Integers, Decimals and Fractions

I. Classify each fraction as decimal or non-decimal:

3 14 10 39 3 6 18 4
; ; ; ; ; ; ;
24 21 25 95 50 22 45 12

II. Perform the following calculations:

3 2 1 2 7 25
1)   7)  
4 3 2 5 10 14
5 3 1 2 1
2)   
6 4 2 8) 3 6
 2  3 4
3) 1    1   1 1
 3  4 9)  2   3
3 2 2 2 3
4)   3
8 5 5 1 1
10)   
3 3 2 2
5) 12  
5 5
2 3 2
6)    
3 4 5

III. Perform the following calculations:

2
1)  12  13 
5
2) 20  4 10  4  4  2  3
3)  3   5  12   52
2

 1 2  10
4)     
 2 5  15
 2
5) 6 1  11  9 1  
 3
2  4  3  2 
2
6)
2
1
3
7) 0.4  31  0.4   0.1
2

2  1
8)   0.75   0.2 
5  4

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IV. Complete the following:

2 .... 4 16 35 .... 25 10
1)  ; 2)  ; 3)  ; 4) 
5 40 9 .... 40 32 90 ....

V. Complete the following table:

Decimal Fraction Percent

0.4
1
4
20%
0.75
30%
4
5
50%
0.01
5%
8
25

VI. Calculate and write the answers in scientific notation:

1)  0.2   0.01  104

3

22  106
2)
5  102
3) 2.5  0.0001  1012
3 1 1
4)  
5 4 10
0.004  1010
5)
0.5  105

VII. Evaluate:
2 3
1) a  b  ab for a  ; b
3 4
1 1 2
2) xy 2  2 x  y for x   ; y 
2 2 3

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VIII. Calculate and write your answers in simplest form.

a) 0.25 + 0.3 × 0.05

b) (16 – 9) × 3 – 1.2 + 0.5 × 6

c) 2(0.3 + 0.5 × 0.07)

d) 2 × 3.8 + 2 × 5.7

e) 6.4 – [7.1 – (6.3 + 7.8) – 4.09]

f) 42 – [ 22 – 4 (3 – 2.4 ) + 10.2] – (5.2 ) + (3.21)

163  315
g)
45 142

2
1 5 3 9
h)       
2 2 4 6

7 2  23
i)
7 25

4 2  5 1 3 5
j)    7   9    
9 3  6 2 4 7

7 2  42
k)
74

7 3 2
l)  
6 4 14

2
 5 2  7 2 
m)     
 3 6  6 3 

4 3 6
n)   
6 4 4

2
1  5 2
o)   3   
2  7 5

8 7 3 4
p)       14
3 6 4 3

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 Algebraic Expressions
1) Reduce when possible:

a) 5x – x g) 3a – 2b – 11b – 9b
b) 3c2 – 4c2 h) 5t4 – 5t3
c) 5t × 4t i) 2u2 – 7u + u2 – 2
d) 4a × 4a j) 3a2 + 7b2
e) 4a + 4a k) 3x – 3y + 5y – 10y + 2x – 2y
f) 3a × 2a2 l) 2mn2 × 4m2n

2) Expand then reduce:

E = 7(5a – 3b) + 3(9a – 7b)

I = 2 + t (3 – t)
F = 2c(3d + 5)  6d(3c + 4)
J = 3.2(2x – 4) 2(0.5x + 1)  0.4(x – 2)
G = xy(2 – y) 3x(y – 8y2 + 3x)
K = 4 + 2(3x – 2y) 3(2x + y) 5(3x  4y + 2)
H = a + b(a + b)
L = x(y + 2)  2(x + y) – y (x – 4) + x(y + 3)

3) Factorize:

a) 5x – 5y h) 12x4 – 3x3
b) 12a – 6b i) 2a + 5ab
c) 9t2 – 3t + 6 j) x2y + xy2 + 5xy
d) t2 – t k) c(3 – b) – d(3 – b)
e) x2 + xy – 5x l) 3x( 3x – 3)  3(3x – 3)
f) 2a + 5ab m) (x + 2)2  4(2 +x)
g) a2b3 + ab2 n) 3x – xy + 3t – yt

4) Solve the following equations:

2x – 16 = 0 4 – 5t = 0 6y + 5 = 1
2z – 1 = 3 2x – 2 = x + 2 3q = 0
2 3
3x – 2 = 5 4a – 2 = 4a + 3 x= 
3 4

5) Translate these statements into equations then solve them:

a) 6 added to a number gives 12.

b) The triple of a number, diminished by 2, is 6.
c) The quotient of the double of a number by 7 is 4.
d) The three quarters of a number makes 2.
e) Ten less than a number is three more than half of that number.

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6) Solve the following equations.

a) 3(y + 3) = 3(6 – 2y)

b) x – 2(3x + 5) + 1 = 4x
c) 4(3t + 2) = 2(3 – t)
d) 4(x – 3) – 2(2x + 2) = 0
e) 4(3y – 1) = 2(2 – 6y)
f) 6(2 – 3t) + 5(4t – 3) = 2t – 4

7) Solve each problem:

a) The area of a rectangle is 192 cm2.

Calculate its dimensions knowing that its length is triple its width.

b) The mixed chorus of grade 7 counts 36 singers. Half of the girls of the class
and the third of the boys participate. Grade 7 counts 7 more girls than boys.
How many boys are there in grade 7?

c) Find three consecutive integers knowing that their sum is 135.

d) Find three consecutive integers knowing that the triple of the middle number is
1 less than the sum of the other two numbers.

e) The sum of triple of a number and 22 is 46. Find this number.

f) Ayman and Rania share 25 pictures. Ayman has 3 more than Rania.
Find the number of pictures of each.

8) Solve the following equations:

x2 x2 x 1 6 3x  4
a)  ; b)  ; c)  5
3 4 2 4 2

9) Knowing that: if AB = 0 then A = 0 or B = 0, factorize then solve each of the

following equations:

a) ( x  3)( x  2)  ( x  3)(2 x  5)  0
b) (2 x  3)( x  5)  3x(2 x  3)  0
c) ( x  4)  ( x  4)(2 x  1)
2

d) 3x( x  7)  2(7  x)( x  2)  0

e) ( x  1)  2 x( x  1)
2

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10) Expand then reduce the following:

a) (x + 2)2 g) (4a – 1)(4a + 1)

b) (x – 5)2 h) (2b + 1)(2b – 1) (4b2 + 1)
c) (x – 2)2 i) (2a – b)(2a + 3b)
d) (x + 3)2 j) x + (5 – y – 7x) – (8 – 2x + y)
e) (3x – 2)2 k) – 8 + ( 3x – 2 ) + (6 – y) + (x + y – 5)
f) (x2 – 2x)2 l) (2x – 3)(2x + 3) – (5x – 1)(5x – 1) + 3(x2 – 3)(x2 + 3)

11) Given : A = (x + 3)(5x – 1) , B = 2x2 – 5x , C = x(2x – 1)2 + (2x – 1)(2x + 1)

a) Expand then reduce A and C.

b) Factorize B and C.
c) Calculate B for x = 1.5.
d) Reduce 2A – B – C.

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 Congruent Triangles
1) Complete.

a) A triangle having two congruent sides is:……………

b) A triangle having three congruent sides is:……………..
c) A right triangle has: …………..
d) In a right triangle, the acute angles are: ………….
e) In an isosceles triangle, the …….. angles are ………..
f) In an equilateral triangle, each angle measures…………

2) ABC is a triangle where [AM] is at the same time a median and a height.

1) What does (AM) represent to [BC]?

2) Deduce the nature of triangle ABC.

3) ABC is a triangle where [AR] is at the same time a height and a bisector of A .

1) Compare the triangles ARB and ARC.

2) Deduce the nature of triangle ABC.

4) ABC is a triangle where [AM] is at the same time the median relative to [BC]
and the bisector of A .
D is the symmetric of A with respect to M.

1) Compare triangles ABM and DMC.

2) What is the nature of triangle ACD? Justify .
3) What is the nature of triangle ABC? Justify.

5) Given: ABCD parallelogram, DS = BT

1) Compare triangles ABC and DAC.
2) Deduce that ADC  ABC .
3) Compare triangles ADS and CBT.

6) Given: ABC isosceles triangle, BI = IF = FC

Prove that triangles AIB and AFC are congruent.

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7) ABC is an isosceles triangle such that AB = AC = 6 cm
and BC = 4cm. Let I be the midpoint of [BC]. Place D and F on (BC)
and outside [BC] such that CD = FB = 3 cm.

1) Prove that triangles AID and AIF are congruent.

2) Prove that ABF and ACD are equal.
3) Compare triangles ABF and ACD.

8) Given: ABC isosceles of vertex A,

I midpoint of [BC], BS = CT.
1) What is the axis of symmetry of triangle ABC?
2) Compare the triangles AIS and AIT.

9) Let ABC be any triangle and [AM] the median relative to [BC]. The
perpendiculars drawn from B and C to (AM) cuts [AM] at I and J respectively.
Show that:
a) The two triangles BMI and CMJ are congruent.
b) IM = IJ.
c) (BI) // (CJ).
d) (BJ) // (CI).

10) Let ABC be any triangle and [BM] the median relative to [AC]. Extend [BM]
from the side of M by a length MN = BM.
a) Show that triangles AMB and MNC are congruent.
b) Deduce that (AB) // (NC).