NCERT Exemplar Solutions: Algebra | Mathematics (Maths) Class 6 PDF Download

Ans: (b)
From the question it is given that, number of match sticks in each match box = 50
Then, the number of matchsticks required to fill n such boxes = 50 × n
= 50n

Ans: (c)

Current age of Amulya = x years
Then, 5 years ago her age was = (x – 5) years

Ans: (a)
Here, 6 × x is equivalent to 6x.
So, option (A) is correct.

Ans: (c)
An expression with a variable, constants and the sign of equality (=) is called an equation.

Ans: (b)
From the question it is given that, value of x is 2.
Now substitute the value of x in x + 10
= 2 + 10
= 12

Ans: (b)
We know that, perimeter of hexagon = number of sides × length of each side
Given, the perimeter of a regular hexagon is x metres
Then, the length of each of its side = (x/6) metres
= (x ÷ 6) metres

Ans: (b)
Transforming – 2 from left hand side to right hand side it becomes 2.
Then, x = 2

Ans: (a)
Let us assume a and b are the two integers,
Then, commutative law of addition = a + b = b + a
Where, a = x, b = y
Therefore, commutative law of addition = x + y = y + x

Ans: (c)
Consider the equation, 2x + 1 = 6
Transforming 1 from left hand side to right hand side it becomes -1.
2x = 6 – 1
2x = 5
X = 5/2

Ans: (c)
In arithmetic 3 × 5 is given by 15.
So, option (C) is correct.

Ans: (b)
In algebra, when a quantity is not known it is represented by using a letter.
So, option (B) is correct.

Ans: (a)

The word ‘variable’ means something that can vary, i.e., change. The value of a variable is not fixed. We use a variable to represent a number and denote it by any letter such as l, m, n, p, x, y, z etc.

Ans: (c)
10 – x means that x is subtracted from the number 10.
So, correct option is (C).

Ans: (a)
From the question it is given that,
Savitri has a sum of Rs x
She spent money on grocery = ₹ 1000
She spent money on clothes = ₹ 500
She spent money on education = ₹ 400
She received gift = ₹ 200
Total money spent by Savitri = 1000 + 500 + 400 = ₹ 1900
Then,
Total money left with her after deducting = ₹ (x – 1900)
Therefore, money left with her after adding gift money = (x – 1900) + 200
= x – 1700

Ans: (a)
Given triangle is an isosceles triangle,
So, perimeter of an isosceles triangle = 2 × x + y
= 2x + y

Ans: (a)
We know that, area of square = side * side
Given, square having a side x.
So, area of a square = x * x
= x²

Ans: (c)
From the question it is given that,
X is multiplied by 2 = x × 2 = 2x
Then, x is multiplied by 2 and then subtracted from 3 = 3 – 2x

Ans: (a)
Consider the given equation q/2 = 3
By cross multiplication we get, q = 6

Ans: (b)
Consider the given equation x – 4 = -2
Transform – 4 from left hand side to right hand side it becomes 4.
x = – 2 + 4
x = 2

Ans: (a)
4/2 = 2
By cross multiplication we get,
4 = 4

Ans: (a)
From the question it is given that,
Kanta has p pencils in her box
She puts q more pencils in the box
The total number of pencils with her are = p + q

Ans: (a)
Consider the given equation 4x = 16
Then, value of x is,
x = 16/4 … [divide both numerator and denominator by 4]
x = 4

Ans: (d)
Let us assume the number be ‘x’,
Then, adding 13 to the number = x + 13
Therefore, x + 13 = 27

The distance (in km) travelled in h hours at a constant speed of 40km per hour is 40h.
From the question,
Time taken to travel a distance = h hours
Travel at a speed of 40 km/h
So, total distance travelled = 40 × h
= 40h

p kg of potatoes are bought for Rs. 70. Cost of 1kg of potatoes (in Rs.) is 70/p.
Given, p kg of potatoes are bought for Rs. 70
Then, cost of 1 kg of potato = 70/p

An auto rickshaw charges Rs 10 for the first kilometre then Rs. 8 for each such subsequent kilometre. The total charge (in Rs.) for d kilometres is 2 + 8d.
From the question it is given that,
An auto rickshaw charges Rs. 10 for the first kilometre
Then Rs. 8 for each such subsequent kilometre.
So, the total charge (in Rs) for d kilometres is = 10 + (d – 1)8
= 10 + 8d – 8
= 2 + 8d

If 7x + 4 = 25, then the value of x is 3.
Consider the equation, 7x + 4 = 25
Transposing 4 from left hand side to right hand side it becomes -4,
7x = 25 – 4
7x = 21
x = 21/7
x = 3

The solution of the equation 3x + 7 = –20 is -9.
Consider the equation, 3x + 7 = –20
Transposing 7 from left hand side to right hand side it becomes -7,
3x = – 20 – 7
3x = – 27
x = -27/3
x = – 9

‘x exceeds y by 7’ can be expressed as x – y = 7.

‘8 more than three times the number x’ can be written as 3x + 8.
As per the condition given in the question, three times the number x = 3x
So, 8 more than three times the number x = 3x + 8

Number of pencils bought for Rs x at the rate of Rs 2 per pencil is x/2.
From the question it is given that, cost of pencil = ₹ x
No. of pencils bought for ₹ 2 = 1
Therefore, number of pencil bought for ₹ x = ₹ x/2

The number of days in w weeks is 7w.
We know that, there are 7 days in a week.
Therefore, number of days in w weeks is 7w.

Annual salary at r rupees per month along with a festival bonus of Rs. 2000 is 12r + 2000.
From the question it is given that,
Salary per month is r rupees
A festival bonus of ₹ 2000
Therefore, Annual salary at r rupees per month along with a festival bonus of Rs 2000 is ₹ 12r + 2000

The two digit number whose ten’s digit is ‘t’ and units’s digit is ‘u’ is 10t + u.
From the question,
Two digit number whose ten’s digit is ‘t’
Two digit number whose unit’s digit is ‘u’
Then, the number = 10 × t + 1 × u
= 10t + u

The variable used in the equation 2p + 8 = 18 is p.
The word ‘variable’ means something that can vary, i.e., change. The value of a variable is not fixed. We use a variable to represent a number and denote it by any letter such as l, m, n, p, x, y, z etc

 x metres = x × 100 centimetres
We know that, 1 metre = 100 centimetre.
Therefore, x metres × 100 centimetres = 100x centimetres

p litres = p× 1000 millilitres
we know that, 1 litre = 1000 millilitres
Therefore, p litres × 1000 millilitres = 1000p milliliters.

r rupees = 100r paise
We know that, 1 rupee = 100 paise
Therefore, r rupees = 100r paise

If the present age of Ramandeep is n years, then her age after 7 years will be n + 7 years.

If I spend f rupees from 100 rupees, the money left with me is 100 – f rupees.

False.
Consider the equation, x + 1 = 0
Then, x = -1

True.
Consider equations x + 1 = 0
So, x = -1
Consider the equation, 2x + 2 = 0
Divide both the side by 2,
Then we get, x + 1 = 0
Therefore, x = – 1

Yes, If m is a whole number, then 2m denotes a multiple of 2.
So, given statement is True.

False.

Additive inverse of x is – x

Yes, If x is a negative integer, – x represents a positive integer.

So, given statement is True.

False.

An expression with a variable, constants and the sign of equality (=) is called an equation.

True

False.

In the equation 7k – 7 = 7, the variable is k.

Yes, a = 3 is a solution of the equation 2a – 1 = 5.
So, given statement is True.

True

Distance between New Delhi and Bhopal is fixed.
Clearly, it is not a variable, so the statement is true.

True

Since 1 minute = 60 seconds 
t minutes = 60t seconds 
So, given statement is True.

False

We have, 3x + 2 = 20
⇒ 3x + 2 – 2 = 20 – 2
[Subtracting 2 from both sides]
⇒ 3x = 18
⇒ x= 18/3 = 6
x = 6, which is the solution of the given equation.

False

Let the number be x.
One third of the number = x/3
According to the given question,
(x/3) + x = 8

False

Difference between the ages of two sisters Leela and Yamini is not a variable because Leela’s and Yamini’s ages are fixed. But the value of a variable is not fixed.

So, the statement false.

No, number of lines that can be drawn through a point is not a variable.
So, given statement is False.

Let the number be x and twice the number x = 2x
Now, according to question,
The expression = 2x + 1
Hence, required expression is 2x + 1.

Let the present temperature be f° C.
∴ Required expression
= Present temperature – 20°C = (f – 20)°C

Let the integer be n.
Successor of n = n + 1
∴ Required expression = n + 1
Note: If 7 is added to a number, we get its successor.

Given, side of triangle is m.
In equilateral triangle, all sides are equal.
∴ Perimeter of an equilateral triangle =Sum of all the sides
= m + m + m = 3m
Hence, the perimeter of an equilateral triangle is 3m.

Given, length of rectangle = k units
Breadth of rectangle = n units
Now, area of rectangle = Length x Breadth = k x n = kn units
Hence, area of the rectangle is kn sq units.

Let sister’s helping hours = x years
Then, Omar’s helping hour = Sister’s helping hour +1 = (x + 1)years
∴ Required expression = (x + 1) years

Let one odd integer be x.
So another consecutive odd integer will be x + 2.
Thus, Two consecutive odd integers are x, x + 2.

Any even integer can be written as 2n, where n is an integer. So, next even integer will be 2n + 2.
Hence, two consecutive even integers are 2n and 2n + 2.

If p is divided by 11, it is given as = P/11.
Now when P/11 is added to 10, it gives = (P/11)+10

x times of 3 = 3x
And smallest natural number = 1.
So, 3x added to 1 = 3x + 1.

6 times q = 6q.
The smallest two-digit number = 10.
6q subtracted from 10 = 10 – 6q.

The required equations are 3y + 4 = 10 and 2x – 3 = 1.
In both equations, 2 is the solution.

The required equation is 2t + 3 = 3, which has the solution t = 0.

The required equation is x + 1 = 0.
Its solution is x = -1, which is not a whole number.

A pen costs 5 times the cost of a pencil.

Amount left with Leela is Rs. 10,000 more than the amount she contributed towards the Prime Minister’s Relief Fund.

The age of Kartik’s father is seven times the age of Kartik.

The minimum temperature on a day in Delhi was 10°C less than the maximum temperature.

Last year, Jay planted 10 more plants than twice the plants planted by his friend John.

Sharad reduced his tea consumption per day by 5 cups after having some health problems.

The number of students dropping out this year is 30 less than the number of students who dropped out last year.

The price of petrol per litre decreased by Rs. 5 this month compared to last month.

Khader’s monthly salary increased by Rs. 1000 in the year 2006 compared to 2005.

The number of girls enrolled this year is 10 less than 3 times the number of girls enrolled last year.

(a) 13 subtracted from twice a number gives 3.
Let the number be x. Twice the number = 2x.
According to the question: 2x – 13 = 3.
(b) Let the number be x.
One-fifth of a number is x=x/5 
5 less than that number x = x-5
 One-fifth of the number = x/5.
According to the question: x/5 = x – 5.
(c) Let the number be x.
Two-thirds of a number is 12.
 Two-thirds of the number x= 2x/3.
According to the question: 2x/3 = 12.
(d) Let the number be x.
Twice the number = 2x.
Now, 9 is added to 2x = 9 + 2x
According to question, 9 + 2x = 13.
(e) Let the number be x.
One-third of the number x= x/3.
Now, 1 is subtracted from x/3 = x/3 - 1
According to the question: x/3 – 1 = 1.

(a) We have given,
Perimeter of an equilateral triangle = 3 (the side of an equilateral triangle)
⇒ p = 3a
(b) We have given,
Diameter of a circle
= 2 (the radius of the circle)
⇒ d = 2r
(c) We have given,
Selling price = cost price + profit
⇒ s = c + p
(d) We have given,
Amount = principal + interest
⇒ a = p + i

Completed table is as follows:

If m is a whole number less than 5, the completed table is as follows:

Now as it can be seen in the table that as m = 2,
2m-5 = -1.
So solution of 2m-5 =-1 is
m = 2.

No. of students in a class = p
Per student fee collected = Rs 50
Money paid in advance = Rs 1800
Money left = Rs (50p – 1800)

No. of tanks = 8
Water collected per tank = x litres
Amount of water in one tank = 100 litres
Total water in all tanks = (8x + 100) litres

Side of the square = m cm
Area of the square = m × m sq. cm

Rule: P = a + b + c where a, b and c are the sides of the triangle in words is expressed as,
Sum of all sides of a triangle is equal to its perimeter.

Rule: The perimeter of a rectangle is twice the sum of its length and breadth.

Last year’s weight = 40 kg
Weight gain after a year = m kg
Present weight = (40 + m) kg

We have given, length of the board = r cm and breadth = t cm.
(i) The length of aluminium strip to frame the board = Perimeter of the board
= 2(length + breadth)
= 2(r + t) cm
But 10 cm extra strip is required to fix it properly.
∴ Total length of the aluminium strip = 2(r + t) cm + 10 cm.
(ii) Number of nails required to repair 1 board = x.
∴ Number of nails required to repair 15 boards = 15 × x = 15x.
(iii) Area of the cloth required for 1 board = Area of the board
= length × breadth
= r cm × t cm
= (rt) sq cm
Area of the cloth required for 8 boards
= 8 × (rt) sq cm
= 8rt sq cm
But 500 sq cm extra cloth per board is required to cover the edges.
∴ For 8 boards we need 8 × 500 sq cm
= 4000 sq cm extra cloth.
Thus, the total area of the cloth required
= (8rt + 4000) sq cm
(iv) The carpenter charges for 1 board = Rs. x.
∴ The carpenter charges for 23 boards
= Rs. (23 × x) = Rs. 23x

Let the present age of Sunita be x years.
The present age of her mother Geeta = 2(Sunita’s present age) = 2x years.
(i) After 4 years,
Sunita’s age = (x + 4) years
Geeta’s age = (2x + 4) years
Before 3 years,
Sunita’s age = (x – 3) years
Geeta’s age = (2x – 3) years.

(i) ➝ (B), (ii) ➝ (E), (iii) ➝ (C), (iv) ➝ (C), (v) ➝(A)
(i) The number of corners of a quadrilateral is 4, i.e., a constant.
(ii) The variable in the equation 2p + 3 = 5 is p.
(iii) We have given, x + 2 = 3
⇒ x + 2 – 2 = 3 – 2
[Subtracting 2 from both sides]
⇒ x = 1, which is the solution of the given equation.
(iv) We have given, 2p + 3 = 5
⇒ 2p + 3- 3 = 5- 3
[Subtracting 3 from both sides]
⇒ 2p = 2
⇒ 2p/2 = 2/2
⇒ p = 1, which is the solution of the given equation.
(v) Equality sign (=) is used in an equation.