6-2 Addition, Subtraction and Estimation with Rational...
Transcript of 6-2 Addition, Subtraction and Estimation with Rational...
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§ 6-2 Addition, Subtraction and Estimationwith Rational Numbers
![Page 2: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/2.jpg)
Defining Addition
Thinking back to how we defined the addition of integers, how can wedefine the addition of rational numbers?
Definition
If ab ,
cb ∈ Q, then a
b + cb = a+c
b .
What if we don’t have common denominators?
Definition
If ab ,
cd ∈ Q, then a
b + cd = ad+bc
bd .
![Page 3: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/3.jpg)
Defining Addition
Thinking back to how we defined the addition of integers, how can wedefine the addition of rational numbers?
Definition
If ab ,
cb ∈ Q, then a
b + cb = a+c
b .
What if we don’t have common denominators?
Definition
If ab ,
cd ∈ Q, then a
b + cd = ad+bc
bd .
![Page 4: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/4.jpg)
Defining Addition
Thinking back to how we defined the addition of integers, how can wedefine the addition of rational numbers?
Definition
If ab ,
cb ∈ Q, then a
b + cb = a+c
b .
What if we don’t have common denominators?
Definition
If ab ,
cd ∈ Q, then a
b + cd = ad+bc
bd .
![Page 5: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/5.jpg)
Defining Addition
Thinking back to how we defined the addition of integers, how can wedefine the addition of rational numbers?
Definition
If ab ,
cb ∈ Q, then a
b + cb = a+c
b .
What if we don’t have common denominators?
Definition
If ab ,
cd ∈ Q, then a
b + cd = ad+bc
bd .
![Page 6: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/6.jpg)
Properties of Addition
What is the statement that would be true if the rational numbers wereclosed under addition?
Closure of the Rational Numbers Under AdditionIf a
b ,cd ∈ Q, then a
b + cd ∈ Q.
Is this true statement? Yes ... why?
If we look at the formula ab + c
d = ad+bcbd , what are ad, bc and bd?
![Page 7: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/7.jpg)
Properties of Addition
What is the statement that would be true if the rational numbers wereclosed under addition?
Closure of the Rational Numbers Under AdditionIf a
b ,cd ∈ Q, then a
b + cd ∈ Q.
Is this true statement? Yes ... why?
If we look at the formula ab + c
d = ad+bcbd , what are ad, bc and bd?
![Page 8: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/8.jpg)
Properties of Addition
What is the statement that would be true if the rational numbers wereclosed under addition?
Closure of the Rational Numbers Under AdditionIf a
b ,cd ∈ Q, then a
b + cd ∈ Q.
Is this true statement?
Yes ... why?
If we look at the formula ab + c
d = ad+bcbd , what are ad, bc and bd?
![Page 9: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/9.jpg)
Properties of Addition
What is the statement that would be true if the rational numbers wereclosed under addition?
Closure of the Rational Numbers Under AdditionIf a
b ,cd ∈ Q, then a
b + cd ∈ Q.
Is this true statement? Yes ... why?
If we look at the formula ab + c
d = ad+bcbd , what are ad, bc and bd?
![Page 10: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/10.jpg)
Properties of Addition
What is the statement that would be true if the rational numbers wereclosed under addition?
Closure of the Rational Numbers Under AdditionIf a
b ,cd ∈ Q, then a
b + cd ∈ Q.
Is this true statement? Yes ... why?
If we look at the formula ab + c
d = ad+bcbd , what are ad, bc and bd?
![Page 11: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/11.jpg)
Properties of Addition
If the associative property held for the addition of rational numbers,what would the statement be?
Associativity of the Rational Numbers Under Addition
If ab ,
cd ,
ef ∈ Q, then a
b +(
cd + e
f
)=
( ab + c
d
)+ e
f .
Is this a true statement? Yes ... why?
What happens when we apply the sum formula on both expressions?
![Page 12: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/12.jpg)
Properties of Addition
If the associative property held for the addition of rational numbers,what would the statement be?
Associativity of the Rational Numbers Under Addition
If ab ,
cd ,
ef ∈ Q, then a
b +(
cd + e
f
)=
( ab + c
d
)+ e
f .
Is this a true statement? Yes ... why?
What happens when we apply the sum formula on both expressions?
![Page 13: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/13.jpg)
Properties of Addition
If the associative property held for the addition of rational numbers,what would the statement be?
Associativity of the Rational Numbers Under Addition
If ab ,
cd ,
ef ∈ Q, then a
b +(
cd + e
f
)=
( ab + c
d
)+ e
f .
Is this a true statement?
Yes ... why?
What happens when we apply the sum formula on both expressions?
![Page 14: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/14.jpg)
Properties of Addition
If the associative property held for the addition of rational numbers,what would the statement be?
Associativity of the Rational Numbers Under Addition
If ab ,
cd ,
ef ∈ Q, then a
b +(
cd + e
f
)=
( ab + c
d
)+ e
f .
Is this a true statement? Yes ... why?
What happens when we apply the sum formula on both expressions?
![Page 15: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/15.jpg)
Properties of Addition
If the associative property held for the addition of rational numbers,what would the statement be?
Associativity of the Rational Numbers Under Addition
If ab ,
cd ,
ef ∈ Q, then a
b +(
cd + e
f
)=
( ab + c
d
)+ e
f .
Is this a true statement? Yes ... why?
What happens when we apply the sum formula on both expressions?
![Page 16: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/16.jpg)
Properties of Addition
If the commutative property held for the addition of rational numbers,what would the statement be?
Commutativity of the Rational Numbers Under Addition
If ab ,
cd ∈ Q, then a
b + cd = c
d + ab .
Is this a true statement? Yes ... why?
What happens when we apply the sum formula on both expressions?
![Page 17: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/17.jpg)
Properties of Addition
If the commutative property held for the addition of rational numbers,what would the statement be?
Commutativity of the Rational Numbers Under Addition
If ab ,
cd ∈ Q, then a
b + cd = c
d + ab .
Is this a true statement? Yes ... why?
What happens when we apply the sum formula on both expressions?
![Page 18: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/18.jpg)
Properties of Addition
If the commutative property held for the addition of rational numbers,what would the statement be?
Commutativity of the Rational Numbers Under Addition
If ab ,
cd ∈ Q, then a
b + cd = c
d + ab .
Is this a true statement?
Yes ... why?
What happens when we apply the sum formula on both expressions?
![Page 19: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/19.jpg)
Properties of Addition
If the commutative property held for the addition of rational numbers,what would the statement be?
Commutativity of the Rational Numbers Under Addition
If ab ,
cd ∈ Q, then a
b + cd = c
d + ab .
Is this a true statement? Yes ... why?
What happens when we apply the sum formula on both expressions?
![Page 20: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/20.jpg)
Properties of Addition
If the commutative property held for the addition of rational numbers,what would the statement be?
Commutativity of the Rational Numbers Under Addition
If ab ,
cd ∈ Q, then a
b + cd = c
d + ab .
Is this a true statement? Yes ... why?
What happens when we apply the sum formula on both expressions?
![Page 21: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/21.jpg)
Properties of Addition
Is there some ‘magical number’ such that when we add it to a rationalnumber a
b , we get back ab ?
Additive Inverse for Rational Numbers
For any ab ∈ Q, there exists the unique rational number 0
b such that
ab+
0b=
ab=
0b+
ab
What property did we apply in the last statement?
Note: There is only one additive identity element for rationalnumbers, but we can change the denominator to get an equivalentform to suit our needs.
![Page 22: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/22.jpg)
Properties of Addition
Is there some ‘magical number’ such that when we add it to a rationalnumber a
b , we get back ab ?
Additive Inverse for Rational Numbers
For any ab ∈ Q, there exists the unique rational number 0
b such that
ab+
0b=
ab=
0b+
ab
What property did we apply in the last statement?
Note: There is only one additive identity element for rationalnumbers, but we can change the denominator to get an equivalentform to suit our needs.
![Page 23: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/23.jpg)
Properties of Addition
Is there some ‘magical number’ such that when we add it to a rationalnumber a
b , we get back ab ?
Additive Inverse for Rational Numbers
For any ab ∈ Q, there exists the unique rational number 0
b such that
ab+
0b=
ab=
0b+
ab
What property did we apply in the last statement?
Note: There is only one additive identity element for rationalnumbers, but we can change the denominator to get an equivalentform to suit our needs.
![Page 24: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/24.jpg)
Properties of Addition
Is there some ‘magical number’ such that when we add it to a rationalnumber a
b , we get back ab ?
Additive Inverse for Rational Numbers
For any ab ∈ Q, there exists the unique rational number 0
b such that
ab+
0b=
ab=
0b+
ab
What property did we apply in the last statement?
Note: There is only one additive identity element for rationalnumbers, but we can change the denominator to get an equivalentform to suit our needs.
![Page 25: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/25.jpg)
Properties of Addition
Is there some magical rational number such that when we add it to ab ,
we get the identity?
Inverse Element for Rational NumbersFor each a
b ∈ Q, −ab is the unique rational number such that
ab+
(−a
b
)=
0b= −a
b+
ab
Remember, we cannot have an inverse element without having anidentity element ...
![Page 26: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/26.jpg)
Properties of Addition
Is there some magical rational number such that when we add it to ab ,
we get the identity?
Inverse Element for Rational NumbersFor each a
b ∈ Q, −ab is the unique rational number such that
ab+(−a
b
)=
0b= −a
b+
ab
Remember, we cannot have an inverse element without having anidentity element ...
![Page 27: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/27.jpg)
Properties of Addition
Is there some magical rational number such that when we add it to ab ,
we get the identity?
Inverse Element for Rational NumbersFor each a
b ∈ Q, −ab is the unique rational number such that
ab+(−a
b
)=
0b= −a
b+
ab
Remember, we cannot have an inverse element without having anidentity element ...
![Page 28: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/28.jpg)
Properties of Addition
Equal Additions
If ab ,
cd ,
ef ∈ Q and if a
b = cd , then a
b + ef = c
d + ef .
We use this property more than we realize when solving equationswith variables.
![Page 29: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/29.jpg)
Properties of Addition
Equal Additions
If ab ,
cd ,
ef ∈ Q and if a
b = cd , then a
b + ef = c
d + ef .
We use this property more than we realize when solving equationswith variables.
![Page 30: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/30.jpg)
Visual Representations
Example
Illustrate 14 + 2
3 .
![Page 31: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/31.jpg)
Visual Representations
Example
Illustrate 14 + 2
3 .
![Page 32: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/32.jpg)
Visual Representations
Example
Illustrate 14 + 2
3 .
![Page 33: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/33.jpg)
Visual Representations
Example
Illustrate 14 + 2
3 .
![Page 34: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/34.jpg)
Visual Representations
Example
Illustrate 14 + 2
3 .
![Page 35: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/35.jpg)
Visual Representations
Example
Illustrate 14 + 2
3 .
+ =
![Page 36: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/36.jpg)
Visual Representations
Example
Illustrate 14 + 2
3 .
+
=
![Page 37: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/37.jpg)
Visual Representations
Example
Illustrate 14 + 2
3 .
+
=
![Page 38: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/38.jpg)
Visual Representations
Example
Illustrate 14 + 2
3 .
+ =
![Page 39: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/39.jpg)
Subtraction of Rational Numbers
How can we define subtraction of rational numbers in terms of whatwe know - addition of rational numbers?
DefinitionIf a
b ,cd ∈ Q, then a
b −cd is the unique e
f ∈ Q such that ab = c
d + ef .
We can also think of subtraction as ...
Subtraction of Rational NumbersFor a
b ,cd ∈ Q,
ab −
cd = ad−bc
bdab −
cd = a
b + −cd
![Page 40: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/40.jpg)
Subtraction of Rational Numbers
How can we define subtraction of rational numbers in terms of whatwe know - addition of rational numbers?
DefinitionIf a
b ,cd ∈ Q, then a
b −cd is the unique e
f ∈ Q such that ab = c
d + ef .
We can also think of subtraction as ...
Subtraction of Rational NumbersFor a
b ,cd ∈ Q,
ab −
cd = ad−bc
bdab −
cd = a
b + −cd
![Page 41: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/41.jpg)
Subtraction of Rational Numbers
How can we define subtraction of rational numbers in terms of whatwe know - addition of rational numbers?
DefinitionIf a
b ,cd ∈ Q, then a
b −cd is the unique e
f ∈ Q such that ab = c
d + ef .
We can also think of subtraction as ...
Subtraction of Rational NumbersFor a
b ,cd ∈ Q,
ab −
cd = ad−bc
bd
ab −
cd = a
b + −cd
![Page 42: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/42.jpg)
Subtraction of Rational Numbers
How can we define subtraction of rational numbers in terms of whatwe know - addition of rational numbers?
DefinitionIf a
b ,cd ∈ Q, then a
b −cd is the unique e
f ∈ Q such that ab = c
d + ef .
We can also think of subtraction as ...
Subtraction of Rational NumbersFor a
b ,cd ∈ Q,
ab −
cd = ad−bc
bdab −
cd = a
b + −cd
![Page 43: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/43.jpg)
Properties of Subtraction
If the rational numbers were closed under subtraction, what would ourstatement be?
Closure of the Rational Numbers Under SubtractionIf a
b ,cd ∈ Q, then a
b −cd ∈ Q.
Is this a true statement?
![Page 44: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/44.jpg)
Properties of Subtraction
If the rational numbers were closed under subtraction, what would ourstatement be?
Closure of the Rational Numbers Under SubtractionIf a
b ,cd ∈ Q, then a
b −cd ∈ Q.
Is this a true statement?
![Page 45: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/45.jpg)
Properties of Subtraction
If the rational numbers were closed under subtraction, what would ourstatement be?
Closure of the Rational Numbers Under SubtractionIf a
b ,cd ∈ Q, then a
b −cd ∈ Q.
Is this a true statement?
![Page 46: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/46.jpg)
Properties of Subtraction
If the commutative property held for subtraction of rational numbers,what would the statement say?
Commutativity for the Subtraction of Rational Numbers
If ab ,
cd ∈ Q, then a
b −cd = c
d −ab .
Is this a true statement?
Can you give a counterexample?
![Page 47: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/47.jpg)
Properties of Subtraction
If the commutative property held for subtraction of rational numbers,what would the statement say?
Commutativity for the Subtraction of Rational Numbers
If ab ,
cd ∈ Q, then a
b −cd = c
d −ab .
Is this a true statement?
Can you give a counterexample?
![Page 48: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/48.jpg)
Properties of Subtraction
If the commutative property held for subtraction of rational numbers,what would the statement say?
Commutativity for the Subtraction of Rational Numbers
If ab ,
cd ∈ Q, then a
b −cd = c
d −ab .
Is this a true statement?
Can you give a counterexample?
![Page 49: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/49.jpg)
Properties of Subtraction
If the commutative property held for subtraction of rational numbers,what would the statement say?
Commutativity for the Subtraction of Rational Numbers
If ab ,
cd ∈ Q, then a
b −cd = c
d −ab .
Is this a true statement?
Can you give a counterexample?
![Page 50: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/50.jpg)
Properties of Subtraction
If the associative property held for subtraction of rational numbers,what would the statement say?
Associativity for the Subtraction of Rational Numbers
If ab ,
cd ,
ef ∈ Q, then a
b −(
cd −
ef
)=
( ab −
cd
)− e
f
Is this a true statement?
Can you give a counterexample?
![Page 51: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/51.jpg)
Properties of Subtraction
If the associative property held for subtraction of rational numbers,what would the statement say?
Associativity for the Subtraction of Rational Numbers
If ab ,
cd ,
ef ∈ Q, then a
b −(
cd −
ef
)=
( ab −
cd
)− e
f
Is this a true statement?
Can you give a counterexample?
![Page 52: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/52.jpg)
Properties of Subtraction
If the associative property held for subtraction of rational numbers,what would the statement say?
Associativity for the Subtraction of Rational Numbers
If ab ,
cd ,
ef ∈ Q, then a
b −(
cd −
ef
)=
( ab −
cd
)− e
f
Is this a true statement?
Can you give a counterexample?
![Page 53: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/53.jpg)
Properties of Subtraction
If the associative property held for subtraction of rational numbers,what would the statement say?
Associativity for the Subtraction of Rational Numbers
If ab ,
cd ,
ef ∈ Q, then a
b −(
cd −
ef
)=
( ab −
cd
)− e
f
Is this a true statement?
Can you give a counterexample?
![Page 54: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/54.jpg)
Adding and Subtracting Mixed Numbers
There are two ways we can go about this. Ideas of what they are?
1 Combining whole number parts and rational parts separately,then combining as necessary
2 Converting to improper fractions, combining and then rewritingthe result as a mixed number
![Page 55: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/55.jpg)
Adding and Subtracting Mixed Numbers
There are two ways we can go about this. Ideas of what they are?
1 Combining whole number parts and rational parts separately,then combining as necessary
2 Converting to improper fractions, combining and then rewritingthe result as a mixed number
![Page 56: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/56.jpg)
Adding and Subtracting Mixed Numbers
There are two ways we can go about this. Ideas of what they are?
1 Combining whole number parts and rational parts separately,then combining as necessary
2 Converting to improper fractions, combining and then rewritingthe result as a mixed number
![Page 57: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/57.jpg)
Adding and Subtracting Mixed Numbers
Example
Find 2 15 + 3 3
4
We first want to separate into whole number parts and rational parts.What property allows us to do this?
215+ 3
34= (2 + 3) +
(15+
34
)= 5 +
(15+
34
)= 5 +
(420
+1520
)= 5
1920
![Page 58: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/58.jpg)
Adding and Subtracting Mixed Numbers
Example
Find 2 15 + 3 3
4
We first want to separate into whole number parts and rational parts.What property allows us to do this?
215+ 3
34= (2 + 3) +
(15+
34
)= 5 +
(15+
34
)= 5 +
(420
+1520
)= 5
1920
![Page 59: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/59.jpg)
Adding and Subtracting Mixed Numbers
Example
Find 2 15 + 3 3
4
We first want to separate into whole number parts and rational parts.What property allows us to do this?
215+ 3
34=
(2 + 3) +(
15+
34
)= 5 +
(15+
34
)= 5 +
(420
+1520
)= 5
1920
![Page 60: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/60.jpg)
Adding and Subtracting Mixed Numbers
Example
Find 2 15 + 3 3
4
We first want to separate into whole number parts and rational parts.What property allows us to do this?
215+ 3
34= (2 + 3) +
(15+
34
)
= 5 +
(15+
34
)= 5 +
(420
+1520
)= 5
1920
![Page 61: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/61.jpg)
Adding and Subtracting Mixed Numbers
Example
Find 2 15 + 3 3
4
We first want to separate into whole number parts and rational parts.What property allows us to do this?
215+ 3
34= (2 + 3) +
(15+
34
)= 5 +
(15+
34
)
= 5 +
(420
+1520
)= 5
1920
![Page 62: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/62.jpg)
Adding and Subtracting Mixed Numbers
Example
Find 2 15 + 3 3
4
We first want to separate into whole number parts and rational parts.What property allows us to do this?
215+ 3
34= (2 + 3) +
(15+
34
)= 5 +
(15+
34
)= 5 +
(420
+1520
)
= 51920
![Page 63: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/63.jpg)
Adding and Subtracting Mixed Numbers
Example
Find 2 15 + 3 3
4
We first want to separate into whole number parts and rational parts.What property allows us to do this?
215+ 3
34= (2 + 3) +
(15+
34
)= 5 +
(15+
34
)= 5 +
(420
+1520
)= 5
1920
![Page 64: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/64.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 3 23 − 2 5
6
323− 2
56= (3− 2) +
(23− 5
6
)= 1 +
(23− 5
6
)= 1 +
(46− 5
6
)= 1 +
−16
=66− 1
6
=56
![Page 65: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/65.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 3 23 − 2 5
6
323− 2
56=
(3− 2) +(
23− 5
6
)= 1 +
(23− 5
6
)= 1 +
(46− 5
6
)= 1 +
−16
=66− 1
6
=56
![Page 66: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/66.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 3 23 − 2 5
6
323− 2
56= (3− 2) +
(23− 5
6
)
= 1 +
(23− 5
6
)= 1 +
(46− 5
6
)= 1 +
−16
=66− 1
6
=56
![Page 67: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/67.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 3 23 − 2 5
6
323− 2
56= (3− 2) +
(23− 5
6
)= 1 +
(23− 5
6
)
= 1 +
(46− 5
6
)= 1 +
−16
=66− 1
6
=56
![Page 68: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/68.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 3 23 − 2 5
6
323− 2
56= (3− 2) +
(23− 5
6
)= 1 +
(23− 5
6
)= 1 +
(46− 5
6
)
= 1 +−16
=66− 1
6
=56
![Page 69: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/69.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 3 23 − 2 5
6
323− 2
56= (3− 2) +
(23− 5
6
)= 1 +
(23− 5
6
)= 1 +
(46− 5
6
)= 1 +
−16
=66− 1
6
=56
![Page 70: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/70.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 3 23 − 2 5
6
323− 2
56= (3− 2) +
(23− 5
6
)= 1 +
(23− 5
6
)= 1 +
(46− 5
6
)= 1 +
−16
=66− 1
6
=56
![Page 71: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/71.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 3 23 − 2 5
6
323− 2
56= (3− 2) +
(23− 5
6
)= 1 +
(23− 5
6
)= 1 +
(46− 5
6
)= 1 +
−16
=66− 1
6
=56
![Page 72: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/72.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 4 27 + 1 3
5
What would our common denominator be here?
We first convert to improper fractions and then combine in the usualway.
427+ 1
35=
307
+85
=15035
+5635
=20635
= 53135
![Page 73: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/73.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 4 27 + 1 3
5
What would our common denominator be here?
We first convert to improper fractions and then combine in the usualway.
427+ 1
35=
307
+85
=15035
+5635
=20635
= 53135
![Page 74: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/74.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 4 27 + 1 3
5
What would our common denominator be here?
We first convert to improper fractions and then combine in the usualway.
427+ 1
35=
307
+85
=15035
+5635
=20635
= 53135
![Page 75: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/75.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 4 27 + 1 3
5
What would our common denominator be here?
We first convert to improper fractions and then combine in the usualway.
427+ 1
35=
307
+85
=15035
+5635
=20635
= 53135
![Page 76: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/76.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 4 27 + 1 3
5
What would our common denominator be here?
We first convert to improper fractions and then combine in the usualway.
427+ 1
35=
307
+85
=15035
+5635
=20635
= 53135
![Page 77: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/77.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 4 27 + 1 3
5
What would our common denominator be here?
We first convert to improper fractions and then combine in the usualway.
427+ 1
35=
307
+85
=15035
+5635
=20635
= 53135
![Page 78: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/78.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 4 27 + 1 3
5
What would our common denominator be here?
We first convert to improper fractions and then combine in the usualway.
427+ 1
35=
307
+85
=15035
+5635
=20635
= 53135
![Page 79: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/79.jpg)
Adding and Subtractive Mixed Numbers
Example
Find 4 27 + 1 3
5
What would our common denominator be here?
We first convert to improper fractions and then combine in the usualway.
427+ 1
35=
307
+85
=15035
+5635
=20635
= 53135
![Page 80: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/80.jpg)
Rational Numbers with Variables
Example
Find 2x −
31−x
There is no difference here as when we have numbers. What is ourcommon denominator?
2x− 3
1− x=
2x· 1− x
1− x− 3
1− x· x
x
=2− 2x
x(1− x)− 3x
x(1− x)
=2− 2x− 3x
x(1− x)
=2− 5xx− x2
![Page 81: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/81.jpg)
Rational Numbers with Variables
Example
Find 2x −
31−x
There is no difference here as when we have numbers. What is ourcommon denominator?
2x− 3
1− x=
2x· 1− x
1− x− 3
1− x· x
x
=2− 2x
x(1− x)− 3x
x(1− x)
=2− 2x− 3x
x(1− x)
=2− 5xx− x2
![Page 82: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/82.jpg)
Rational Numbers with Variables
Example
Find 2x −
31−x
There is no difference here as when we have numbers. What is ourcommon denominator?
2x− 3
1− x=
2x· 1− x
1− x− 3
1− x· x
x
=2− 2x
x(1− x)− 3x
x(1− x)
=2− 2x− 3x
x(1− x)
=2− 5xx− x2
![Page 83: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/83.jpg)
Rational Numbers with Variables
Example
Find 2x −
31−x
There is no difference here as when we have numbers. What is ourcommon denominator?
2x− 3
1− x=
2x· 1− x
1− x− 3
1− x· x
x
=2− 2x
x(1− x)− 3x
x(1− x)
=2− 2x− 3x
x(1− x)
=2− 5xx− x2
![Page 84: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/84.jpg)
Rational Numbers with Variables
Example
Find 2x −
31−x
There is no difference here as when we have numbers. What is ourcommon denominator?
2x− 3
1− x=
2x· 1− x
1− x− 3
1− x· x
x
=2− 2x
x(1− x)− 3x
x(1− x)
=2− 2x− 3x
x(1− x)
=2− 5xx− x2
![Page 85: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/85.jpg)
Rational Numbers with Variables
Example
Find 2x −
31−x
There is no difference here as when we have numbers. What is ourcommon denominator?
2x− 3
1− x=
2x· 1− x
1− x− 3
1− x· x
x
=2− 2x
x(1− x)− 3x
x(1− x)
=2− 2x− 3x
x(1− x)
=2− 5xx− x2
![Page 86: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/86.jpg)
Rational Numbers with Variables
Example
Find 2x −
31−x
There is no difference here as when we have numbers. What is ourcommon denominator?
2x− 3
1− x=
2x· 1− x
1− x− 3
1− x· x
x
=2− 2x
x(1− x)− 3x
x(1− x)
=2− 2x− 3x
x(1− x)
=2− 5xx− x2
![Page 87: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/87.jpg)
Estimation
What is the value of estimation with rational numbers?
Reasonability of solution ...
Example
Estimate 2 13 + 1
10 − 1 78 .
Roughly speaking, we have ...
(using halves) 2 12 + 0− 2 ≈ 1
2
(using wholes) 2 + 0− 2 = 0
(using quarters) 2 14 + 0− 2 = 1
4
![Page 88: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/88.jpg)
Estimation
What is the value of estimation with rational numbers?
Reasonability of solution ...
Example
Estimate 2 13 + 1
10 − 1 78 .
Roughly speaking, we have ...
(using halves) 2 12 + 0− 2 ≈ 1
2
(using wholes) 2 + 0− 2 = 0
(using quarters) 2 14 + 0− 2 = 1
4
![Page 89: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/89.jpg)
Estimation
What is the value of estimation with rational numbers?
Reasonability of solution ...
Example
Estimate 2 13 + 1
10 − 1 78 .
Roughly speaking, we have ...
(using halves) 2 12 + 0− 2 ≈ 1
2
(using wholes) 2 + 0− 2 = 0
(using quarters) 2 14 + 0− 2 = 1
4
![Page 90: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/90.jpg)
Estimation
What is the value of estimation with rational numbers?
Reasonability of solution ...
Example
Estimate 2 13 + 1
10 − 1 78 .
Roughly speaking, we have ...
(using halves)
2 12 + 0− 2 ≈ 1
2
(using wholes) 2 + 0− 2 = 0
(using quarters) 2 14 + 0− 2 = 1
4
![Page 91: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/91.jpg)
Estimation
What is the value of estimation with rational numbers?
Reasonability of solution ...
Example
Estimate 2 13 + 1
10 − 1 78 .
Roughly speaking, we have ...
(using halves) 2 12 + 0− 2 ≈ 1
2
(using wholes) 2 + 0− 2 = 0
(using quarters) 2 14 + 0− 2 = 1
4
![Page 92: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/92.jpg)
Estimation
What is the value of estimation with rational numbers?
Reasonability of solution ...
Example
Estimate 2 13 + 1
10 − 1 78 .
Roughly speaking, we have ...
(using halves) 2 12 + 0− 2 ≈ 1
2
(using wholes)
2 + 0− 2 = 0
(using quarters) 2 14 + 0− 2 = 1
4
![Page 93: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/93.jpg)
Estimation
What is the value of estimation with rational numbers?
Reasonability of solution ...
Example
Estimate 2 13 + 1
10 − 1 78 .
Roughly speaking, we have ...
(using halves) 2 12 + 0− 2 ≈ 1
2
(using wholes) 2 + 0− 2 = 0
(using quarters) 2 14 + 0− 2 = 1
4
![Page 94: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/94.jpg)
Estimation
What is the value of estimation with rational numbers?
Reasonability of solution ...
Example
Estimate 2 13 + 1
10 − 1 78 .
Roughly speaking, we have ...
(using halves) 2 12 + 0− 2 ≈ 1
2
(using wholes) 2 + 0− 2 = 0
(using quarters)
2 14 + 0− 2 = 1
4
![Page 95: 6-2 Addition, Subtraction and Estimation with Rational Numbersbtravers.weebly.com/uploads/6/7/2/9/6729909/6-2... · x6-2 Addition, Subtraction and Estimation with Rational Numbers.](https://reader030.fdocuments.in/reader030/viewer/2022041106/5f08d5357e708231d423f0cd/html5/thumbnails/95.jpg)
Estimation
What is the value of estimation with rational numbers?
Reasonability of solution ...
Example
Estimate 2 13 + 1
10 − 1 78 .
Roughly speaking, we have ...
(using halves) 2 12 + 0− 2 ≈ 1
2
(using wholes) 2 + 0− 2 = 0
(using quarters) 2 14 + 0− 2 = 1
4