Prime Mathematics 4

Concept of Algebra

We know that we always use number in arithmetic but we use letters in place
of number in algebra. So, algebra is the generalization of arithmetic.

There are x number of books in the figure.
∴ the value of x = 3

There are y number of birds in the figure.
∴ the value of y = 5

There are z number of pencils in the figure.
∴ the value of z = 8

In the above examples, we used x, y and z letters in place of 3, 5 and 8. So, in
arithmetic we use only numerals but in algebra we use numerals and letters both.

Constant and variable:
There are many pairs of numbers whose sum is 12, as in the following:
5 + 7 = 12, 8 + 4 = 12, 2 + 10 = 12, 3 + 9 = 12, 11 + 1 = 12
Here two numbers vary but their sum is always same. If the two numbers
which vary can be written as letters x and y, then the values of x and y are
different numbers whose sum is same as 12. So, 12 is called constant. The
value of constant does not change. Some examples of constant are:
The number of districts of Nepal.
The number of legs of a cow.
The number of sides of a quadrilateral.
The values of letters x and y vary, that is their values do not remain same
in the above examples. So, x and y are called variables. The values of the
variables are different. Some example of variable are:
The number of days of months.
The weight of students of a class IV.
The price of books of class IV.
We use letters like a, b, c, ...................., x, y, z in the places of numbers
to represent different values. So, these letters are called variables. But
sometimes these letters are used to represent constants. Why?

196 Prime Mathematics Book − 4

Look at the number line in the following:

xy
0 1 2 3 4 5 6 7 8 9 10
In the above number line, the letters x and y are used in place of 3 and 7. It
means x has only one value 3 and y also has only one value 7. So, x and y are
constant.

If a letter is used to represent
only one value, it is called a

constant.
If a letter is used to represent
many values, it is called a
variable.

Exercise: 10.1
State whether the following letters are constants or variables:
1. x represents the number of districts of Bagmati zone; x is a .............
2. y represents the number of sides of a triangle; y is a ....................
3. x represents the height of students of your class. x is a ..............
4. z represents the number of days of a week; z is a ................
5. z represents the number of students of Nepal; z is a ................
6. y represents the price of different types of rice; y is a ...............
7. x represents the weight of your friends in your class; x is a ..............
8. z represents the number of legs of a lion; z is a ....................
9. y represents the number of letters in the word 'Table'; y is a ............
10. α represents the number of pages of your book; α is a ................

Prime Mathematics Book − 4 197

Operations on constants and variables
We know that there are four fundamental operations in mathematics which
are addition (+), subtraction (−), multiplication (×) and division (÷). These
four fundamental operations also used between constants and variables.
Let’s study and learn

How many marbles are there
altogether in two boxes?

4 marbles + 4 marbles
= 2 × 4 (two times 4)
= 8 marbles

How many marbles are there xxx
altogether in three boxes?

x marbles + x marbles + x marbles
= 3 × x (three times x)
= 3x

Do the following activities like the above examples:

1. x x x x 2. x x x 3 3. x x x 3 2

===
===
===

4. y y y 7 5. y y y 5 4 6. z z z z 2

= = =
= = =
= = =

198 Prime Mathematics Book − 4

Exercise: 10.2

1. Express the following statements in mathematical form by using
mathematical operations.
(a) a is added to 6.
(b) 4 is subtracted from x.
(c) x is multiplied by 4.
(d) Four times y.
(e) Three times x is added to 7.
(f) Two times y is subtracted from 9.
(g) Four times b is increased by 5.
(h) Three times x is decreased by 12.
(i) Three times the sum of x and 4.
(j) Two times z is divided by 3.
(k) 7 is subtracted from three − fourth of y.

2. Hari has x pencils. His brother gives him 5 pencils. How many pencils
will he have now?

3. A man is y years old now. How old will he be after 7 years?
4. Kaji Sherpa is x years old now. How old was he before 5 years?
5. The price of 1 pen is Rs. x. What is the price of such 7 pens?
6. Nilma is x years old now. The age of her mother is 7 years more than

three times her age. How old is she now?
7. The number of pages in 5 books is x. How many pages are in 1

book?

Prime Mathematics Book − 4 199

Base and coefficient of a term Yes, it is 4 times 3.
i.e. 4 × 3 = 12
Can you tell me that 3 +
3 + 3 + 3 means?

+ + + = 4 groups of 3 balls = 4 × 3 = 12

x+x+x+x

Now, can you tell me what Yes, it is 4 groups
x + x + x + x means? of x balls.

Well done!

It is written as 4 times of x. In
mathematics, 4 × x = 4x

In the product 4x, 4 is called What is x called ?
the coefficient of x.

x is called In −7y, −7 is coefficient of
the base. y and y is base.

Well done!

Coefficient is a number placed before the base and base is the variable
in an algebraic term.

Exercise: 10.3

Write the base and coefficient in the following terms:

1. 5x 2. 3y 3. −2x

4. x 5. −15xy 6. −ab
7. 4abc 9.
10. 3(a + b) 8. 2 x − 4 y
7 9

200 Prime Mathematics Book − 4

Algebraic terms and expressions
Mohan had some money.
Let us say he had Rs. x. He spent Rs. 3 to buy a pencil. Now he has Rs. (x−3)
left.
Indumani's age is 8 years more than two times her daughter's age. Let us
suppose daughter's age be y years. Then her mother's age is (2y + 8) years.
From the above statements, x, 3, 8 and y are called algebraic terms. The
algebraic terms x and 3 are connected by '−' sign. i.e. x − 3 is called algebraic
expression. Similarly, 2y + 8 is also algebraic expression.
Therefore, the result which is obtained after connecting the algebraic terms
by '+' or '−' sign is called the algebraic expression.
Examples of algebraic terms are x, 5, 2x, 3b, 4c, 5ab, −7xyz etc.
Examples of algebraic expressions are x + 3, 2x − 7, 3xy + 2x, 2x − 3y + 7z
etc.

Types of algebraic expressions:
There are various types of algebraic expressions.
Monomial expression: An algebraic expression which contains only one term
is called monomial expression. For examples; x, 3xy, 2y, x/y, 3/2 xy, etc.
Binomial expression: An algebraic expression which contains two terms is
called binomial expression. For examples; a + b, 2a − 3, 4x − 3y etc.
Trinomial expression: An algebraic expression which contains three terms is
called trinomial expression. For examples; a + b − c, 2a + 3x + 5 etc.
Multinomial expression: An algebraic expression which contains more than
three terms is called multinomial or polynomial expression. For examples; 2x
+ y − 3x + 7, a + b − 2c + 5x − 8 etc.

Exercise: 10.4

1. Write down the terms and number of terms in the following alge-

braic expressions.

(a) 3x (b) x + 3y

(c) 2a − b (d) 4xy − 2yz

(e) 2x − 7 (f) a − b + c

(g) 2x + y − 3z (h) 5a − 3b + c − 8

(i) 3x + 4xy − 2y + 7 (j) 7 − 3x − 2yz + 4c + 5a

Prime Mathematics Book − 4 201

2. Identify the terms and expressions in the following:

(a) 4x (b) 2x − 7

(c) 3abc (d) 2x + 3y − 7

(e) 5 − 3y (f) a + b − 3

(g) 2xy2 (h) 5a − 2b + c

3. Write the type of expressions in the following expressions:

(a) 2xy (b) 3x + 4y

(c) x + 4ab (d) 7xyz

(e) 2x − 3b − c (f) 4x − 5y + 7z − 3

(g) x + y − c + 8 (h) 2a − 3b + c

4. Write two examples of each algebraic expressions.
5. Write an algebraic expression for the following statements. Also

write the name of the expression.
(a) A class has 30 students. If x number of them are present in a day,

how many of them are absent?
(b) Five times x is added to 4 times y.
(c) There were 20 birds in a tree. If y of them flew away, how many of

them remained in the tree?
(d) 5 is subtracted from the sum of 2x and 3z.
(e) Mahim had x apples. His father gave him 5 more apples. How

many apples does he have now?

Evaluation of terms or expressions:

Mohan, can you find the
value of 3x if x = 3 ?

Mohan, can you find Yes sir! I can find the
the value of 2x + y value of 3x by replacing
when x = 3 and y = 5 ?
x with value 3.
3x = 3 × 3 = 9

202 Prime Mathematics Book − 4

Yes sir, I can ! I put the value of x = 3
and y = 5 in place of x and y and then
simplify by using fundamental operation.

2x + y = 2 × 3 + 5
=6+5
= 11

Replacing the variables of a term or
expression with given numeral values of the variables
and simplifing using fundamental operation called the

evaluation of terms or expressions.

Class Work

1. If x= 3, y = 2, z = 5 and w = 1, find the value of the following:

(a) 2x = 2 × 3 = 6 (b) 4x =
(c) 3y = (d) 3xy =
(e) 4wy = (f) 3xz =

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

(a) x + y = 2 + 4 = 6 (b) y - z =

(c) 3y + 2z = (d) 2x + z =

(e) 4y - 3z = (f) 3(x + z - y) =

Exercise: 10.5

1. If x = 2, y = 3, a = 1 and b = 4, find the values of the following:
(a) 3x (b) 2ay (c) 5ab (d) axy
(e) 2xy (f) 12y (g) abx (h) 16ax
(i) 2axy (j) 5xy

2. If x = 3, y = 1, z = 4 and w = 2, find the values of the following:
(a) x + 5 (b) x - 2 (c) w + 7
(d) x + z (e) Y + z (f) 3x + y
(g) 7x - 3z (h) 5w + 2z (i) 7z - 2y
(j) x + 2y - z (k) 2x + y + w (l) 4x - 5y + z
(m) 2x + (5y - z) (n) (5w - 2z) + (3x - y)

Prime Mathematics Book − 4 203

3. If x = 5, y = 2, a = 3 and b = 4, find the length of the following line
segments.

(a) A x cm B 4 cm C (b) A 3 cm C x cm B

AC = AB + BC (c) P 2y R 4 cm Q
=x+4
=5+4 (d) X 4 cm A a cm B y cm Y
=9
(e) D 2x A 7cm E (f) A 3xcm C 2a cm D b cm B

4. Find the perimeter of the following figures if x = 3cm and y = 2cm.

(a) A (b) M 2y cm (c) A 5ycm D

(4y+1)cm x cm

x cm

(4x+3)cm

x cm
3xcm
3xcm
B y cm C N 3x cm P B 5ycm C

(d) P 12cm S (e) 2xcm D (f) A 4xcm B
E 2xcm
H ycm G D ycm C
C xcm xcm

Q 4xcm 2xcm 2xcm 2xcm 2xcm
R A 2xcm B F ycm E

5. (a) Find the value of x + y if x = 5cm and y = 3cm.

(b) Find the value of 3a - 2b if a = 4cm and b = 3cm.

(c) Find the value of x if y = 4 and x + y = 9
p + q
(d) Find the value of p if q = 5 and 3 = 4.

Like and unlike terms:

Hari, can you tell, Yes, sir ! In 3x and
what is alike in 3x 5x, the variable

and 5x ? (base) x is the same.

So, the algebraic terms having
the same variable (base) are
called like terms. For example;

5x, 4x, -7x etc.

204 Prime Mathematics Book − 4

Hari, are 7ab and Yes, sir ! 7ab and 12ab are
12ab like terms ? also like terms, because, the
Good !
variables (base) ab in 7ab
and 12ab are same.

Hari, are 2x and 3y No, 2x and 3y are not
like terms ? like terms because the
variables in both are
Very good ! Such
terms are called unlike different.

terms.

Thus, the algebraic terms having the different variables (base) are called
unlike terms. For example; 2x, a, 5y, 3ab, etc.

Hari, can you say what are
like and unlike terms?

Yes, sir ! The terms having the same
variable (base) are like terms and
different variable (base) are unlike

terms. For example, the like terms are
x, 3x, 4x and the unlike terms are
x, 2a, 5b

Exercise: 10.6

1. State whether the following terms are like or unlike:

(a) y, 3y (b) 2ab, 3ab (c) x, b
(d) 2x, 5y, 3z (e) 2ab, 3bc (f) pqr, prq
(g) 2xuz, 3xyz, 5xyz (h) 2b, 2a, 2c (i) 3abc, 2ab, 6bc

2. List the like terms from the following:
(a) 2x, 3y, a 4x, 2a, 5y, x, 9y
(b) 6a, 7b, 6c, 9a, 13b, 4c, 21b, 19a
(c) xy, yz, ax, 3yz, 2ax, 3xy
(d) 7ab, 5xy, 3bc, 8ab, 21xy, 2ab, bc, 4xy

Prime Mathematics Book − 4 205

Addition and subtraction of like terms

Ram, you have 3 apples. I
gave you 5 apples again. Now,
how many apples do you have

altogether ?

Let an apple be Sir, now I have 8 apples.
denoted by x. 3 apples + 5 apples
= 8 apples
xx xx x
Then, 3 apples = 3x
xxx x x x x x 5 apples = 5x
3x + 5x x x x
∴ 3apples + 5 apples
= 8x = 3x + 5x
= 8x

Ram, do you know Yes, sir ! I know.
that addition of like The addition of like
terms means we add the
terms ?
Very good ! coefficients of
like terms.

Rita, can you Yes sir, I can!
subtract 2x from 7x - 2x

7x ? = 5x

The
subtraction of like terms means,

we subtract the coefficient of
like terms.

xx

x x x x x xx 7x- 2x = 5x xxxxx

206 Prime Mathematics Book − 4

Class Work

1. Add the following: (b) 2x and 5x (c) 9b and 7b
(a) 2a and 4a = =
= 2a + 4a =
= 6a =
(f) 8yz and 4yz
(d) 3xy and 8xy (e) 5ab and 7ab =
= = =
(i) -x and -4a
== =
=
(g) 3a, 5a and 7a (h) -2a and -4a
(c) 5x from 12x
== =
=
== (f) 6xy from 19xy
2. Subtract the following: =
=
(a) 3a from 5a (b) 2b from 9b
(i) 2x from -6x
== =
=
==

(d) 4y from 7y (e) 2ab from 9ab

==

==

(g) 5bc from 13bc (h) x from -4x
= =
= =

Addition and subtraction of unlike terms

Rita is 5x + 4y = No, sir ! We can’t
9xy? add 5x and 4y.

Why? 5x and 4y are unlike
terms. So, the coefficient
of x and y can’t be added.

Therefore, the addition of 5x and 4y is 5x + 4y. The addition of unlike
terms means the unlike terms are just connected by ‘+’ sign. Similarly, the
subtraction of unlike terms means the unlike terms are just connected by ‘-’
sign.
For example; subtraction of 3x from 5y means 5y - 3x.

Prime Mathematics Book − 4 207

Exercise: 10.7

1. Do the following sums: (b) 2a + a
(d) 3y + 7y
(a) a + a (f) 17b + 6b
(h) 9bc + 15bc
(c) 2x + 5x (j) 6ab + 9ab
(e) 8x + 3x (l) 7a + 3a + 15a
(g) 2xy + 5xy (n) 5xy + 9xy + 21xy
(i) 12abc + 27abc (p) -12abc - 3abc

(k) x + 2x +5x
(m) 12abc + 27abc

(o) -4a - 5a

2. Do the following sums :

(a) 5a - 3a (b) 6x - 2x (c) 7y - 4y

(d) 26z - 12z (e) 18p - 12p (f) 29b - 14b

(g) 12m - 8m (h) 9xy - 5xy (i) 28ab - 13ab

(j) 32bc - 21bc (k) 12xyz - 5xyz (l) 39abc - 14abc

(m) 17pqr - 12pqr (n) 2a - 5a (o) 8xy - 15xy
3. Add the following:

(a) 2x and 3x (b) 5a and 7a

(c) 6a and 3b (d) 2x and 5y

(e) 4x, 5x and 7x (f) 3a, 4a, 6a an 7a

(g) 2xy, 7xy and 9xy (h) 15abc, 12abc, 9abc and 6bc

(i) 2a, 4a, 9a and 4x (j) -4a, -2a and -8a

4. Subtract the following:

(a) 4x from 7x (b) 9abc from 15abc (g) 5x from -8x
(h) 7x form -10x
(c) 12b from 25x (d) 6xy from 13xy

(e) 4mn from 19 mn (f) 8yz from 12xy

5. Write down the total length of each of the line segments:

(a) A 2xcm C 3xcm B (b) P xcm Q 4xcm R

(c) M 5acm N 7acm P (d) A 2ycm B 4ycm C 3ycm D

(e) P acm Q 6acm R 3acm S (f)A 2xcm C xcm D 3xcm E 2ycmB

208 Prime Mathematics Book − 4

6. If x = 3cm, find the actual length of MN in each line segment:

(a)M 7x N 2x O (b) M 6x O
N 9x

(c) M x (d) M 12x
O
12x N O 30xN

7. If y = 3cm and b = 2cm, find the actual perimeter of the following
figures:
6b D
(a) A 3y (b) A

2y C 2b 2b

B 5y B 6b C

(c) 3y A 3y (d) 2b b
BE 7b 6b
3y 3y
C 3y D 9b

Simplification

Rita, Can you simplify Yes sir, I can ! First I arrange the
7a - 2a + 3a - 4a ? terms having same sign together.
Then simplify by using the sign.
Ok ! Show your
solution.

Very good. 7a - 2a + 3a - 4a
= 7a + 3a - 2a - 4a
= 10a - 6a
= 4a
Rita, Can you simplify
5x - 2y - 3x + 5y ?
Yes sir, I can ! First I
arrange the like terms
together. Then, simplify by
Very good! using the sign.

5x - 2y - 3x + 5y
= 5x - 3x + 5y - 2y
= 2x + 3y

Prime Mathematics Book − 4 209

Class Work

Simplify: (b) 2z - 4a + 9a - 3a (c) 8y + 5z - 4y + 6z
(a) 5x - 3x + 9x = =
= = =
= = =
= (e) 6a - 2b - 4a + 5b (f) 7x - 3x + 9x - 5x
= =
(d) 5x + 7x - 4x = =
= = =
=
=

Exercise: 10.8

1. Simplify: (b) 2x + 7x + 5x
(a) 3a + 5a (d) 9y + 8y - 5y
(c) 4x - 5x + 7x (f) 12b + 5b - 4b - 3b
(e) 14y - 3y - 8y (h) 8c - 9d + 15c
(g) 15x - 7x + 5y + 4y

2. Simplify:

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

(c) 9b + 7b - 8c + 19c (d) 8x + 5z - 4x + 6z

(e) 2a + 4b - 7 + 5a + 2 + 3b (f) 11x - 3y - 8x + 7y

(g) 8x + 5z - 4x + 6z (h) 15x + 6y + 3x - 4z - 2y + 7z

Addition and subtraction of algebraic expressions
Study and learn from the following examples how to add or subtract the
algebraic expressions having two or more terms.

Example 1: Add: 7x + 5y and 4x + 3y.
Solution: Vertical arrangement method:

7x + 5y

4x + 3y First, we arrange the given algebraic
11x + 8y expressions in vertical column with
like terms in the same column. Then,
add the coefficients of like terms.

210 Prime Mathematics Book − 4

Horizontal arrangement method:

(7x + 5y) + (4x + 3y)

= 7x + 5y + 4x + 3y First, we write the given algebraic
= 7x + 4x + 5y + 3y expressions in separate brackets with ‘+’
= 11x + 8y
sign. Then, we open the brackets and
arrange the like terms together. Then,

add the coefficients of like terms.

Example 2: Subtract 5x + 2y from 9x + 7y.
Solution: Vertical arrangement method:

9x + 7y
(+)5x (+) 2y

4x + 5y

First, we arrange the given algebraic
expressions in vertical column with like
terms in the same column. Then change the

sign of the terms of the expression
(5x + 7y) which is to be subtracted. Then
add or subtract the coefficient of like terms

and put ‘+’ or ‘-’ sign.

Horizontal arrangement method:

(9x + 7y) - (5x + 2y) First, we write the given algebraic
expressions in separate brackets with ‘-’ sign.
= 9x + 7y - 5x - 2y

= 9x - 5x + 7y - 2y Then, we open the brackets and arrange the like
terms together. At last, we add or subtract the
= 4x + 5y coefficient of like terms.

Exercise: 10.9

1. Add the following algebraic expressions by vertical arrangement
method.

(a) 3x + 7y and 5x + 2y (b) 9x + 3y and 6x + 11y
(c) 7a + 3b and 5a + 2b (d) 4b + 9c and 3c + 5b

(e) 5p + 7q and 9q + 3p (f) 7x - 3y and 2x + 5y

(g) 4x + 7y - 3 and 5y + 2x + 7 (h) 8a - 2b + 4c and 3b + 2a - 3c

Prime Mathematics Book − 4 211

2. Add the following algebraic expressions by horizontal arrangement
method:

(a) 5a + 3b and 6a + 7b (b) 5p + 7q and 3p + 2q

(c) 11x + 13y and 3x + 5y (d) 17p + 12q and 5q + 3p

(e) 7m + 5n and 3m - 2n (f) 8a - 3b and 3a - 5b

(g) 2a + 3b + 4 and 3b + 5a - 2 (h) 4x - 3y + 5z and 7y + 3x - 2z

3. Subtract by vertical arrangement method:
(a) 3a + 4b from 5a + 7b
(b) 4x + 7y from 9x + 15y
(c) 2x - 3y from 8x + 5y
(d) 5m - 4n from 7m - 2n
(e) 4p - 3q from 7p + 6q
(f) 10a + 12b - 3 from 15a - 2b + 7
(g) 7x - 3y from 19x - 9y
(h) 12x - 5y + 2z from 17x + 2y - 3z
(i) -3x + 4y -5z from -4x + 5y - 6z

4. Subtract by horizontal arrangement method:
(a) 2x + 3y from 5x + 7y
(b) 3a + 2b from 7a + 5b
(c) 7a - 5b from 15a - 2b
(d) 5m + 8n from 10m - 2n
(e) 5y + 3x from 7x - 4y
(f) 4x + 3y - 2 from 8x - 2y + 7
(g) 10p + 6q from 21p + 13q
(h) -4x +5y from -5x - 6y
(i) -x - y from -2x - 5y

212 Prime Mathematics Book − 4

Open Sentence

The sum of 9 and
6 is 15.
This statement is true

So, this sentence is called a
true sentence.

The difference of

26 and 15 is 14. This statement is false.

So, this sentence is called a
false sentence.

Such numerical sentences which might be true or false are called mathematical
sentences.

The sum of x and 6
is 14. Sir, which sentence is
this?
We can’t say this
statement is true or false. So,
this sentence is called and
open sentence.

Mathematically it can be written as
x + 6 = 14.
This is true when x = 8, otherwise it is safe. Some more examples of open
sentences are x + 3 > 12, x + 2 < 6, x - 4 > 6, x - 2 = 7 etc. Among these open
sentences, x - 2 = 7 is called an equation.

Exercise: 10.10

1. State whether the following sentences are true, false or open
sentence:
(a) The sum of 12 and 5 is 17.
(b) The sum of 25 and 12 is 47.
(c) The difference between x and 5 is 17.

Prime Mathematics Book − 4 213

(d) The difference between 19 and 8 is 11.
(e) The product of 5 and 6 is equal to the sum of x and 19.
(f) 20 is exactly divisible by 5.
(g) The product of 9 and 7 is an odd number.
(h) A quadrilateral is a closed figure bounded by four straight line

segments.
(i) 2x + 9 = 15.
(j) The product of 6 and 7 is greater than 40.

2. Make each of the following open sentences true by putting the
correct number in the boxes:
(a) The sum of 32 and 17 is .

(b) is a prime number greater than 15 and less than 19.

(c) The difference between 39 and 12 is .

(d) × 5 = 45 (e) + 19 = 58

(f) ÷ 6 = 12 (g) 17 × = 119

(h) - 14 = 16 (i) 24+ = 59

(j) ÷ 23 = 3

3. Make each of the following open sentences true by substituting the
correct value of the variables:

(a) x + 7 = 21 (b) 15 – x = 7

(c) 9 + x = 21 (d) y – 10 = 18

(e) 5 × y = 40 (f) 16 ÷ z = 4

(g) z ÷ 6 = 9 (h) x + 14 = 40

214 Prime Mathematics Book − 4

Equation

In the given figure alongside, the weight on
the left hand side is more than on the right
hand side. That is why the balance is tilting
towards the left. We know that 5kg + 1kg is
not equal to 2kg + 2kg.

5kg 1kg 2kg 2kg

Let’s take 1kg, weight from the left hand

side and place it on the right hand side. Now

the balance is straight. This means that the

weights on the both sides are equal.

So, 5kg = 2kg + 2kg + 1 kg

5kg 2kg 2kg 1kg

In the third figure alongside, the balance is
straight. It means the weights on the both
sides are equal

So, x + 2kg = 5kg xkg 2kg 5kg

This is an open sentence. Such open sentence is also known as an equation.

This open sentence x + 2 = 5 is true only when x = 3. So the value 3 of

variable x is called the solution of the equation.

An open sentence that

contains the equal to sign (=) is
called an equation.
I have Rs.25 .How

much money do you
If Rs. 12 is added to have ?
the money which I have,
then the total money will
also be equal to Rs.25.

Ok! If you have Rs.
x and we add Rs.12 to it, then it

will be Rs. 25 so, x+12 = 25

Prime Mathematics Book − 4 215

x + 12 = 25 is an open sentence which is also called an equation. If we put
x = 13, then the equation becomes

13 +12 = 25
Or, 25 = 25, which is true.
So, the correct value of the variable x in this equation is 13.
Hence, Shyam has Rs. 13.
In the equation x+12 = 25 , the value of x is only one i.e. 13. So, 13 is called
the solution of the equation.

Class Work
1. Find the values of the variables in the following equations by gvessing

so that the equations are true.

(a) x + 5 = 9, ∴x=

(b) x + 7 = 15, ∴x=

(c) x - 4 = 5 , ∴x=

(d) y – 7 = 6, ∴y=

(e) y + 3 = 21, ∴y=

(f) z + 6 = 17, ∴z=

Solving of the equations
The process by which we find the value of the variable contained in the given
equation so that the equation is true is called the solving of the equations.
Study and learn the following examples.

Example 1: Solve: x + 3 = 9 To find the value of x,
Solution: Here x + 3 = 9 3 should be taken away from the

or, x + 3 - 3 = 9 - 3 left side of the equation, i.e.
∴x=6 x+3. So,3 is subtracted from both

sides of the equation.

Example 2: Solve x - 4 = 19 To find the value of x, 4
Solution: Here, x - 4 = 19 should be taken away from x - 4.

or, x - 4 + 4 = 19 + 4 So, 4 is added to both sides of
∴ x = 23 the equation.

216 Prime Mathematics Book − 4

Example 3: Solve : 4y = 20 To find the value of y, the
Solution: Here 4y = 20 coefficient 4 of 4y should be
taken away so, both sides of the
or, 4y = 20 = 5
4 4 equation is divided by 4.

∴y=5

Example 4: Solve y =7 y 7 To find the value of y, the
2 2 = tadkeennoamwianya.toSor ,2boofth2y should be
7 sides of the

y equation is multiplied by 2.
2
Solution: Here, = 7 77

y y × 2 = 7×2
2 2

or, × 2 = 7 × 2 i.e. y = 14

∴ y = 14 To find the value of x, first 4
should be taken away from 3x + 4.
Example 5: Solve : 3x + 4 = 22 So, 4 is subtracted. Then 3 is to be
Solution: Here 3x + 4 = 22 taken away from 3x. So, the both

or, 3x + 4 - 4 = 22 - 4 sides of 3x = 18 is divided by 3.

or, 3x = 18 6
3x 18
or, 3 = 3

∴x=6

Exercise: 10.11

1. Solve the following equations:

(a) x+2 = 7 (b) x+5 = 13 (c) x+3 = 10
(f)
(d) x+7 = 20 (e) x- 3 = 17 (i) Y-2 = 19
(l)
(g) Z-7 =15 (h) 3x = 15 5y = 35

(j) x+9 = 21 (k) 3y = 21 Z – 17 = 6

(m) x = 4 (n) 7m = 42 (o) z = 9
6 4y = 13
7
(p) Y + 21 = 32 (q) 9x = 72 (r)

2. (s) x – 9 = 35 e(qtu) ation1x3s:= 4 (u) -2x + 3 =5
solve the following

(a) 2x+1 = 7 (b) 3y3x+– 2 = 13 (c) 2x = 5 = 25
(d) 4y – 7 = 21 (e) 2 = 5 (f) 2xx-4-
3 = 5
(g) 7x + 10 = 31 (h) 6z – 5 = 43 (i) 2 = 7

(j) 3y = 21 (k) 3x + 5 = 13 (l) 2x - 12 = 7
5 2 4

Prime Mathematics Book − 4 217

Word Problems

In the given word problems, we need to find the value of the unknown
quantity. To find the value of unknown quantity , we should consider the
unknown quantity as a variables like x, y, z etc. The given word problems
should be expressed in mathematical equation. Then we use the process of
solving equation to find the value of variable which is the unknown quantity
of the given problems. Study and learn the following examples:

Example 1: The sum of two number is 32. If one of the numbers is 17,
find the other number.

Solution: Let the other number be x. The given number is 17, and sum of
two numbers is 32.
So, x + 17 = 32
or, x + 17 – 17 = 32 - 17
or, x = 15
Hence, the other number is 15.

Example 2: Ram is 6 years older than his sister. If Ram is 14 years old,
how old is his sister?

Solution: Let his sister’s age be y years.
Ram’s age is 14 years and he is 6 years older than his sister.
So, y + 6 = 14
or, y + 6 - 6 = 14 – 6
or, y = 8
Hence, his sister‘s age is 8 years.

Example 3: Find the value of x from the given figure.
15cm

Solution: x-3 7
From the given figure,

x – 3 + 7 = 15
or, x + 4 = 15
or, x + 4 - 4 = 15 - 4
∴ x = 11

218 Prime Mathematics Book − 4

Exercise: 10.12

1. Write the equation of the following word problems and solve them:
(a) If the sum of x and 5 is 19, find the value of x.
(b) If three times y is added to 7, the sum becomes 19, find the value
of y.
(c) The difference of two numbers is 9. If one of the smaller numbers
is 7, find the bigger number.
(d) A number increased by 12 is 23. Find the number.
(e) 7 times a number is 49. Find the number.
(f) If a number is divided by 7 , the result is 5, what is the number?

2. Find the value of the variables in the given problems:

(a) If x-5 = 9 , what is the value of x ?
2x-3
(b) If x+251= = 3, what is the value of x ?
(c) If 7 , what is the value of x ?

3. Write the equation of the following word problems and solve them:

(a) There are 40 students in a class. If the number of boys is 25 and
the number of girl is x, find the number of girls.

(b) Rajesh is 7 years younger than Mohan. If Rajesh is 19 years old,
how old is Mohan?

(c) Adarsh had few marbles. When he bought 12 more marbles, he had
27 marbles altogether. How many marbles did he have at first?

(d) Ansari is 13 years old. Bijesh is 6 years older than Ansari. How old
is Bijesh ? Find it out.

(e) If 7 is added to the double of a number, then the result is 19. Find

the number.

4. Make the equations from the given figures and solve them:

(a) 12cm (b) 10cm
2x 4cm
x+2 x-4

(c) 17cm (d) 15cm
y 2y 3x+1
2cm x+2

Prime Mathematics Book − 4 219

(e) x+2 5cm (f) 3x+2 x+4
x+4
Perimeter Perimeter
=14cm =26cm

x-1 3x+2

Unit Revision Test

1. If a = 2 and b = 3, find the value of
(a) 3a + 2b - 5 (b) 5b - 2a + 2ab

2. Identify the base and coefficient in
(a) 7x (b) 2(a+b)

3. Add: (b) 2ab , 7ab, 6ab (c) 2x + 7y and 5x + 2y
(a) 7x, 3x, 5x

4. Subtract : (b) 5ab from 12ab (c) 3x + 2y from 9x + 7y
(a) 4x from 9x

5. Simplify : (b) 19y - 4y + 3y - 6y (c) 14x + 4y - 3x + 2y
(a) 5x + 7x - 3x

6. Solve : (b) y - 5 = 14 (c) 3x – 2 = 13
(a) x + 9 = 19

(d) y - 5 = 12 (e) 2z + 7 = 5 (f) 2x - 5 = 5
3 3 7
7. Write the equations and solve them:
(a) The sum of two numbers is 29. If one of the numbers is 13,find
the other number.
(b) Rinku is 4 years older than Maya. If Maya is 13 years old, how old
is Rinku ?
(c) A boy had some money. If he spent Rs.15 to buy a copy, then he
had Rs. 12 left. How much money did he have at first?

8. Find the value of x from the given figures: 2x

(a) 19cm (b)

2x+1 x 2x Perimeter 2x
= 24cm

2x

220 Prime Mathematics Book − 4

Answers:

Exercise 10.1

1. Show to your teacher.

Exercise 10.2

1) a) a + 6 b) x - 4 c) x × 4 d) 4 × y e) 323xz + 7 43
h) 3x - 12 i) 3(x + 4) j) k)
f) 9-2y g) 4b + 5 y-7

2) x - 5 3) y + 7 4) x - 5 5) Rs. 7x 6) 3x + 7 7) 5x

Exercise 10.3

Base Coefficient Base Coefficient

1. x 5 6. . ab -1

2. y 3 7. . abc 4

3. x -2 8. x 72

4. x 1 9. y -54

5. xy -15 10. (a + b) 3

Exercise 10.4

1. Show to your teacher.

2. Show to your teacher.

3. a) Monomial b) Binomial c) Binomial d) Monomial e) Trinomial
i) Trinomial
f) Multinomial g) Multinomial h) Trinomial
d) (2x + 3z) - 5 e) x + 5
4. Show to your teacher

5. a) 30 - x b) x + 4y c) 20 - y

Exercise 10.5
1. a) 6 b) 6 c) 20 d) 6 e) 12 f) 36 g) 8 h) 32 i) 12 j) 30

2. a) 8 b) 1 c) 9 d) 7 e) 5 f) 10 g) 9 h) 18 i) 26 j) 1
k) 9 l) 11 m) 7 n) 10

3. a) 9cm b) 8cm c) 8cm d) 9cm e) 17 cm f) 25 cm
4. a) 8cm b) 16cm c) 38cmd) 48cm e) 30 cm f) 36cm
5. a) 8cm b) 6cm c) 5 d) 7

Exercise 10.6
1. a) like b) like c) unlike d) unlike e) unlike f) like g) unlike h) unlike i) unlike

2. Show to your teacher.

Exercise 10.7

1. a) 2a b) 3a c) 7x d) 10y e) 11x f) 23b g) 7xy h) 24bc
n) 35xy o) -9a p) -15abc
i) 39abc j) 15ab k) 8x l) 25a m) 39abc

Prime Mathematics Book − 4 221

2. a) 2a b) 4x c) 3y d) 14z e) 6p f) 15b g) 4m h) 4xy
o) -7xy
i)15ab j) 11bc k) 7xyz l) 25abc m) 5pqr n) -3a g) 18xy

3. a) 5x b) 12a c) 6a + 3b d) 2x + 5y e) 16x f) 20a f) 12xy - 8yz

h) 36abc + 6bc i) 15a + 4x j) -14a

4. a) 3x b) 6abc c) 25x - 12b d) 7xy e) 15mn

g) -13x h) - 17x

5. a) 5x cm b) 5xcm c) 12a cm d) 9y cm e) 10a cm f) 6x + 2y cm

6. a) 15cm b) 9cm c) 33cm d) 54 cm

7. a) 30cm b) 32cm c) 45cm d) 50cm

Exercise 10.8

1. a) 8a b) 14x c) 6x d) 12y e) 3y f) 10b g) 8x + 9y
d) 4x + 11z e) 7a + 7b - 5
h) 23c - 9d

2. a) 11a+b b) 3x + 4y c) 16b + 11c
h) 18x + 4y + 3z
f) 3x + 4y g) 4x + 11z

Exercise 10.9

1. a) 8x + 9y b) 15x + 14y c) 12a + 5b d) 9b + 12c e) 8p + 16q f) 9x + 2y

g) 6x + 12y + 4

h) 10a + b + c

2. a) 11a + 10b b) 8p + 9q c) 14x + 18y d) 20p + 17q

e) 10m + 3n f) 11p - 8b

g) 7a + 6b + 2 h) 7x + 4y + 3z

3. a) 2a + 3b b) 5x + 8y c) 6x + 8y d) 2m + 2n e) 3p + 9q

f) 5a - 14b +10 g) 12x - 6y h) 5x + 7y - 5z i) -x + y - z

4. a) 3x + 4y b) 4a + 3b c) 8a + 3b d) 5m - 10n e) 4x - 9y

f) 4x - 5y + 9 g) 11p + 7q h) -x -11y i) -x - 4y

Exercise 10.10

1. a) true b) false c) open d) true e) open f) ture g) true h) true i) open
j) true.
b) 17 c) 27 d) 9 e) 39 f) 72 g) 7 h) 30 i) 35 j) 69
2. a) 49 b) 8 c) 12 d) 28 e) 8 f) 4 g) 54 h) 26
3. a) 14

Exercise 10.11

1. a) 5 b) 8 c) 7 d) 13 e) 20 f) 21 g) 22 h) 5
i) 7 j) 12 r) 91 s) 44 t) 52
m) 24 k) 7 l) 23 f) 18 g) 3 h) 8
u) -1 n) 6 o) 36
p) 11 q) 8
2. a) 3 b) 5 c) 15
i) 23 j) 35 k) 7 d) 7 e) 9
l) 20

Exercise 10.12

1. a) 14 b) 4 c) 16 d) 11 e) 7 f) 35

2. a) 14 b) 9 c) 13

3. a) 15 b) 26 c) 15 d) 19 e) 6 f) 47
4. a) 7 b) 3 c) 5 d) 3 e) 4

222 Prime Mathematics Book − 4

Model Question

Final Examination

Subject : Mathematics F.M. : 100
Grade : IV P.M.: 10
Time : 2hrs.

Group ‘A’ (10×1 = 10)

1. Choose the correct answer.

a) If two sides of a triangle are equal, it is called ............

(i) an isosceles triangle ii) an equilateral triangle

iii) a scalene triangle.

b) The algebraic expression for the sum of x and 1 is 3 is ...................

i) x-1 = 3 ii) x + 1 = 3 iii) 1x = 3

c) If x - 2 = 3, then the value of x is .............

i) 5 ii) 3 iii) 1

d) There are ............ grams in 1 kg.

i) 10 ii) 100 iii) 1000

e) The formula to calculate the volume of cube is ............

i) l×l×l ii) l+l+l iii) l×b×h

f) The cost of 1 book is Rs. 100, what is the cost of 5 books ?

i) Rs. 50 ii) Rs. 500 iii) Rs. 5

g) An angle which is equal to 90o is called .............. angle

i) Acute ii) abtuse iii) right.

h) The product of 0.5×3 is ...............

i) 15 ii) 1.5 iii) 0.15

i) The smallest 5 digit number is .............

i) 55555 ii) 10000 iii) 99999

j) The place value of 8 in 8790 is ..............

i) 8000 ii) 80 iii) 8

Prime Mathematics Book − 4 223

Group ‘B’ (17×2 = 34)

2. Draw an angle of 70o.
3. Convert 85 into mixed fraction.
4. Add : 12 + 15

5. Convert : 5kg into grams.

6. Write the Roman number for 68.

7. Write the expanded form of 897532.

8. Multiply : 0.7×3

9. Simplify : 8÷4+3

10. Find the HCF of 12 and 15.

11. Add : 0.87 + 0.68

12. Find the value of x + y + z if x = 2, y = 1, z = 3.

13. Solve : x - 3 = 5

14. Add : 9L 460 ml + 3L 875ml
15. Convert into percentage : 25
16. Represent by listing method. A= {odd number less than 10}

17. Find the area of rectangle whose length is 5m and breadth is 3m.

18. Write the number name in international system.

8, 729, 462

Group ‘C’ (14×4 = 56)

19. Solve : 4y - 3 = 9 b) 4 0 8 3 4 5
20. Perform the following -2 9 9 3 5 7

a) 1 4 4 5 2
+8 2 5 4 8

224 Prime Mathematics Book − 4

21. The cost of a chair is Rs. 575.
What is the cost of such 75 chairs ?

×
.............................
+ ............................
.............................

22. The cost of 9 watches is Rs. 7020.
What is the cost of 1 watch ?
Quotient = .....................

23. Simplify : {48 - (5×12÷4)}+ 27
24. Find the volume of cuboid whose length is 5cm, breadth is 4cm and height is 3cm.
25. If y = 2cm, find the actual perimeter of the following triangle.
26. Simplify : 15x + 6y + 3x - 4z - 2y + 7z
27. Simplify : 54 x 1165 x 31

28. Subtract: 20l 330ml from 23l 250 ml

29. find the LCM of 15 and 20.

30. The table given below shows the marks obtained by Bibek in an examination in various subjects.

showthe bar graph

Subject English Nepali Maths Science Social

Mrks 60 50 80 70 40

31. Measure the lengths of sides of the following triangle and write there type

(e) A AB=

BC=

CA=

B C ABC is ................................................

Prime Mathematics Book − 4 225

32. Write the ordered pair numbers of the points shown in the graph. (4)

Y

C
B

X’ O AD X

Y’

Some additional questions.

1) Classify the following angles as right angle, acute angle, abtuse angle, straight angle and reflect

angle.

a) 72° b) 172° c) 342° d) 120° e) 90° f) 180° g) 155° h) 21o i) 810°

j) 28° k) 102° l) 181°

2) Write in expanded form :

a) 6,98,24,034 = + + ++

+ ++

b) 3,12,80,56,254 = + + + +

+++

3) Write in short form.
a) 5000000 + 700000 + 80000 + 70 + 8 =

b) 40000000 + 500000 + 300 + 40 + 1 =

4) Find the HCF and LCM of the given numbers.

25 and 30

5) Write the given Hindu-Arabic numerals in Roman numerals :

a) 795 b) 192

6) Write the given Roman numerals in Hindu-Arabic numerals :

a) CCIX b) DCLXII

7) Write the greatest and smallest numbers of 5 digits formed by using the digits 7, 9, 0, 8, 2 : Then
find their sum.

8) Find the difference between the greatest number and the smallest number of 6 digits by using
the digits 7, 3, 4, 0, 5, 1 .

226 Prime Mathematics Book − 4

9) A cartoon of soaps contains 30 packets of soaps. If each packet contains 12 soaps, how many
soaps are there in such 4 cartoons ?

10) 648 plants are planted in 8 columns. How many plants are planted in each column ?

11) Simplify :

42 - 24 × 88 ÷ 11 + 25

12) Simplify :

25 - [20 - {3 × 6 - (5 × 3) + 6} + 9] + 4

13) Add :

3 hours 45 minutes and 5 hours 30 minutes.

14) Subtract :

6 hours 15 minutes from 8 hours 37 minutes.

15) Perform the following :

years months days

13 4 20

+8 8 15

16) Perform the following :

years months days

87 4

-3 6 24

17) Add :

Rs. P

15 75

+9 50

18) Subtract :

cm mm

12 4

-4 8

19) Find the perimeter of :

4cm 6cm

8cm

Prime Mathematics Book − 4 227

20) Find the perimeter of the rectangle having length 7cm and breadth 5 cm.

21) Find the area of the rectangle having length 8cm and breadth 6cm.

22) Calculate the volume of the cuboid having length 2cm, breadth 3cm and height
4cm.

23) Perform the following :

8 - 5 + 1 3
3 3 3

24) Find any two equivalent fraction of 5 .

25) Calculate the size of angles shown by the letter.

80o

x 45o

26) Simplify :

5.169 + 3.258 - 2.805

27) If 10 kg of potatoes cost Rs. 250. Find the cost of 1 kg of potatoes.

28) 48 out of 60 students of a class are passed. What percent of the students are
passed ?

29) Represent the following data on bar graph.

The number of students enrolled in classes 1-5 are as follows :

Class I II III IV V
45
No. of students 20 40 25 30

30) a) Simplify :

15x + 6y + 3x - 4z - 2y + 6z

b) Subtract :

3a + 5b from 5a + 8b

31) Solve :

a) 7x + 10 = 31

b) 2x + 5 = 5
3

228 Prime Mathematics Book − 4

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