Download Grade 8 Chapter 4 Monomials

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Grade 8 Chapter 4 Monomials
Simplify the following expressions :
a)
12.2y + 4  3y  6.25y  8
b)
(6(x + 1) + 3x  3)  3
Given the triangle illustrated at the right.
The length of each side is represented by an algebraic
expression.
(5a  7) cm
(9  3a) cm
3 cm
(2a + 4) cm
a)
To calculate the perimeter of the triangle, do the following :
5a  7 + 9  3a + 2a + 4
b)
To calculate the area of the triangle, do the following :
3(2a + 4)  2
3
Valerie, her brother Matthew and their friend Philip collected money to help homeless teenagers in
their neighbourhood.
Philip collected a quarter of the amount Matthew collected. On her own, Valerie collected $97.35. All
together they collected $300.00.
How much money did Matthew and Philip collect individually?
Show your work.
4
There are 3 members in the Swanson family – Father, Mother and daughter Lynn.
Father is 5 years older than Mother. Lynn is 23 years younger than her father. The sum of their ages is
110 years.
If x is Mother’s age, write the equation that represents this situation.
5
A ribbon measures 4.8 m. It is cut into 5 pieces.
The 2nd piece is 3 cm longer than the 1st piece and 1 cm longer than the 4th piece. Ribbon pieces 1, 3
and 5 are the same length.
What is the length in metres of the longest piece of ribbon?
Show your work.
6
Simplify the following expressions .
a)
2x + 3 + 2(4x  5)
b)
(9x  8)  (6x  4)
c)
-6(5  4x)
d)
(15x  3)  3
7
Three friends have a total of $60. Jennifer has $5 less than Lucy. Silvia has twice as much money as
Jennifer.
How much money does each of them have?
Show your work.
8
The sum of the ages of two persons is 100 years. In five years, the sum of their ages will be equal to 5
times the age of the younger person.
How old is each of them presently?
Show your work.
9
The perimeter of the plot of land illustrated below is 56 m.
How many metres does the variable a represents?
Show your work.
10
The last Mathematics test comprised 14 questions. Each question was worth the same number of
marks. Partial marks were allotted for some questions.
Karl's final mark was 49.
He got...
- full marks on 6 of the questions;
- one fourth of the marks on 4 questions;
- one third of the marks on 2 questions;
- half the marks on 1 question;
- no marks on 1 question.
What was the maximum number of marks allotted for each question?
Show your work.
11
If I add $3.00 to 5 times the money I have, the result is the same as if I were to subtract $18.00 from 6
times what I have.
How much money do I have?
Show your work.
12
Simplify the following expression :
2a  4a + 3a + 8a  5a
13
Simplify the following expression :
4y  8y + 13y  8.6
14
Simplify the following expression :
12x  8x  7x
15
Given the following figure :
b+6
b
b
12
What algebraic expression represents the perimeter of this quadrilateral?
16
The measures of the sides of the rectangle below are given in units.
x+2
7x  1
The dimensions of the rectangle are doubled.
What algebraic expression, reduced to its simplest terms, can be used to represent the perimeter of the
new rectangle?
17
The price of a bicycle is 3 times that of a pair of skis.
The cost of both the bicycle and the pair of skis is $540.
What is the price of the bicycle?
Show your work.
18
Kim is 6 years younger than Sylvia. The sum of their ages is 70.
How old is each?
Show your work.
19
Simplify the following expression :
2x + 3x + x  5x  -2x + 3x
20
A school principal bought several tennis rackets and 3 times as many dictionaries to be handed out as
the end-of-year prizes. The price of each tennis racket is $72 while a dictionary costs half the price of
a racket.
If the school sets aside a $720 for these prizes, find the number of rackets and dictionaries that will be
awarded.
Show your work.
21
Martin, Louise and Denis have $12 400 to share among themselves. If Martin is to get twice Louise's
share and Denis gets $8500 more than Louise, find the amount of money each of them will receive.
Show your work.
22
Simplify the following algebraic expression:
-2(x + 2)  (x  2)
23
An apple farmer employs two people who receive equivalent salaries. One receives $78.60 and
7 bottles of cider. The other receives 5 bottles of cider and $84.00.
How much does one bottle of cider cost?
Show your work.
24
If you simplified the algebraic expression
3y  8  y
 2 , you would get:
4
y
C)
y+
B)
y+2
D)
y+6
25
Simplify:
26
Simplify the following algebraic expressions.
27
28
3
2
A)
a)
4m + 6 + 5m  7  2m
b)
3x + 4  2(2x + 1) + 5
12 x  8
 52 x  3
-4
Simplify the following algebraic expressions:
a)
3x + 5y  x + 2 + 2x  4y + 6
b)
(3x  4)  (4x  8)
c)
3x  6 y  12
3
d)
2x  3 + 2(5x  8)
Simplify the following algebraic expressions:
a)
3x  3
x
3
b)
2(x + 2) + 3x  5
29
a)
Simplify:
b)
Given:
46 x  3  2 x
2
a = 5.5
b = 2
Find the value of the following algebraic expression:
2a + b2 + 5c.
30
On Monday, a group of friends went biking. On Tuesday, they
biked 5 km more than three times the distance they had biked
on Monday.
They also went biking on Wednesday and travelled twice the
distance they had travelled on Tuesday.
The group knows that they covered a total of 105 km during
the three days.
How many kilometres did they bike on Tuesday?
Show all your work.
c = 4
Answer Sheet
1
a)
b)
2.95y  4
3x + 1
2
a)
The perimeter of the triangle is (4a + 6) cm long.
b)
The area of the triangle is (3a + 6) cm2.
3
Work : (example)
Let x be the amount of money Matthew collected
x
x   97.35 = 300.00
4
5x
= 202.65
4
x = 162.12 (Matthew)
40.53 (Philip)
Result
4
Matthew collected $162.12.
Philip collected $40.53.
The equation representing this situation is :
x + (x + 5) + (x + 5  23) = 110
or
x + (x + 5) + (x  18) = 110
or
any other equivalent equation.
5
Work : (example)
Length of the 5 pieces, in cm
4.8  100 = 480 cm
Mathematizing the problem
If
x, the length of one of the 3 equal pieces
x + 3, the length of the 2nd piece
x + 2, the length of the 4th piece
x + (x + 3) + x + (x + 2) + x = 480
5x + 5 = 480
x = 95 cm
Length of each piece
1st, 3rd and 5th piece: 95 cm
2nd piece: 98 cm
4th piece: 97 cm
Length of the longest piece, in metres
98 cm = 0.98 m
Result
6
a)
b)
c)
d)
The longest piece of ribbon is 0.98 m long.
10x  7
3x  4
-30 + 24x
5x  1
7
Example of an appropriate solution
Let
x, be the amount of money Lucy has
x  5, amount of money Jennifer has
2(x  5), amount of money Silvia has
Mathematization
x + x  5 + 2(x  5) = 65
Solve the equation
4x  15 = 65
4x = 80
x = 20
Answer
8
Lucy has $20.
Jennifer has $15.
Silvia has $30.
Example of an appropriate solution
Let
x, be the age of the younger person
100  x, the age of the older person
Mathematize
x + 5 + 100  x + 5 = 5(x + 5)
Solve the equation
100 = 5x + 25
85 = 5x
17 = x
Age of the older person : 100  17 = 83
Answer
The two persons are 17 years old and 83 years old.
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Work : (example)
Perimeter
a + 2a + a + a + a + 3a + 3a + 4a = 56
16a = 56
Solution
16a = 56
56
a=
16
a = 3.5
Result
10
The variable a represents 3.5 m.
Work : (example)
Let x be the maximum number of marks per question
Mathematize the situation
4x 2x x
6x 

  0  49
4
3
2
Solve the equation
2x x

3
2
2 1

 6  1   x
3 2

 36  6  4  3 

x
6


49x
6
6x  x 
= 49
= 49
= 49
= 49
x=
Result
49  6
6
49
Each question was worth 6 marks.
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Work : (example)
Let x represent my money
Mathematize the situation
5x + 3 = 6x  18
Solve the equation
6x  5x = 18 + 3
x = 21
Result
$21
12
4a
13
9y  8.6
14
-3x
15
The algebraic expression for the perimeter of the quadrilateral is 3b + 18.
16
The perimeter of the new rectangle is represented by (32x + 4) units.
17
Work : (example)
Let
x be the price of a pair of skis
3x be the price of the bicycle
Mathematize the situation
x + 3x = 540
Solve the equation
4x = 540
x = 135
Price of the bicycle
135  3 = 405
Result
18
The price of the bicycle is $405.
Work : (example)
Let
x be Sylvia's age
(x  6) be Kim's age
Mathematize the situation
x + (x  6) = 70
Solve the equation
2x  6 = 70
2x = 76
x = 38
Result
19
6x
Sylvia is 38 years old and Kim is 32 years old.
20
Work : (example)
Let
x, number of tennis rackets
3x, number of dictionaries
72x + 36(3x) = 720
72x + 108x = 720
180x = 720
x=
720
=4
180
3x = 4  3 = 12
Result
21
4 rackets and 12 dictionary will be awarded.
Work : (example)
Let
x, Louise's share
2x, Martin's share
x + 8500, Denis's share
x + 2x + x + 8500 = 12 400
4x + 8500 = 12 400
4x = 3900
x=
3900
= 975
4
2x = 1950
x + 8500 = 9475
Result
22
Martin will receive $1950.
Louise will receive $975.
Denis will receive $9475.
The simplified expression is: -3x  2
23
Example of an appropriate solution
Let x, be the price of one bottle of cider
78.60 + 7x = 5x + 84
2x = 54
x = 27
Answer
The price of one bottle of cider is $2.70.
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A
25
7x  13
26
a)
b)
7m  1
-x + 7
27
a)
b)
c)
d)
4x + y + 8
-x + 4
x + 2y  4
12x  19
28
a)
b)
-1
5x  1
29
a)
Examples of appropriate solutions
Example 1
Example 2
24 x  12  2 x
2
26 x  12
=
2
26 x
12

=
2
2
= 13x  6
46 x  3 2 x

2
2
Answer:
13x  6 or -6 + 13x
Deduct 1 mark for each error in calculation.
b)
Example of an appropriate solution
2(5.5) + (-2)2 + 5(-4)
= 11 + 4  20
= 15  20
= -5
Answer:
-5
= 2(6x  3) + x
= 12x  6 + x
= 13x  6
30
Example of an appropriate solution
Let
then
and
x = Number of kilometres biked on Monday
3x + 5 = Number of kilometres biked on Tuesday
2(3x + 5) = Number of kilometres biked on Wednesday
x  3 x  5  23 x  5  105
x  3 x  5  6 x  10  105
10 x  15  105
10 x  15  15  105  15
10 x  90
90
10
x9
x
Distance travelled on Tuesday
3x  5  39  5
 27  5
 32
Answer:
On Tuesday they biked 32 kilometres.
Note:
Students who determined have correctly defined variables but have not set up an
appropriate equation have shown that they have a partial understanding of the problem.