Prime Math 6 - Final rabi_final

\ 3 is called the solution of the equation y + 2 = 5.

The true value of the variable of an equation
is the solution of the equation.

The facts for solving equations

Solving an equation means to find the value of the variables that makes the given
equation true. To solve the given equation, we use generally four facts which are:

i. When we add an equal number on the both sides of two equal quantities, the
sum will remain equal. For example,

If x = 3, then x + 2 = 3 + 2

ii. When we subtract an equal number on the both sides of two equal quantities,
the difference will remain equal. For example.

If y = 7, then y - 4 = 7 - 4

iii. When we multiply by the equal number to the both sides of two equal quantities,
the product will remain equal. For example,

If x = 6, then x × 5 = 6 × 5

iv. When we divide by the equal number to the both sides of two equal quantities,
the quotient will remain equal. For example,

If y = 8, then y ÷ 2 = 8 ÷ 2

Exercise 13.11

1. x = {0, 1, 2, 3, 4, 5, 6, ………….} is the solution set of the variables of the
following equations. Substitute the variables by each element of set x and
find the solution of the following equations.

(a) x + 3 = 8 (b) x + 4 = 6 (c) y - 4 = 2
(d) 2y = 6 (e) X ÷ 3 = 2 (f) Z - 2 = 3

Prime Mathematics Book - 6 245

2. What should be added to both sides of the following equations to find the
value of the variables? Find it.

(a) x - 2 = 12 (b) y - 4 = 9 (c) x - 7 = 14

3. What should be subtracted from both sides of the following equations to
find the value of the variables? Find it.

(a) x + 8 = 12 (b) y + 11 = 20 (c) z + 4 = 12

4. By what should the both sides of the following equations be multiplied to
find the value of the variables? Find it.

(a) x = 9 (b) y = 12 (c) z = 3
4 5 7

5. By what should the both sides of the following equations be divided to find
the value of the variables? Find it.

(a) 7x = 21 (b) 3x = 36 (c) 8x = 32

Working rule of solving equations
Study and learn the following examples.

Example 1: Solve: x - 7 = 15.
Solution: x - 7 = 15
or, x - 7 + 7 = 15 + 7
x = 22 to remove 7 from the L.H.S.,
add 7 both sides.

Example 2: Solve: y + 5 = 21.
Solution: y + 5 = 21
or, y + 5 - 5 = 21 - 5
y = 16 to remove 5 from the L.H.S.,
subtract 5 both sides.

246 Prime Mathematics Book - 6

Example 3: Solve: 4x = 12
Solution: 7

4x = 12
7

or, 4x × 7 = 12 × 7 [multiply both sides by 7]
7 [dividing both sides by 4]

or 4x = 84

or, 4x = 84
4 4

\ x = 21

Example 4: Solve: 4x-5 = 7
7

Solution: 3x-5 = 7
7

or, 3x-5 × 7 =7×7 [multiplying both sides by 7]
7 [adding 5 both sides]
[dividing both sides by 3]
or, 3x - 5 = 49

or. 3x - 5 + 5 = 49 + 5

or, 3x = 54

or 3x = 54
3 3

\ x = 18

Example 5: Solve: 5y + 3 = 3y + 15.
Solution:
5y + 3 = 3y + 15
or, 5y - 3y = 15 - 3 3y is transposed to L.H.S and
or, 2y = 12 3 is transposed to R.H.S.

or, 2y = 12 [dividing both side by 2]
2 2

\y=6

Prime Mathematics Book - 6 247

Example 6: Solve: x - 2x = 2 1 .
Solution: 2 5 2
x 2x = 221
2 - 5

or, 5x - 4x = 5 [L.C.M. of 2 and 5 is 10]
10 2

or, x = 5
10 2

or, 2x = 50 [by cross multiplication]

or, 2x = 50 [dividing both sides by 2]
2 2

\ x = 25

Exercise 13.11 (A)

1. Solve: (b) 7 + x = 12 (c) y + 4 = 17
(a) x + 5 = 14 (e) a - 4 = 7 (f) x - 3 = 8
(d) y + 6 = 14 (h) 8 - x = 3 (i) 7 - y = 9
(g) y - 3 = 12

2. Solve:

(a) x = 6 (b) x =8 (c) y =9
3 7 5

(d) a =7 (e) 8x = 96 (f) 3x = 21
4

(g) 7y = 84 (h) 2y = 8 (i) 5x = 15
3 2

(j) 3z = 33 (k) 4z = 16 (l) 2x = 12
4 5 5

3. Solve: (b) 3x - 17 = 46 (c) 15 + 2y = 19
(e) 3.z + 2 = 14 (f) 5y - 3 = 4y + 8
(a) 2x - 3 = 9 (h) 2x + 14 = 5x + 17 (i) 4z + 6 = 7z + 15
(d) 3x - 7 = 5
(g) 7x - 9 = 2x + 1

248 Prime Mathematics Book - 6

4. Solve:

(a) x + 3 = 7 (b) y-2 = 3 (c) 2x - 3 = 3
4 5 5

(d) 2x - 3 = 5 (e) 72 = 8 (f) 3 + 4 = 7
5 x x

(g) 3y - 2 = 10 (h) 2x - 1 = 3x - 4
4 3 2

5. Solve:

(a) x + 2x = 13 (b) x - 2x = 4 (c) 2x + 3x = 52
4 5 3 7 3 2

(d) y + y = 331 (e) 3x + 4 = 2x (f) 5z - 3z = 1721
2 3 2 2 4

6. Solve the following problems by making an equation.

(a) x 4cm (b) 5cm x

14cm (d) 9cm x
3x 3cm 2x
(c)
21cm 24cm
(e) 2x

19cm

7. Make an equation for the area of the rectangle and then solve the equation.
Also find the length and breadth of each rectangle.

(a) (b) (c) (3x-2)cm
60cm2 15cm2 28cm2
6cm
(x+7)cm (x+3)cm (2x-1)cm 7cm

2xcm
3cm
8. Make an equation for the perimeter of the rectangle and then solve the
equation. Also find the actual length and breadth of each rectangle.

(a) (b) (c) (x+2)cm

P = 32cm P = 26cm P = 44cm

(2x+1)cm (4x+1)cm (2x+5)cm

Prime Mathematics Book - 6 249

Verbal problems:

We use an equation to solve the verbal (word) problems. First we should write the
given word problems into the mathematical form with consideration of the unknown
quantity as variable like x, y, z etc. the mathematical form is an equation. We solve
the equation and find the value of the variable which is the required solution of the
given word problem.

Study and learn the following examples.

Worked Out Examples.

Example 1: When 3 is added to one-fourth of a number, the result is 10.
Find the number.

Solution: let the number be x.

Then according to question, one - fourth
1
1 of x + 3 = 10 means 4
4
x
or, 4 + 3 = 10

or, x = 10 - 3 [subtracting 3 both sides.]
4
x
or, 4 = 7

or, x = 7 × 4 [multiplying both sides by 4.]

\ x = 28

The required number is 28.

Example 2: If one number is greater than other by 9 and their sum is 25, find
the numbers.

Solution: let a number be x.

Then, the greater number is x + 9

According to question,

x + x + 9 = 25

or, 2x + 9 - 9 = 25 - 9

or, 2x = 25 - 9

or, 2x = 16
2x 16
or, 2 = 2

\ x = 8 One number is 8 and greater number is 8 + 9 = 17.

250 Prime Mathematics Book - 6

Example 3: The father's age is 20 years more than his son's age. If the sum of
Solution: their ages is 48 years, find their ages.

Let the age of son be x years

Then, the age of father is (x + 20) years

According to question,

x + 20 + x = 48

or, 2x = 48 - 20 Father's age is 20 years
more than the son's age.
or, 2x = 28 So, we suppose the age of

or, x = 28 son be x years.
2

\ x = 14

Therefore, the age of son = x = 14 years.

the age of father = x + 20 = 14 + 20 = 34 years.

Example 4: The perimeter of a rectangle is 56cm. If its length is 4cm more than
its breadth, find the length and breadth of the rectangle.

Solution: Let the breadth of the rectangle be x cm.

Then, its length is (x + 4)cm.

Now, perimeter of the rectangle = 2(l + b).

So, 2 (l + b) = 56

or, 2(x + 4 + x) = 56

or, 2(2x + 4) = 56

or, 2 (2x + 4) = 56
2 2
or, 2x + 4 = 28
Length of the rectangle is
or, 2x + 4 - 4 = 24 - 4 greater than breadth. So, we
suppose the breadth of the
or, 2x = 24
rectangle be x cm.
or, 2x = 24
2 2

\ x = 12 The length of the rectangle = x + 4 = 12 + 4 = 16cm
The breadth of the rectangle = x = 12cm.

Prime Mathematics Book - 6 251

Exercise 13.12

1. Make the equations of the following each condition and solve them.
a) The sum of x and 7 is 18. Find the value of x.
b) The difference of x from 19 is 12. Find the value of x.
c) If the product of y and 5 is 35, find the value of y.
d) The quotient is 5 when P is divided by 3. Find the value of P.

2. Make the equations of the following condition and solve them.
a) 3 is added to one-fouth of x, the result is 7. Find the value of x.
b) The difference of two numbers is 17. If the greater number is 39, find the other
number.
c) If 21 is subtracted from the product of x and 5, the result is 9. Find the value
of x.
d) If 7 is added to the product of x and 6, the result is 25. Find the value of x.

3. Make the equations of the following given verbal problems and solve them.
a) If one number is greater than other by 12 and their sum is 28, find the numbers.
b) If one number is less than other by 7 and their sum is 37, find the numbers.
c) A boy thought a number. He doubled it and added 9. If he got 23, find the number
which he thought at first.

4. Make the equation of the following condition and solve them.
a) If x students are absent out of 420 students in a school and 385 students are present,
find the number of absent students.
b) A student had 25 apples. He gave x apples to his friend. Now he had 13 apples only.
How many apples did he give to his friend ?
c) A box contains x good oranges and 35 rotten oranges. If the box contains total
oranges is 75, find the number of good oranges.
d) A stick of length x meter can measure 36 meters in 6 times. Find the length of the
stick.
e) The number of boys and girls in a school are x and 70 respectively. If the total
students in the school is 135, find the number of boys.

252 Prime Mathematics Book - 6

5. Solve the following problems.
a) The father's age is 24 years more than his daughter's age. If the sum of their ages is
44, find their ages.
b) Among two brothers, the younger brother is 12 years younger than his elder brother.
If the sum of their ages is 30, find their ages.
c) The perimeter of a rectangle is 48 cm. If its length is 5 cm. more than its breadth,
find the length and breadth of the rectangle.
d) The perimeter of a square is 36 meter. Find the length of a side of the square.

Inequality:

Trichotomy Properties:
Let's take any two whole numbers and compare them. 7 and 9 are two whole numbers and

compare between them. We say that 9>7 (9 is greater than 7) or 7<9 (7 is less than 9). When

we compare a number 7 with the sum of two numbers 5 and 2, we say that 7 is equal to
5 + 2. We can write, 7 = 5 + 2. Here, the symbol ‘=’ is used instead of is equal to.

If a and b are any two integers, then any one of the following mathematical statements
can be true at a time between them.
Either a < b or a > b or a = b.
Such property of integers is called Trichotomy property. The symbols '<' (less than), '>'
(greater than) and '=' (equal to) are the symbols of Trichotomy.
We illustrate the trichotomy property of intergers by taking an example.

Let's take a = 6 and b = -3. Then a = b does not hold true. Again, if we take a = 7 and
b = 4, then a < b does not hold true. Again, if we take a = 6 and b = 4, then
a > b is true because 6 > 4.

Negation of Trichotomy Property:
Let's consider any two integers 8 and 5. We say, '8 is greater than 5', which is true.
This statement can write by using the trichotomy symbol as 8>5.

If we use the symbol of negation for >, then we write 8>5 means 8 is not greater
than 5, Which is false. If we use the other symbols of negation < and ¹ , we can
write 8<5 means 8 is not less than 5, which is true. 8 ¹ 5 means 8 is not equal to
5, which is also true. The symbols >, < and ¹ are called the symbols of negation of

trichotomy property of integers.

Prime Mathematics Book - 6 253

Exercise 13.13

1. Insert the appropritate trichotomy symbols (>, < or =) in the blank spaces.

a) 3 .......... 5 b) -6 ........... 2 - 8 c) -4 ........... 7

d) 9 .......... 12 e) -6 ........... 5 f) 3 ............. -2

2. Identify the true or false in the following statements:

a) 7 > 4 ............ b) 5 > -3 ............ c) -8 > + 12 ...........

d) -7 < -9 ........... e) 5 = 4 + 1 ......... f) 6 - 5 < 5 - 6 .........

g) 3 × (-4) < 5 × (-3) ..... h) 3 < 2 ......... i) -5 > -3 ...........

3. Using the trichotomy property of integers write down the correct mathematical
statements for each case.
a) If a = 4 and b = 6, then ..................
b) If a = -4 and b = 4, then .................
c) If a = 6 and b = 5 + 1, then ..............
d) If a = 9 and b = 5, then ..................

4. Rewrite the following sentences by using appropriate symbols of negation.

a) 14 is not less than 17. b) 5 is not greater than 9.

c) 12 is not less than 9. d) -6 is not equal to 6.

e) -7 is not greater than -9. f) 11 is not less than 17.

5. Rewrite the following statements using trichotomy symbols. Also write the

negation of each statement and represent them by using negation of trichotomy

symbols.

a) 12 is greater than 9. b) x is less than 6.

c) y is equal to 7. d) a is greater than -6.

e) x + 9 is equal to 19. f) 12 + y is less than x - 6.

6. Insert the appropriate symbol of negation used in trichotomy property so that

each sentences will be a true sentence.

a) 9 .......... 5 b) -4 ............... 4 c) -10 ........... 9

d) -5 .......... 7 e) 12 .............. 24 f) 19 ............ 8

7. Write the negative statement for each statement given below:

a) 9 is an odd number. b) Birgunj is the district of Nepal.

c) 3 is a factor of 134. d) 81 is the square of 2.

e) If a,b,c are the three sides of a triangle, then (b + c) > a

f) x + y = x + y g) 281 is a prime number.

254 Prime Mathematics Book - 6

Trichotomy property of Integers in a Number line:

Tsering ! Can you make Why not sir ! This work
a list of the numbers is very simple for me.
which are greater than 3
by using the symbol of The list of the numbers
Trichotomy ? greater than 3 is 4>3,
5>3, 6>3, 7>3, 8>3.
Ok ! Then you can do your job.

For this work, we can use Oh! Sir, it is very
a number line to show all difficult to write all
the numbers which are the numbers greater
greater or smaller than than 3. Because the
the given number. list grows longer and
longer.

The list of the numbers which are greater
than 3 can show in the number line as follows.

-4 -3 -2 -1 0 1 2 3 4 5 6

All the numbers which are greater than 3 lie on the right side of 3, so the mark
denoting 3 is circled and the oblique line moves up to the mark of 3 and then a
straight line is drawn with an arrow head heading towards the right. He denotes any
number greater than 3 by x and write x > 3.

To Show the list of numbers equal to 2 or more than 2, we use the number line
as follows:

-4 -3 -2 -1 0 1 2 3 4 5
Prime Mathematics Book - 6
255

All the numbers which are equal to 2 or more than 2 lie to the right side of 2.
Therefore, the circle denoting 2 is blackened and a straight line is drawn from just
above this mark with an arrow head heading towards the right. Here, we use the
symbol ³ to mean greater than or equal to. If we denote any number greater than
or equal to 2 by x, then we write as x ³ 2.

What does the coloured part
in the number lines denote?

-3 -2 -1 0 1 2 3 4
fig: I

-3 -2 -1 0 1 2 3 4
fig: II

In the fig. I, -1 is circled and the oblique line moves up to the mark -2 and then a
straight line is drawn with an arrow head heading towards the left. So it denotes
all the numbers less than -1. Therefore, the coloured part of the number line
represents x < -1
In the fig. II, -1 is circled with blackened and a straight line is drawn from just above
this mark with an arrow head heading towards the right. So it denotes all the
numbers greater or equal to -1. Therefore it represents x ³ -1.

Rules of Trichotomy
1. Consider a true statement -1 < 4. Adding 2 to both sides, we get

-1 + 2 < 4 + 2 i.e. 1 < 6, which is true.
Also, 4 > -1 is a true statement. Adding 2 to both sides, we get
4 + 2 > -1 + 2 i.e. 6 > 1, which is true.
If a and b are two integers such that a < b or b > a and c is another
integer, then a + c < b + c > a + c.

256 Prime Mathematics Book - 6

2. Consider a true statement 4 > -1. Subtracting 2 from both sides, we get
4 - 2 > -1 - 2 i.e. 2 > -3, which is true. Also, -1 < 4 is a true statement.
Subtracting 2 from both sides, we get - 1 - 2 < 4 - 2 i.e. - 3 < 2, which
is true. If a and b are two integers such that a < b or b > a and c is
another integer, then a - c < b - c or b - c > a - c.

3. Consider a true statement such that 7 > 3 or 3 < 7.
Multiplying both sides by 2, we get

7×2<3×2 or 3 × 2 < 7 × 2
ð 14 > 6 or 6 < 14

Above statements are also true.
If we multiply both sides by -2, then we get

7 × -2 > 3 × -2 or 3 × -2 < 7 × -2
ð -14 > -6 or -6 < -14

Above statements are false. The statement becomes true when we
reverse the trichotomy symbol. This is -14 < -6 or -6 > -14, which is
true. It means, when we multiply both sides of an inequality by any
negative integers, then the symbol of the inequality becomes reverse.

If a and b are two integers such that a > b or b < a and c is another
integer, then ac > bc or bc < ac where c is greater than o and
ac <bc or bc > ac where c is less than o.

4. Consider the true statements such that 12 < 15 or 15 > 12. Dividing both

sides of the both statements by 3, we get

12 < 12 or 12 > 12
3 3 3 3

ð 4 > 5 or 5 < 4, where both are true.

If we divide both sides of the both statements by -3, we get

12 < 12 or 12 > 12
-3 -3 -3 -3

ð -4 > -5 or -5 < -4, where both are false.

The statements become true when we reverse the symbol of trichotomy

in each case.

That is -4 > -5 or -5 < -4, which are true.

Prime Mathematics Book - 6 257

If a and b are two integers such that a > b or b < a and c is another

integer, then a > b or b < a where c is greater than o and
c c c c

a < b or b > a where c is less than o.
c c c c

It means, if we divide both sides of a true inequality by any negative interger,

then we reverse the symbol.

5. If a and b are two integers such that a = b and c is any other integers,
than

(i) a + c = b + c (addition axiom)

(ii) a - c = b - c (subtraction axiom)

(iii) ac = bc (multiplication axiom)

(iv) a = b , where c ¹ o. (division axiom)
c c

Exercise 13.14

1. Show each of the following inequalities in the separate number line:

a) x > 2 b) x > 5 c) x > -4 d) x < 4

e) x < -2 f) x < 6 g) x ­ 3 h) x £ -5

i) x ³ -4 j) x ³ 7 k) x ³ -1 l) x £ ­4

2. Identify whether each of the following statements is true or false by using the
rules of trichotomy.

a) 5 + (-3) < 2 + (-3) b) 5 × (-3) = 2 × (-3)

c) 2 + (-3) > 5 + (-3) d) 5 ÷ (-3) > 2 ÷ (-3)

e) 5 - (-3) < 2 - (-3) f) 3 + (-5) < 2 + (-5)

g) 5 × (-7) < 3 × (-7) h) 3 ÷ (-7) < 5 ÷ (-7)

3. Write the inequalities represented by the following number lines.

(a) (b)

-3 -2 -1 0 1 2 3 4 -3 -2 -1 0 1 2 3 4
(c) (d)

-3 -2 -1 0 1 2 3 4 -3 -2 -1 0 1 2 3 4

258 Prime Mathematics Book - 6

Unit Revision Test

1. Identify the types of algebraic expression in the following:

(a) 7xy (b) 2x+7 (c) 3x2-2xy+4y2 (d) 4x2+7xy+y2-z

2. Write the coefficient, base and power in 7x3.

3. Rewrite the following statements in algebraic expressions:

(a) The sum of twice y and x.

(b) Three times the product of a and b is increased by 2xy.

4. If x=2, y=-3, and z=1, find the value of:

(a) 2x2-3y (b) 4x2z-y2

5. Add:(a) 2x2-3xy+7, 3x2+y2-3 and 4xy +3y2 (b) 4a2+3ab and 2a2-ab

6. Subtract 4x2-3xy+7 from 7x2+2xy-2

7. What should be added to 7x-3y+z to get 11x+2y+3z?

8. If a=2x+3y and b=x-2y, find 2a+3b and 4a-5b.

9. Multiply:

(a) 5a3 x 4a2b (b) (3x2-2xy) by 2xy2 (c) (3x+2y) b (x-3y)

10. If x=p2+pq+q2, y=p-q and z=p3+q3, find xy-z.

11. Divide the sum of 3x2y and 12 xy3 by 3xy.

12. Divide:

(a) (x2+x-12) by (x+4) (b) (2x2-3xy-2y2) by (2x+y)

13. Solve:

(a) 2x-3=15 (b) 3x-7=15-x (c) 3y-4 =y
2
14. If one number is greater than other by 7 and their sum is 67, find the numbers.

15. The perimeter of a reactangle is 48 cm. If its breadth is 2 cm less than its

length, find the area of the rectangle.

16. Show each of the following inequalities in the separate number line.

(a) x>3 (b) x£2 (c) x³-2

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PRIME Questions for More Practice

1. i. In the given figure 2A L = LB= 8 cm what is the measure of AB.

AL B

ii. In the given figure AC = 7 cm, BD = 8 cm and BC = 3 cm then what is the measure
of AD ?

A BC D

iii. From the given figure, Write the perpendicular line AB
segments and parallel line segments separately HG D C

FE
iv. Draw a line segment EF = 9cm and construct a perpendicular bisector of the line

EF.
v. Construct a square having side 5 cm.

2 i. Find the value of 'x' from the given figure.
A
D

8x C

7x
B

ii. find the value of x from the given figure.

B 6x A
5x 7x E

D
C

Prime Mathematics Book - 6 275

iii. Find the value of unknown angles. 50° 2x
3x
iv. From the following figures , Find the value of x
B
and also the acute angle ÐABC 8x

2x

AD

v. Find the value of angles marked in the adjoining diagram. 70° a 15°
b 45° c

3. i. Find the value of x from the given figure. D
5x

3x F
4x

E

ii. Find the value of y and z from the adjoning diagram. 85°
z

115°

35°
y

iii. Find the value of x from the given figure. 3x 3x
2x 2x

276 Prime Mathematics Book - 6

A
70°

iv. Find the value of x from the given diagram. 130° x E
D B C

v. Plot the points A (2, 2), (2, 5), (6, 2) and (6, 5) in graph paper and join them one
after another.

4. i. Length of the sides AB and AC are each 6cm A
and perimeter is 20cm. Find the length of the
side BC.

B C

ii. Find the area of the adjoining diagram. 2 cm

2 cm
3 cm
5 cm 2 cm

iii. Find the area of the shaded region given in diagram. 6 cm

2 cm

8 cm

iv. Find the perimeter and area of a square having side 6 cm.
v. Find the length of the side of a square havin perimeter 56 cm. Also find its area.

5. i. Find the perimeter of a square whose area is 49 m².
ii. Breadth of a rectangle is one third of its length and perimeter is 80 cm . Find the
length and breadth of rectangle. Also find its area.
iii. Area of rectangle is 36 m² and breadth is one fourth of its length. Find the length,
breadth and perimeter of rectangle.
iv. If the length and breadth of a cuboid are 12 cm and 10 cm respectively, What will
be the height of the cuboid of volume 960cm3? Value of the cubed is 96 cm³.
v. A cuboid has the volume 648 cm³. If the height and breadth of the cuboid are 9 cm
and 6 cm respectively, What is the length ? Also find its total surface area.

Prime Mathematics Book - 6 277

6. i. Represent the set of number, between 10 to 50 which are divided by 5 in description

method, listing method and set builder method.

ii. Represent the set of number between 5 to 30 which are divided by 3 in description

method, listing method and set builder method.

iii. In the adjoining diagram, pick out the dissimilar ones and represent the set of similar

elements as the following . Pa e q
a) Description method u
b) Listing method oi m
c) Set builder method

iv. If A = {2, 4, 6, 8}, then show the following sets by listing method.

a) B = { the set obtained by subtracting 1 from each member of set A}

b) P = { The set obtained by multiplying each element of A by 2}

v. If A = { 1, 2, 3} write down all possible sub - sets of A

7. i. If A = { 2, 4, 6, 8, 10}, B = {1, 3, 5, 7, 9, 11}, C = {a, b, c}, Find :
a) n(A) + n(B) + n(C)
b) n(A) + n(B) – n(C)

ii. If A = {prime number less than 10 }, B = {Vowel alphabets} and C = {1, 4, 8, 9},
Find: n(A) + n(B) + n(C)

iii. Find the sum of the greatest and the smallest number of 8 digits and rewrite the
sum in words according to national system ?

iv. Find the sum of the greatest and the smallest number formed by the digits 2, 6, 8,
4, 3, 5, 9, 0 and rewrite the sum in words according to international system.

v. Find the difference of the largest and smallest number formed by 4, 2, 3, 2, 1. Write
down the result in words.

8. Simplify the followings
i. [96÷8{33 ÷ (14 – 3)}] + 46
ii. (16 × 4) ÷ 8 [27 ÷ 9 ( 8 × 5 – 42 + 6) + 3]
iii. 9 – {8 – 6 (6 – 5 – 3)}
iv. Divide the sum of 36 and product of 18 and 6 by the difference of 50 and 14.
v. The 18 times of the difference of 15 and 12 and divided by 16.

9. i. Express each of the following numbers as a product of the prime numbers.

a) 1600 b) 1100 c) 1300

ii. Express each of the following numbers as a product of any three composite numbers.

a) 672 b) 1080 c) 4500

iii. Find the prime factor of 144 by factor tree and by division method.

278 Prime Mathematics Book - 6

iv. Find the common prime factors of 150 and 425.
v. List the common multiples less than 60 of m(6) and m(9).

10. i. Two watches are kept in a room. One watch rings at an interval of 3/3 hrs. and
another watch rings at an interval of 5/5 hrs. At what interval of with they ring
together ?

ii. Find the least number which is exactly divided by 80 and 96.
iii. Find the least number which is exactly divisible by 24 and 36.
iv. Find the square root of 12100 by prime factorization method.
v. What number multiplied by itself gives the product of 5625 ?

11. i. In a plantation programme, everybody planted as many plants as there were

participants so that planted 1225 plants in total. How many participants were there ?

ii. Find any three two rational numbers between 1/4 and 1/5.

iii. Draw the number line and represent the following rational numbers on it.

a)2/5 b) 6/11

iv. Rabin wrote 1/4 part of a big register. How much part is left to write?

v. A Swosthani book is reading by Reena, Meena and Tina as 1/3 part, 1/6 part and

1/2 part respectively. What part is left to read for Pranisha ?

12. i. Multiply : 2 1 ×2 4 × 7
7 9 12

ii. Simplify : 1 1 ×1 1 —: 1 1
2 3 6

iii. Divide : 15 3 by 3 3
4 8

iv. A small rabbit can cover 2/3 m distance by jumping. How many times, should be
jump to cover 16 m distance ?

v. A bottle can hold 1 1/2 liter of milk. How many such bottles can be filled by 20
liters of milk ?

13. Simplify the followings:
11 7 9
i. 6 – 8 + 4 –2

ii. 3 2 ×2 1 —: 1 2 × 2
5 2 15 15

( )iii. 4 – 3 —: 3 – 1
5 4 3

Prime Mathematics Book - 6 279

11 5 1 and subract 2 from the result.
iv. Divide the sum of 2 and 6 by 6 3

1 1
v. Multiply the sum of 2 and 3 by 3 and divided by 2.

14. i. Add : 10.113

20.658

33.165

42.125
+ 31.312

ii. Madhav wants to buy a pair of basket ball shoes of cost Rs.640.95. How much more
money is needed to him where he has only Rs. 338.50 in the pocket?

iii. Simplify : 24.54 ¸ 6 +6.25 ¸ 5.
iv. One bag peanuts costs Rs. 45.5. How many bags can buy with Rs. 227.5 ?
v. A shirt can be made with 2.5 meter of cloth. How many shirts can be made from

1387.5 meter of cloth ? Also find the cloth is left at last.

15. i. Find the sum of 60% Rs. 1200 and 40% of Rs. 800.
ii. Ramesh obtainede 35 marks in mathematics out of full marks 50. How much
percentage did he get ?
iii. In a class of 80 students, 16 were failed, What percentage were passed ?
iv. If x:y = 3:4, a:b = x:y and a = 12, What is the value of b ?
v. Ratio of share of Ram and Shyam is 3:8. If Ram has Rs. 56, How much money does
Shyam have ?

16. i. Length and breadth of a room are in the ratio 3:2 where the length 15m, Find the
area and perimeter of the room.

ii. Samyak bought 1dozen of Banana for Rs. 60 and sold each Banana at Rs. 6. Find the
total gain ?

iii. A merchant bought 24m of cloth at the rate of Rs. 75 per meter and wants to make
a tatol profit Rs. 600. At ehat price should 24m of cloth be sold ?

iv. If Bishal bought 2 dozen copies for Rs. 1200 and made the profit of Rs. 8 in each
copy, What would be the tatol selling price ?

17. Unitary Method:
i. If the cost of 4 books is Rs. 540, find the cost of 2 dozen of such books.
ii. If the cost of 20 kg rice is Rs. 1620, what is the cost of 2 quintal of rice ?

280 Prime Mathematics Book - 6

iii. Kushal bought 5kg of mangoes for Rs. 200. How much mangoes could he buy at Rs.
800 ?

iv. A car needs 5 liters of petrol to cover the distance of 75 km. How many km will it
cover by 20 liters of petrol ?

v. 20 men can do a work in 5 days. How many men is required to complete the same
work in 20 days ?

18. i. Calculate the simple interest where principal is Rs. 50,000, time is 7 years and rate
of interest is 6% per annum.

ii. Calculate the simple interest when P = Rs.1600, Time (T) = 30 months and R = 4.5%.
Also find the amount.

iii. Ramesh borrowed Rs. 50,000 from a bank at 10% per year simple interest. How much
interest does he pay at the end of 5 years.

iv. Find the interest and amount on Rs. 3000 for 24 months at 5% per year.
v. Find the sum of money borrowed when interest in two years at the rate of 10% is

Rs. 1000.

19. i. Construct a frequency table for the age of 20 students from the following
10, 10, 11, 12, 10, 13, 12, 10, 14, 12, 14, 12, 13, 10, 11, 11, 11, 12, 10

ii. Construct the frequency table for the follwing marks.
5, 10, 15, 25, 20, 20, 25, 5, 5, 5, 10, 15, 15, 15, 10, 10, 10, 5, 5, 15, 15, 20, 20,
25, 10, 25, 10, 15

iii. Construct a simple bar diagram.

class I II III IV V

Students of students 30 25 20 25 15

iv. Represent the data in a simple bar diagram.

Education House Fooding Clothing Others
Rs.400 Rs.600
Rs.800 Rs.1200 Rs.1000

20. i. If a = 2, b = 3 and c= 4, find the value of ab + bc + ca + 20
ii. If x = 3, y = 1 and z = 2, find the value of x2 + y2 + z2
iii. Add : a +2b +7a, 2b +3c and 3c +2a
iv. Subtract : 5x + 6y + 7 from the sum of 2x + 3y + 4 and 6x + 8y + 9
v. Simplify : 2 (2a + 3b) – 4(a +2b) + 3(3a + b)

Prime Mathematics Book - 6 281

21. i. Simplify : (a + 2b + 3c) – 5(a + 4b + 3c) + (a + 12b + 8c)
ii. Simplify : x (x + 2y) + y (x – 2y)
iii. Multiply : (7x² + 5x + 6) and 2x
iv. Divide : (4x² + x – 14) by (x + 2)
v. Area of a rectangle is (3a²b + 12ab) square units and length is 3ab units. Find the
breadth of the rectangle.

22. Solve the following:
i. x + 7 = 12
ii. 3x – 1= x + 5
iii. x - 2(x – 1) = 1 - 2x
iv. Find the number, when twice the number is equal to less than the sum of it with 7
by 1.
v. Find the number whose sum with its double is equal to 12.

282 Prime Mathematics Book - 6

Model Questions

FIRST TERMINAL EXAMINATION

Group A [10 x1=10]
1. a) Write down one example of finite set.

b) Write down the cardinality of set {3, 4, 5, 6}.

2. a) Write the formula to calculate perimeter of rectangle.
b) Find the perimeter of the following figure.

ab

3. a) Identify the coefficient, base and power of the algebric term 9x7. c
b) Write the algebraic expression of: x is added to 5

4. a) Write the algebric complement anlge of 50°. xx x
b) Find the value of x from the given figure.

5. a) Find the profit percentage if CP is 'a' and profit is 'b'.
b) If CP is Rs. 300 and SP is Rs. 200, find the loss.

Group – B [17×2 = 34]
6. a) An article is bought at Rs. 400 with a profit of Rs. 100, find the selling price.

b) Find the loss percentage if CP = Rs.150 and SP = Rs.135.

7. a) Write two examples of the Monomial expression.
b) If x = 3, find the value of 3x+7

8. a) Find the length of a square whose perimeter is 48cm
b) Find the perimeter of a rectangle when its length is 5cm and breadth is 3cm.

9. a) Convert the unlike fractions 3 and 5 to like fractions.
48

b) Perform the following subtraction:- 9 – 1 .
12 12

10. a) Find the measurement of the unknown angle. 40°
Prime Mathematics Book - 6 x

283

b) Find the value of x from the given figure. x 120°
c) Find the value of x from the adjoing diagram. 50°

x+10°
70°

11. a). Draw the angle of 75° using compass.
b) Find the value of x from the given equalateral triangle.

x

12. a) Find the sum of the angle of the regular pentagon. 100°
b) Find the size of the unknown angle of the following polygon. 110°

x

95°
105°

13. a) Write down the set A = {x: x € N, N = first 5 natural numbers} in listing form.
b) Draw the venn digrame to show A= {1, 2, 3, 4} and B = {4, 5, 6}.

284 Prime Mathematics Book - 6

Group C [14x4=56]
14. Mr. Shrestha bought a shirt for Rs.450. At what price did he sell it so that his loss

was Rs.75?
15. Dolma bought a watch for Rs.625 and sold it for Rs.550. Find the profit or loss percentage.
16. Simplify : 2ab + 3bc - ab - 2bc.
17. If a = 3cm, find the perimeter of the given triangle.

2a a

18. Find the perimeter of the adjoing diagram. 3a

2 cm

4 cm6 cm
2 cm

19. Write down the sets A and B from the given venn diagram.

AB
3 57
4 68

9

20. Arrange the fractions 1 , 1 and 1 in ascending order.
23 4

21. What should be added to 4 7 to make it 8 5 ?
12 12

22. The perimeter of a rectangle is 60cm and its length is 17cm. Find its breath.
23. You are running around a football ground of length 75m and breadth 50m. How many

meters will you cover when you compete 8 rounds around it?

Prime Mathematics Book - 6 285

24. If one angle of a right angle triangle is 42°,find the size of other acute angle.
25. Find the size of the unknown angle from the given figure.

A

50°

? 100° E
B C

D

26. Construct a square of side 6cm.
27. Find the value of ÐBAO from the following diagram.

DC

E B O
F
x
A

THE END

286 Prime Mathematics Book - 6

Model Questions

SECOND TERMINAL EXAMINATION
Group A [10 x 1 =10 ]

1. a) In the given trapezium, write down a pair of a parallel lines.

b) Write down the point on positive direction of X-axis at 4 units far from the origin.
2. a) Find the area of shaded region from the adjoining

diagram by counting the unit square.

b) If A = {3, 9, 27, 18, 24}, write the cardinal number of this set.

3. a) If cost price = CP, selling price = SP, express the Loss (L) in terms of SP and CP.
b) Identify the coefficient of x in the monomial 8x.

4. a) Write down complementary angle of 75°.
b) Simplify 8x+7y+12x+2y

5. a) Write down one example of variable quantity.
b) Is 3x – 2y an expression ?

Group 'B' [17x2=34] A
6. a) In the figure AB and BC are perpendicular lines.
D
If ÐDBC = 20° and ÐABD = x,
then what is the value of x?

20° C
B

b) In the given figure find the value of x.

2x+30°
110°

Prime Mathematics Book - 6 287

c) Find the value of x from the adjoing diagram. x
2x x

7. a) If the co-ordinates of the point is (-2, -3), in which quadrant does the point lies and
what are x and y-coordinates.

b) The length of a rectangle is 10cm and breadth is 8cm. Find its perimeter.

8. a) If P = {Factors of 18}, Q = {1, 2, 4, 5 } and R = {1, 3, 6, 9, 10, 15}. Which set is
equivalent to the set P?

b) If A = {Even numbers from 21 to 39}, find n (A).

9. a) Find the prime factors of 384.

b) Simplify: 3 1 _ 1
2 4

10. a) If 15 pens costs Rs 180 how many pens can be bought by Rs 864.
b) What is the interest of Rs 2000 at the rate of 8% per year for 5 years?

11. a) Find the perimeter of the adjoing diagram. 7 cm
4 cm
3 cm

2 cm

b) Find the area of the shaded region. 3 cm
4 cm
5 cm

20 cm

12. a) Subtract 2x+3y from 4x+5y
b) If x=2 and y=3, determine the value of x2+2xy+y2

13. a) Multiply : (p x q x 2p x 3q)
b) Solve: 5x + 5 = 25

Group 'C' [14x4=44] A
14. In the adjoining figure, find the measurement of ÐABD. 40°

C BD
288 Prime Mathematics Book - 6

15. Construct the equilateral triangle of side 4.5cm. AD
110° 100°
16. In the adjoining figure ABCD is a quadrilateral
find the value of angle marked with the letter x .

70° x
B C

17. Find the ÐQPR from the adjoing diagram. P

? 110°
RT
130°

SQ

18. A cuboids has the volume 768cm3. If the length and height of the cuboids are 12cm and
8cm respectively. What is the breadth?

19. From the adjoining figure, represent The set by following methods. a e
i) Description method for the set A. o
ii) Set-builder method for the set A. iu
iii) Write down cordinality of set A.

20. Simplify : 7 _ 5 + 1 1
8 12 12

21. Find the HCF of 24 and 36.

22. Find the sum of the greatest and the smallest numbers formed by the digits 2, 6, 8, 7,
4, 3, 5, 9 and write the sum in words.

23. Simplify : [{(70+50)¸10 }+13]¸ 5 - 4.

24. Reejan bought 1 dozon of apple for Rs. 60 and sold each apple at Rs. 6, find his
total gain.

25. What should be subtracted from 9a+11b-2c to get 4a+7b-5c ?

26. Divide (2p2q-6pq2) by 2pq.

27. Solve 4(x+3)=2(x+7)

THE END

Prime Mathematics Book - 6 289

Model Questions

FINAL TERMINAL EXAMINATION

Group - A [10 x 1 = 10]

1. (a) Write an example of finite set and infinite set.
(b) Write the composite numbers between 25 to 50.

2. (a) Simplify: 8x - 17y + 12x
(b) Divide: 27a3 by 3a2

3. (a) Write the inequality from the given number line.

-4 -3 -2 -1 0 1 2 3 4 5 ? C

(b) What is the number represented by the tally bars

4. (a) Write the acute angle and obtuse angle from the given figure. O B
(b) Write the name of the triangle according to sides. A

5. (a) In which quadrant does a point lie if its co-ordinates is (-3, 2)?
(b) Draw the possible axes of symmetry in
the given adjoining figure.

Group - B [17 x 2 = 34]

6. (a) Write the elements of a set of letter of the word "Mathematics". Also write the
cardinal number of the set.

(b) Find the H.C.F. of 12 and 20.

7. (a) If 3, 4, 12 and x are in proportion, find the value of x. 4 cm
(b) Find the area of the shaded region
in the adjoining diagram. 8 cm

8. (a) If x=2 and y=3, find the value of 2x2+3xy. 12 cm
(b) Subtract 2ab-3bc+5 from 4ab+2bc-3.

9. (a) Multiply: (4a2-3b2) by 2ab

(b) Solve: 3x-2 = 10
4

10. (a) Write the list of the numbers greater than 3. Also represent it in a number line
by using the symbol of Trichotomy.

290 Prime Mathematics Book - 6

(b) Marks obtained by 30 students of class VI in an examination is given below:
29, 28, 29, 26, 25, 26, 29, 30, 28, 29, 30, 31, 30, 31, 30, 31, 30, 31, 25, 25, 26, 31, 30,
28, 27, 28, 28, 29, 28, 26.
Construct a frequency distribution table with tally marks.

11. (a) Find the value of x from D x C
the given adjoining figure. A B
200 350
(b) Find the measure of ÐC in
the adjoining DABC. O
A

B 470 C

12. (a) If one angle of a right angled triangle is 480, find the size of the other acute angle

of the triangle.

(b) Find the sum of the interior angles of a regular pentagon.

13. (a) Find the value of x from A 1250 D

the adjoining figure. x C
(b) Draw and angle of 400 using protractor.
B 470

(c) Plot the points (0,-2), (4,1) and (3,6) on the graph paper. Join the points in order

by using ruler and write the name of the figure.

Group - C [14 x 4 = 56]
14. Define universal set with an example. Write the all possible subsets of the set A = { 2, 4, 6 }.
15. Simplify: 62+2 { 56 ¸ (4 x 2) - 5 }

16. Add the expressions x2+3xy+2y2 and 4x2+3y2-xy then subtract the result from 7x2-4xy-y2.

17. Multiply: ( 3x + 2y - 6 ) by ( 2x - y )
18. Divide: ( 6a2 - 13a + 6 ) by 2a
19. If one number is greater than other by 14 and their sum is 40, find the numbers.
20. The length of a rectangular block is thrice its breadth and its height is 6 cm. If the volume of

the block is 1152 cu. cm, find its length and breadth.
21. If the cost of 1 dozen of pencils is Rs. 60, find the cost of 18 pencils.
22. A man buys a cycle for Rs. 7200 and sells at 15% profit.

(i) How much is the profit amount?
(ii) How much is the selling price of the cycle?
23. Mr. Sherpa deposited a sum of Rs. 2500 in a bank at 4% per year simple interest. How much
money would he get at the end of 3 years? Find it.

Prime Mathematics Book - 6 291

24. The table given below shows the numbers of different fruits. Draw a bar graph to show their numbers.

Fruits Mango Orange Banana Apple Guava
Numbers 16 20 14 10 24

25. Copy the given figure approximately and reflect A P
DABC to DA'B'C' under reflection on the line PQ. B
C

26. From the adjoining triangle, find the A y Q
values of x and y.
740 CD

Bx

27. Construct an equilateral triangle of a side 5 cm.

THE END

292 Prime Mathematics Book - 6