Wize High School Grade 9 Math Textbook > Rational Numbers

Adding & Subtracting Integers & Decimals

You can think of adding and subtracting integers and decimals as temperatures.

Wize Tip
+++ → adding
−-− → removing

Positive numbers → "heat".
Negative numbers  = "cold".

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Example 1
14.1−(+5.3)=14.1-(+5.3)=14.1−(+5.3)=  8.8 

It's 14.1 degrees, and you're "removing 5.3 degrees of heat" → the final temperature will be 8.8 degrees.

Example 2
14.1+(−5.3)=14.1+\left(-5.3\right)=14.1+(−5.3)=  8.8 

It's 14.1 degrees, and you're "adding 5.3 degrees of cold" → the final temperature will be 8.8 degrees.

Example 3
14.1−(−5.3)=14.1-\left(-5.3\right)=14.1−(−5.3)=  19.4 

It's 14.1 degrees, and you're "removing 5.3 degrees of cold" → the final temperature will be 19.4 degrees.

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Remember these Shortcuts!
Wize Tip
Same signs side-by-side make a positive+ + →+− − → +Ex.  3+(+2)=3+2=5Ex.  3−(−2)=3+2=5Opposite signs side-by-side make a negative+ − →−− + → −Ex.  3+(−2)=3−2=1Ex.  3−(+2)=3−2=1\begin{array}{|c|c|}
\hline \\
\begin{array}{c}\text{Same signs side-by-side make a positive}\\+\ +\ \to +\\-\ -\ \to\ +\end{array}
&
\begin{array}{l}\text{Ex. \ }3\bcf+(\bcf+2)=3\bcf+2=5\\
\text{Ex. \ }3\bcf-(\bcf-2)=3\bcf+2=5 \end{array}
\\ \\

\hline \\
\begin{array}{c}\text{Opposite signs side-by-side make a negative}\\+\ -\ \to -\\-\ +\ \to\ -\end{array}
&
\begin{array}{l}\text{Ex. \ }3\bcf+(\bcf-2)=3\bcf-2=1\\
\text{Ex. \ }3\bcf-(\bcf+2)=3\bcf-2=1 \end{array}
\\ \\
\hline
\end{array}Same signs side-by-side make a positive+ + →+− − → +​Opposite signs side-by-side make a negative+ − →−− + → −​​Ex.  3+(+2)=3+2=5Ex.  3−(−2)=3+2=5​Ex.  3+(−2)=3−2=1Ex.  3−(+2)=3−2=1​​​

Examples 4 (revisited)
14.1−(+5.3)=14.1−5.3=8.814.1\bcf-(\bcf+5.3)=14.1\bcf-5.3=8.814.1−(+5.3)=14.1−5.3=8.8
14.1+(−5.3)=14.1−5.3=8.814.1\bcf+\left(\bcf-5.3\right)=14.1\bcf-5.3=8.814.1+(−5.3)=14.1−5.3=8.8 
14.1−(−5.3)=14.1+5.3=19.414.1\bcf-\left(\bcf-5.3\right)=14.1\bcf+5.3=19.414.1−(−5.3)=14.1+5.3=19.4

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Example:  Adding & Subtracting Integers & Decimals

A bird flies off its nest that is 11.3 meters above the ground to catch a fish in the pond that is 4.7 meters deep. The bird dives a total of 13.4 meters.

a) What is the distance this bird dove below the surface of the pond water?

b) What is the distance from the bottom of the pond to the bottom of the bird's dive?

Part a)
The bird's total diving distance subtract the distance it dove above the water will give us the distance the bird dove below the surface:

13.4−11.3=2.113.4-11.3=2.113.4−11.3=2.1 

Therefore, the bird dove 2.1 meters below the surface of the pond.

Part b)
The distance from the bottom of the pond to the bottom of the bird's dive is 4.7−2.1=2.64.7-2.1=2.64.7−2.1=2.6 meters.

Alternatively, we can complete this problem by treating the ground level (or surface of the pond) as our 0 reference point:
If we let the surface of the water be at height 0, the distance above the water be positive, and the distance below the water be negative, then the distance is −4.7−(−2.1)=−4.7+2.1=−2.6-4.7-(-2.1)=-4.7+2.1=-2.6−4.7−(−2.1)=−4.7+2.1=−2.6 meters.

Practice: Adding & Subtracting Decimals
Evaluate −3.5−(−0.7)+(−1.2)+1.4-3.5-\left(-0.7\right)+\left(-1.2\right)+1.4−3.5−(−0.7)+(−1.2)+1.4.

Answer

I don't know

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Multiplying & Dividing Integers & Decimals
When multiplying and dividing integers and decimals, remember the following rules.

Multiplication       Division(+)(+)→(+)(+)(+)→(+)(−)(−)→(+)(−)(−)→(+)(+)(−)→(−)(+)(−)→(−)(−)(+)→(−)(−)(+)→(−)\begin{array}{|lc|lc|}
\hline
\text{Multiplication}&~~~~~~~&\text{Division}\\

\hline\\
(\bcf+)(\bcf+)\to(\bcf+)&&
\displaystyle\frac{(\bcf+)}{(\bcf+)}\to(\bcf+)\\\\

(\bcf-)(\bcf-)\to(\bcf+)&&
\displaystyle\frac{(\bcf-)}{(\bcf-)}\to(\bcf+)\\\\

(\bcf+)(\bcf-)\to(\bcf-)&&
\displaystyle\frac{(\bcf+)}{(\bcf-)}\to(\bcf-)\\\\

(\bcf-)(\bcf+)\to(\bcf-)&&\displaystyle\frac{(\bcf-)}{(\bcf+)}\to(\bcf-)\\\\
\hline



\end{array}Multiplication(+)(+)→(+)(−)(−)→(+)(+)(−)→(−)(−)(+)→(−)​       ​Division(+)(+)​→(+)(−)(−)​→(+)(−)(+)​→(−)(+)(−)​→(−)​​

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Example 1
a) −4.7×(−2)=-4.7\times(-2)=−4.7×(−2)=    9.4  

Remember: (−)×(−)→(+)(\bcf-)\times(\bcf-)\to(\bcf+)(−)×(−)→(+)

So the answer is +(4.7×2)=+9.4\bcf+(4.7\times2)=\bcf+9.4+(4.7×2)=+9.4

b) 5.5(−3)=5.5(-3)=5.5(−3)=   - 16.5  

Remember: (+)×(−)→(−)(\bcf+)\times(\bcf-)\to(\bcf-)(+)×(−)→(−)

So the answer is −(5.5×3)=−16.5\bcf-(5.5\times3)=\bcf-16.5−(5.5×3)=−16.5

c) −10÷2.5=-10\div2.5=−10÷2.5=   - 4  

Remember: (−)÷(+)→(−)(\bcf-)\div(\bcf+)\to(\bcf-)(−)÷(+)→(−)

So the answer is −(10÷2.5)=−4\bcf-(10\div2.5)=\bcf-4−(10÷2.5)=−4

d) 8(−2.5)−5=\dfrac{8(-2.5)}{-5}=−58(−2.5)​=   4  

This can be rewritten as 8(−2.5)÷(−5)8(-2.5)\div(-5)8(−2.5)÷(−5)
Remember: (+)×(−)÷(−)  →  (−)÷(−)  →  (+)(\bcf+)\times(\bcf-)\div(\bcf-)~~\to~~(\bcf-)\div(\bcf-)~~\to~~(\bcf+)(+)×(−)÷(−)  →  (−)÷(−)  →  (+)

So the answer is +[8×2.5÷5]=+4\bcf+[8\times2.5\div5]=\bcf+4+[8×2.5÷5]=+4

e) (−3)2×(−2)3=(-3)^2\times(-2)^3=(−3)2×(−2)3=   - 72  

Rewriting this as multiplication:
(−3)(−3)×(−2)(−2)(−2)(-3)(-3)\times(-2)(-2)(-2)(−3)(−3)×(−2)(−2)(−2)

Since every two negatives make a positive, we get
+(3×3)×−(2×2×2)\bcf+(3\times3)\times\bcf-(2\times2\times2)+(3×3)×−(2×2×2)
=−(3×3×2×2×2)=\bcf-(3\times3\times2\times2\times2)=−(3×3×2×2×2)

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Example 2
Evaluate 2×(−3.5)×(−3)−2(−1.1)2\times\left(-3.5\right)\times\left(-3\right)-2\left(-1.1\right)2×(−3.5)×(−3)−2(−1.1) 

Simplify this using BEDMAS

(+2)×(−3.5)×(−3)⏟MultiplicationThe first two numbers make a −Then this −and the third number make a+−(+2)(−1.1)⏟Multiplication
\underbrace{(\bcf+2)\times(\bcf-3.5)
\times(\bcf-3)}_
{\begin{array}{c}
\scriptsize{\text{Multiplication}}\\
\scriptsize\text{The first two numbers make a }-\\
\scriptsize\text{Then this }- \text{and the third number make a}+ 
\end{array}}
-\underbrace{(\bcf+2)(\bcf-1.1)}_\text{Multiplication}MultiplicationThe first two numbers make a −Then this −and the third number make a+​(+2)×(−3.5)×(−3)​​−Multiplication(+2)(−1.1)​​

=(+21)−(−2.2)⏟Subtraction=\underbrace{(\bcf+21)-(\bcf-2.2)}_\text{Subtraction}=Subtraction(+21)−(−2.2)​​

=+21+2.2=\bcf{+}21\bcf+2.2=+21+2.2

=+23.2=\bcf+23.2=+23.2

=23.2=\boxed{23.2}=23.2​

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Example 3
Evaluate [−1.5(−4)−10.2]÷(−2)−2.1(−3)(−2.5)−5\displaystyle [-1.5\left(-4\right)-10.2]\div{(-2)}-\frac{2.1\left(-3\right)\left(-2.5\right)}{-5}[−1.5(−4)−10.2]÷(−2)−−52.1(−3)(−2.5)​

[−1.5(−4)−10.2]⏟BracketDeal with multiplication first÷(−2)−2.1(−3)(−2.5)−5\underbrace{[\bcf-1.5\left(\bcf-4\right)-10.2]}_{\begin{array}{c}\scriptsize\text{Bracket}\\\scriptsize\text{Deal with multiplication first}\end{array}}\div{(-2)}-\dfrac{2.1\left(-3\right)\left(-2.5\right)}{-5}BracketDeal with multiplication first​[−1.5(−4)−10.2]​​÷(−2)−−52.1(−3)(−2.5)​

=[+6−10.2]⏟Bracket÷(−2)−2.1(−3)(−2.5)−5=\underbrace{[\bcf+6-10.2]}_\text{Bracket}\div{(-2)}-\dfrac{2.1\left(-3\right)\left(-2.5\right)}{-5}=Bracket[+6−10.2]​​÷(−2)−−52.1(−3)(−2.5)​

=−4.2÷(−2)⏟Division−2.1(−3)(−2.5)−5⏟Multiplication and DivisionThe 3 negatives make a negative=\underbrace{\bcf-4.2\div{(\bcf-2)}}_\text{Division}-\underbrace{\dfrac{2.1\left(\bcf-3\right)\left(\bcf-2.5\right)}{\bcf-5}}_{\begin{array}{c}\scriptsize\text{Multiplication and Division}\\\scriptsize\text{The 3 negatives make a negative}\end{array}}=Division−4.2÷(−2)​​−Multiplication and DivisionThe 3 negatives make a negative​−52.1(−3)(−2.5)​​​

=+2.1−(−3.15)=\bcf+2.1-(\bcf-3.15)=+2.1−(−3.15)

=+2.1+3.15=\bcf+2.1\bcf+3.15=+2.1+3.15

=5.25=\boxed{5.25}=5.25​

Practice: Multiplying & Dividing Integers & Decimals
Evaluate the following without using a calculator.
If the answer is negative, make sure to include - in your answer (for example, if the answer is negative ten, you should type -10). 
If the answer is positive, don't include + in your answer (for example, if the answer is positive 10, you should just type 10).

a) (−25)(−6)=(-25)(-6)=(−25)(−6)=

b) 4×(−13)=4\times(-13)=4×(−13)=

c) −497=\dfrac{-49}{7}=7−49​=

d) −125÷(−5)=-125\div(-5)=−125÷(−5)=

I don't know

Practice: Muliplying & Dividing Integers & Decimals
What is the simplified answer to (−2)(5.5−5)−(−0.5)(−2)4(-2)\left(\dfrac{5.5}{-5}\right)-(-0.5)(-2)^4(−2)(−55.5​)−(−0.5)(−2)4?

Answer

I don't know

Practice: Working with Integers & Decimals
Evaluate (−x)3(y3+(x)(−y))−(−x×2.2y)+(−y)4(-x)^3\left(y^3+(x)(-y)\right)-\left(\dfrac{-x\times2.2}{y}\right)+(-y)^4(−x)3(y3+(x)(−y))−(y−x×2.2​)+(−y)4 when x=3x=3x=3 and y=2y=2y=2.

Answer

I don't know