}{$(+7) + (-3)$};\qquad} E: \latex{\fcolorbox{f6b900}{fef6e3}{$4$}};
- A: \latex{\fcolorbox{008a43}{eaf4f2}{$(-11)-(+8)$};\qquad} B: \latex{\fcolorbox{008a43}{eaf4f2}{$-3$};\qquad} C: \latex{\fcolorbox{008a43}{eaf4f2}{$(-11) + (-8)$};\qquad} D: \latex{\fcolorbox{008a43}{eaf4f2}{$-11-8$};\qquad} E: \latex{\fcolorbox{008a43}{eaf4f2}{$-19$}};
- A: \latex{\fcolorbox{ba0217}{fbeae5}{$-15 + 13$};\qquad} B: \latex{\fcolorbox{ba0217}{fbeae5}{$(-15)-(-13)$};\qquad} C: \latex{\fcolorbox{ba0217}{fbeae5}{$(-15) + (+13)$};\qquad} D: \latex{\fcolorbox{ba0217}{fbeae5}{$-2$};\qquad} E: \latex{\fcolorbox{ba0217}{fbeae5}{$-15-13$}}.
{{exercise_number}}. Write down the following operations in a shorter form in your notebook and then perform them.
- \latex{(+48) + (-12)-(+11)- (+33)}
- \latex{(-48) + (-12)-(-11)- (+33)}
- \latex{(-48)-(+12) + (-11) + (+33)}
- \latex{(+48)-(-12)-(+11) + (-33)}
- \latex{(-48)-(+12) + (+11) + (-33)}
- \latex{(-48)-(-12) + (-11)- (+33)}
{{exercise_number}}. Choose two of the numbers \latex{ +36, –144, +180 } and –\latex{ 72 }. Add the two numbers, and then add the difference of the other two to the sum. Write down several results. How many different results can you get? Come up with similar exercises.
{{exercise_number}}. Write the following operations in a shorter form, and then perform them.
- \latex{(+10) + (-34)-(+49)- (-25)}
- \latex{(-42) + (+27)-(-15)- (+38)}
- \latex{(-9-6)-(42-50)}
- \latex{(+28-13-9)-(-7)-(+2)}
{{exercise_number}}. Replace the symbols with signs or operators to obtain the greatest and the smallest possible results. Find all the solutions.
- \latex{(-17)\triangle(-43)- (\square25) + (\square17)}
- \latex{(\square39)\triangle (+107) + (\square58)\triangle (-72)}
- \latex{(+57)-(\square13)\triangle (+98)\triangle (\square7)}
- \latex{(\square50)\triangle (\square39)\triangle(\square27)\triangle(\square14)}
{{exercise_number}}. Determine the results of the following operations in the simplest way possible.
- \latex{12-(+9) + (-11)- (+2)}
- \latex{-115 + (+36)-(-52)- (-64)-(-15)}
- \latex{(+87)-(-500)-(+87) + (-75)-(+425)}
{{exercise_number}}. Write the numbers \latex{ +18, +13, –15 }, and\latex{ –17 } in the squares, and combine them.
-
-
+
- How many different series of operations can you make?
- Calculate the results. How many different results can you get?
- Write down the series of operations with the greatest and smallest results.
Quiz
What will the sum be when combining \latex{ 2,000 } terms? \latex{\quad (–1) – (+2) + (–3) – (+4) + ...}