Algebra

It is given that a box of matchsticks contains 50 matchsticks.

1 box = 50 matchsticks

For n boxes, number of matchsticks = n × 50

⇒ the number of matchsticks required to fill n such boxes = 50n

The present age of Amulya is given to be x years.

⇒ her age before 5 years = (x – 5) years

The product of 6 and x = 6x

An equation contain two mathematically expressions which are connected by an equal.

From the above given options, only C satisfy this definition.

Given that x = 2

In x + 10, put given value of x

⇒ x + 10 = 2 + 10

⇒ x + 10 = 12

We know that perimeter is the sum of the length of sides of the figure.

Given figure is a regular hexagon which states that all the six sides of it are equal to each other.

So, the perimeter of the hexagon = 6× length of side

But, given that perimeter = x meters

⇒ 6× length of side = x

Taking 6 on the other side it will be divided.

⇒ length of side = (x ÷ 6) meters

Given x = 2

Putting this value in each of the options, we check where LHS = RHS

A. x + 2 = 5

LHS = x +2 = 2 +2 =4

RHS = 5

LHS ≠RHS

B. x - 2 = 0

LHS = x - 2 = 2 - 2 = 0

RHS = 0

LHS = RHS

C. 2x + 1 = 0

LHS = 2x +1 = 2× 2 + 1=4 + 1 =5

RHS = 0

LHS ≠RHS

D. x + 3 = 6

LHS = x +3 = 2 +3 = 5

RHS = 6

LHS ≠RHS

Hence, x =2 is the solution of x-2 =0.

Commutative means changing the order of the operands does not change the result.

From the given options

x +y = y + x, shows the commutative property in addition.

Solving each of the equations given.

A. x + 1 = 1

Taking 1 to RHS and subtracting from 1.

⇒ x = 1- 1

⇒ x = 0

The solution is an integer.

B. x – 1 = 3

Taking 1 to RHS and adding to 3.

⇒ x = 3+1

⇒ x = 4

The solution is an integer.

C. 2x + 1 = 6

Taking 1 to RHS and subtracting from 6

⇒ 2x = 6 – 1

⇒ 2x = 5

Now taking 2 to RHS and dividing 5

⇒ x = 2.5

The solution is not an integer.

D. 1 – x = 5

Taking 1 to RHS and subtracting from 5

⇒ -x = 5 – 1

⇒ - x = 4

Multiplying both the sides by -1

⇒ x = - 4

The solution is an integer.

We know that 3 × 5 = 15

In algebra letters may stand for unknown quantities.

For example, in x + 2 the value of x is unknown and can be anything.

A variable is a quantity that may change its value according to the equation or problem.

Hence it can take different values.

10 – x means x is subtracted from 10.

Given sum of money Savitri has = Rs x

Expenses made by her:

Grocery = Rs 1000

Clothes = Rs 500

Education = Rs 400

Total expenses = 1000 + 500 + 400 = 1900

Money left with her after deducting expenses = Rs (x – 1900)

Money received as gift = Rs 200

Hence, money left with her after adding gifted money = (x – 1900) + 200

⇒ Money left with Savitri = Rs (x – 1700)

We know that perimeter is the sum of the length of sides of the figure.

Given sides of the triangle are x, x and y.

Perimeter of triangle = x + x + y

⇒ perimeter = 2x + y

We know that the area of a square is length of side multiply by length of side

Given that the side of the square is x

⇒ Area of the square = x × x

Given that x is multiplied by 2 = 2x

Then, this expression is subtracted from 3

= 3 – 2x

Hence, 3 – 2x is the required expression

Given

Take 2 to the other side and multiply with 3

⇒ q = 2× 3

⇒ q = 6

Given x – 4 = -2

Take 4 to the other side and add to -2

⇒ x = -2 + 4

⇒ x = 2

The equation does not contain any variable.

Also, it is a true statement

Given no. of pencils with Kanta = p

No. of pencils added = q

Adding above number,

Total no. of pencils = p +q

Given equation: 4x = 16

Taking 4 on the other side it will divide 16,

⇒ x = 4

Hence, x = 4 satisfy the equation.

Let the number being think of is x.

Now, adding 13 in it gives

x + 13

So, it is becomes 27.

⇒ x + 13 = 27

So the equation is x + 13 = 27.

(40h) km

Given that speed is 40 km per hour.

This means

Distance travelled in 1 hour = 40 km

Distance travelled in h hours = 40 × h km

⇒ Distance travelled in h hours = 40h km

Given that p kg of potatoes are bought for Rs 70.

This means

Cost of p kg of potatoes = Rs 70

Cost of 1 kg of potatoes

Auto rickshaw charges for 1^st KM = Rs.10

Charges after 1^st KM (i.e., 2^nd, 3^rd, 4^th) = Rs.8

Charges for d KMs = charges for 1^st, 2^nd, 3^rd, 4^th, …. d

= charges for 1^st + charges for 2^nd, 3^rd, 4^th, …. d-1

(∵ charges for 1 km is separated we write d-1 in the second term instead of d)

= 10 + 8 × (d-1)

We will be charged same amount after 1^st that is Rs.8

= 10 + 8d – 8

= 2 + 8d

Total charges for d KMs is 2 + 8d

Given equation 7x + 4 = 25

We have to find the value of X (i.e., for what value of X the equation will be satisfied)

7x + 4 = 25

Taking 4 in L.H.S to R.H.S

(R.H.S = The part which is Right side to the equity in an equation

L.H.S = The part which is Left side to the equity in an equation)

By transferring the number to the other side of the equation. Its sign will be changed.

7x = 25 – 4

7(x) = 21

In the above statement 7 is multiplied by X

If that goes to other side it will be divides the number on other side

X =

X = 3

(or)

7x + 4 = 25

Subtracting 4 on both sides of the equation

7x +4 – 4 = 25 – 4

7x = 21

Dividing X on both sides of the equation

X = 3

Given equation 3x + 7 = –20

Sending 7 to R.H.S

3x = -20 – 7

3x = -27

Taking 3 to R.H.S

x =

x = -9

‘X exceeds by 7’ means ‘value of x is increased by 7’

∵ x value increases we use ‘+’ operator

Mathematically we can write the given statement as X + 7

The number is x

Three times the number is 3x

8 more than the three times the number is 3x + 8

(we use + operator since ‘more than’ is given)

Let the number of pencils bought be m

Cost of pencils is Rs. X

Cost of each pencils is Rs.2

Cost of pencils = number of pencils × cost of each pencils

X = m × 2

m =

number of pencils bought (m) =

Number of days in a week = 7 day

Number of days in w weeks = w × number of days in a week

= w × 7 days

= 7w days

Annual means 12 months (a year) + bonus

Monthly salary = Rs. r

Festival bonus= 2000

Annual salary = 12-month salary + bonus

= 12 × r + 2000

= 12r + 2000

e.g.

Let 29 be a number in that 2 is in ten’s place

9 is in unit’s place

Given

‘t’ is ten’s digit of a two-digit number

‘u’ is unit’s digit of a two-digit number

A two-digit number consists of two digits only

∴ The two-digit number is ‘tu’

In this question ‘p’ is a variable.

Every other value is constant in the equation (i.e., 8, 18,2)

P don’t have any constant value, so it is a variable.

We know that

1 meter = 100 centimeter

Then, the value of x meters is

X (1 meter) = x (100 centimeter)

X meter = 100x centimeters

We know that

1 liter = 1000 milliliters

Then, the value of p liters is

P (1 liters) = P (1000 milliliters)

P liters = 1000P milliliters

We know that

1 rupee = 100 paise

Then, the value of r rupees is

r (1 rupee) = r (100 paise)

r rupees = 100r paise

The present age of Ramandeep is n years

Every year her age will be increasing (adding)

After 7 years, her age will be added by 7 to his present age

After 7 years her age = present age + 7

= n +7

Every time when you spend some money from amount the money will be subtracted from the amount

If I have 100 rupees

After spending f rupees from 100

The remaining amount will be 100 – f

Given x + 1 = 0

Taking ‘1’ to R.H.S

X = -1

So, value of x is -1 not 0

So, given statement is false

Given x + 1 = 0 ⇒ (a)

2x + 2 = 0 ⇒ (b)

In the second equation we are going to take 2 as common since all the term in it is multiple of 2

2(x + 1) = 0

Sending 2 to R.H.S

X + 1 =

X + 1 = 0 ⇒ (c)

We can see that equation a & c are same

So, they will have same solutions.

So, given statement is True

Given m is a whole number

2m denotes a multiple of 2

∵ m is multiplied by 2

So, all values of 2m is multiplied by 2

So, given statement is True

For any number its additive inverse is negative sign of that number

For integer x is -1(x)

= -x

The additive inverse of x is -x

So, given statement is false

For any integer if we multiply -1 we will get opposite sign for that integer

e.g.,

n → -1 (n) → -n

if X is negative number

then we can make it positive number if we multiply it by -1

x → -1 (x) → -x

∴ given statement is correct.

A equation must contain equity symbol in the above equation we don’t find any such symbol.

So, given one is not an equation.

So, given statement is false

If L.H.S. = R.H.S. then that is said to be an equation.

∴ given statement is correct.

We know that 7 is a constant not variable.

In the above equation k is the variable.

So, given statement is false

Given

2a – 1 = 5

Taking 1 to R.H.S.

2a = 5 + 1

2a = 6

Taking 2 to R.H.S.

a =

a = 3

∴ given statement is correct.

The distance between New Delhi and Bhopal will be a constant

∵ every time it won’t change. The distance between them is fixed.

So, that is not a variable.

∴ given statement is correct.

True

Given: t minutes = 60 t seconds

⇒ 1 × t minutes = 60 t seconds

⇒ 1 minute = 60 seconds

False

Given equation is

3x + 2 = 20

⇒ 3x = 20 – 2

⇒ 3x = 18

⇒ x = 6

Hence, solution of given equation is x = 6

But, according to question x = 5 is the solution of the given equation.

False

Let the number be x

Now, according to given statement

But, given statement is

False

Difference between the age of two sister 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.

True

Infinite number of lines can be drawn through a point.

Let the number be x

Twice the number x = 2x

According to question,

∴ The expression is 2x + 1

Let the present temperature be x°c

∴ required expression is (present temperature – 20°c)

∴ required expression is (x-20)°c

Let the integer be n

Successor of n = n + 1

∴ required expression = n + 1

Given, side of a triangle = m

In an equilateral triangle, all sides are equal = m

∴ perimeter of an equilateral triangle = sum of all sides

Thus, perimeter of equilateral triangle = m + m + m = 3m

Given, length of rectangle = k units

Breadth of rectangle = n units

Area of rectangle = length × bredth

= k × n

Area of rectangle = kn sq units

Let sister’s helping hours = x hours

Then, Omar’s helping hour = sister’s helping hour + 1 = (x+1) hours

Thus, required expression = (x+1) hours

Any odd integer can be written as 2n+1, where n = integer

So, the next odd integer will be (2n+1) + 2 = 2n + 3

Hence, two consecutive odd integers are 2n + 1 and 2n + 3

Any even integer can be written as 2n, where n = integer

So, the next even integer will be 2n+ 2

Hence, two consecutive even integer are 2n and 2n +2

Multiples of 5 are

Multiply 5 by 1 = 5× 1 = 5

Multiply 5 by 2 = 5× 2 = 10

Multiply 5 by 3 = 5× 3 = 15

And so on

Multiply 5 by n = 5× n = 5n, where n is any whole number

Let the numerator be x

Then denominator = x + 1

Hence, required fraction is

Let height of Empire State be h metre

Height of Mount Everest = 20 × h = 20h metre

Hence, required expression is 20h

Cost of one notebook = Rs p

Cost of 2 notebook = 2 × p = Rs 2p

Cost of one pencil = Rs 3

Total cost = Cost of 2 notebook + Cost of 1 pencil

∴ Total cost = Rs (2p+3)

Given: z is multiplied by –3 = (-3) × z = -3z

Now, result is subtracted from 13 = 13 – (-3z) = 13 + 3z

Hence, required equation is 13 + 3z

Given:

Now, result is added to 10

⇒

Hence, required equation is

Given: x times of 3 = 3 × x = 3x

Smallest natural number = 1

Thus, resulting expression = 3x + 1

Given: 6 times q = 6 × q = 6q

Smallest two digit number = 10

Thus, resulting expression = 10 – 6q

Let the two number be x and y, which has solution 2 in equation

1. For getting first equation, the number x is multiplied by 2, then the number is 2x

After that, 3 is subtracted from it which results into 1

Hence, 2x – 3 = 1

⇒ 2x = 3+1

⇒ 2x = 4

⇒x = 2

2. For getting second equation, the number y is multiplied by 3, then the number is 3y

After that, it will be added to 4 which results into 10

Hence, 3y + 4 = 10

⇒ 3y = 10-4

⇒ 3y = 6

⇒y = 2

Hence, required equation are 2x - 3 = 1 and 3y + 4 = 10

Let the one number be x, which has solution 0 in equation

For getting equation, the number x is multiplied by 2, then the number is 2x

After that, it will be added to 3 which results into 3

Hence, 2x + 3 = 3

⇒ 2x = 3-3

⇒ 2x = 0

⇒x = 0

Hence, required equation is 2x + 3 = 3

We know that, whole number are 0, 1, 2,…

Now, let one number be x whole solution is not a whole number

For getting equation, the number x will be added to 1 which results into 0. Then,

⇒ x + 1 = 0

⇒ x = 0-1 = -1

Where -1 is not a whole number

So, required equation is x + 1 = 0

Given: A pencil costs Rs p

a pen costs Rs 5p

⇒ Cost of pen = Rs 5 × Rs 1p

⇒ Cost of pen = Rs 5 × Rs p

⇒ Cost of pen = Rs 5 × Cost of pencil

Thus, Cost of pen is 5 times the cost of pencil.

After contributing Leela is left with 10000 more than the contributed amount.

Kartik’s father is 7 times older than Kartik.

The difference between maximum and minimum temperature on a day in Delhi is 10° C.

OR

On a day in Delhi the maximum temperature is greater by 10° C than the minimum.

Last year Jay planted 10 trees more than twice the number of trees planted by John.

After having some health problem Sharad takes 5 tea cups less than what he used to take before per day.

The number of students dropping out of school this year has reduced by 30 than last year.

The price of petrol is reduced by 5 Rs per litre from last month.

In 2006 Khader’s monthly salary was increased by 1000 Rs than his salary in 2005.

The number of girls enrolled in a school this year is 10 less than thrice the number of girls enrolled last year.

(a) Let the variable number be ‘x’

Twice of that number = 2x

13 subtracted from twice of that number = 2x – 13

Equation: 2x – 13 = 3

(b) Let the variable number be ‘x’

One fifth of the number = x

5 less than that number = x – 5

Equation: x = x – 5

(c) Let the variable number be ‘x’

Two-third of number = x

Equation: x = 12

(d) Let the variable number be ‘x’

Twice of number = 2x

9 added to twice a number = 2x + 9

Equation: 2x + 9 = 13

(e) Let the variable number be ‘x’

One-third of number = x

1 subtracted from one-third of number = x – 1

Equation: x – 1 = 1

(a) three times of side (a) = 3a

Equation: p = 3a

(b) twice of radius (r) = 2r

Equation: d = 2r

(c) Equation: s = c + p

(d) Equation: a = p + i

(i) Her brother is 2 years younger.

which means smaller than Kanika by 2 years hence we should subtract 2 from Kanika’s age to get her brother’s age brother’s age = (x – 2) years

(ii) Her father’s age exceeds her age by 35 years. Which means father’s age is 35 years more than Kanika’s age hence we should add 35 to Kanika’s age to get father’s age father’s age = (x + 35) years

(iii) Mother’s age is 3 years less than that of her father. Which means mother is smaller than father by 3 years hence we should subtract 3 from father’s age calculated above mother’s age = (x + 35) – 3 = x + 35 – 3 = (x + 32) years

(iv) Her grand father’s age is 8 times of her age. Grandfather’s age = 8 × x = 8x

Hence the table

m is a whole number less than 5 Whole numbers less than 5 are 0, 1, 2, 3 and 4 hence m can take values 0, 1, 2, 3 and 4

When m = 0:

2m – 5 = 2 × 0 – 5 = -5

When m = 1:

2m – 5 = 2 × 1 – 5 = 2 – 5 = -3

When m = 2:

2m – 5 = 2 × 2 – 5 = 4 – 5 = -1

When m = 3:

2m – 5 = 2 × 3 – 5 = 6 – 5 = 1

When m = 4:

2m – 5 = 2 × 4 – 5 = 8 – 5 = 3

The table becomes

By looking at table the equation 2m – 5 = – 1 holds true when m = 2 hence solution of equation 2m – 5 = – 1 is m = 2

number of students = p

amount collected per student = 50 Rs

total amount collected = amount collected per student × amount collected per student

total amount collected = 50 × p = 50p

amount paid for transport = 1800 Rs

to find the money left we must subtract the amount paid from total amount collected

∴ money left = total amount collected - amount paid for transport

Therefore, money left with them to spend on other items = 50p – 1800

Number of water tanks = 8

Water collected by 1 tank = x litres

Water collected by 8 tanks = 8 × Water collected by 1 tank

= 8 × x

= 8x litres

On a day 100 litres of water was already there in one of tank which means we should add 100 to total water collected by 8 tanks

∴ total amount of water = Water collected by 8 tanks + 100

Therefore, total amount of water in the tanks on that day is 8x + 100 litres

side of square = m cm

Area of square = side × side

∴ area of square = m × m

Therefore, area of a square whose side is m cm is m × m

the perimeter of a triangle is sum of length all the three sides of that triangle

the perimeter of rectangle is twice the sum of its length and breadth

weight on last birthday = 40 kg

Increase in weight after a year = m kg

Present weight = weight last year + increase in weight

= (40 + m) kg

Therefore, my present weight is (40 + m) kg

(i) We have to frame the perimeter of the board hence length of aluminium strip required to frame the board will be same as perimeter of board Perimeter of board = 2 × (r + t)

But 10 cm extra strip is required to fix it properly hence we should add 10 to perimeter of board

Therefore, aluminium strip required to frame the board is 2 × (r + t) + 10 cm

(ii) Number of nails required to repair 1 board = x

Number of nails to repair 15 boards = 15 × Number of nails required to repair 1 board

Number of nails to repair 15 boards = 15 × x = 15x

(iii) Area of one board = r × t sq. cm

500 sq. cm extra cloth is required per board so we should add 500 in area of one board to get the total area of cloth required for one board

Area of cloth required for one board = Area of one board + 500 sq. cm

Therefore, Area of cloth required for one board = (r × t) + 500 sq. cm

Area of cloth required for 8 boards = 8 × Area of cloth required for one board

Area of cloth required for 8 boards = 8 × [(r × t) + 500]

= 8(r × t) + 4000 sq. cm

Therefore, Area of cloth required for 8 boards = 8(r × t) + 4000 sq. cm

(iv) Expenditure of making 1 board = x Rs

Expenditure of making 23 boards = 23 × Expenditure of making 1 board = 23 × x = 23x

Therefore, Expenditure of making 23 boards = 23x

Let current age of Geeta be x years

By given current age of Sunita = x

(i) after 4 years?

Age after 4 years = current age + 4

Therefore, Geeta’s age after 4 years = x + 4 years

Therefore, Sunita’s age after 4 years = x + 4 years

(ii) before 3 years?

Age before 3 years = current age – 3

Therefore, Geeta’s age before 3 years = x – 3 years

Therefore, Sunita’s age before 3 years = x – 3 years

(i) The number of corners of a quadrilateral

Let it be any quadrilateral the number of corners will always be constant example for a square the number of corners are 4 which is constant which won’t change

(ii) The variable in the equation 2p + 3 = 5

A variable is a letter which can take any value in equation here ‘p’ can take any value hence the variable is ‘p’

(iii) The solution of the equation x + 2 = 3

To find x take the 2 on the right hand side which will become -2 on going to the right as shown

x = 3 – 2

x = 1

therefore, solution of the equation x + 2 = 3 is x = 1

(iv) solution of the equation 2p + 3 = 5

To find x take the 3 on the right hand side which will become -3 on going to the right as shown

2p = 5 – 3

⇒ 2p = 2

⇒ p =

⇒ p = 1

Therefore, solution of the equation 2p + 3 = 5 is p = 1

(v) A sign used in an equation

From the options given in column II its equal to sign (=)