digital text book

A TEXT BOOK OF

MATHEMATICSGRADE VIII

MATHEMATICS DIGITAL

TEXT BOOK

CLASS VIII

Shareena. YB.Ed. Mathematics

Reg.No. 18014350013

CONTENT

Chapter 1

Equal Triangles

1. Sides and Angles

2. One side and two angles

3. Bisector of an Angle

Chapter 2

Equations

1. Addition and Subtraction

2. Multiplication and Division

3. Different Changes

4. Algebraic Method

A TEXT BOOK OF

MATHEMATICSGRADE VIII

MATHEMATICS DIGITAL

TEXT BOOK

CLASS VIII

Shareena. YB.Ed. Mathematics

Reg.No. 18014350013

CONTENT

Chapter 1

Equal Triangles

1. Sides and Angles

2. One side and two angles

3. Bisector of an Angle

Chapter 2

Equations

1. Addition and Subtraction

2. Multiplication and Division

3. Different Changes

4. Algebraic Method

Chapter 1

Equal Triangles

Sides and Angles

We know how to draw a triangle if the lengths of thesides are given. We can draw triangle with sides 4 centimetres,5 centimetres, 7 centimetres in various method.

We can draw with the 4 centimetres side as base.

Like this, with the 5 centimetres side as base.

with the 7 centimetres side as base.

Consider the triangles given below.

Since sides are equal, angles must also be equal. That is, eachangle of ABC is equal to some angle of PQR.

∠A is the largest angle in ABC

∠R is the largest angle in PQR

So, ∠A = ∠R

∠C is the smallest angle in ABC.

∠Q is the smallest angle in PQR.

So, ∠C = ∠Q

∠B is the medium sized angle in ABC.

∠P is the medium sized angle in PQR.

So, ∠B=∠P

In other words,

The longest side of ABC is BC, and the angle oppositethis side is ∠A, which is the largest angle.

The longest side of PQR is PQ and the angle oppositethis side is ∠R, which is the largest angle.

If the sides of a triangle are equal to the sides of anothertriangle, then the angles of the triangles are also equal.

Equal triangles 1 Equal triangles 2

Chapter 1

Equal Triangles

Sides and Angles

We know how to draw a triangle if the lengths of thesides are given. We can draw triangle with sides 4 centimetres,5 centimetres, 7 centimetres in various method.

We can draw with the 4 centimetres side as base.

Like this, with the 5 centimetres side as base.

with the 7 centimetres side as base.

Consider the triangles given below.

Since sides are equal, angles must also be equal. That is, eachangle of ABC is equal to some angle of PQR.

∠A is the largest angle in ABC

∠R is the largest angle in PQR

So, ∠A = ∠R

∠C is the smallest angle in ABC.

∠Q is the smallest angle in PQR.

So, ∠C = ∠Q

∠B is the medium sized angle in ABC.

∠P is the medium sized angle in PQR.

So, ∠B=∠P

In other words,

The longest side of ABC is BC, and the angle oppositethis side is ∠A, which is the largest angle.

The longest side of PQR is PQ and the angle oppositethis side is ∠R, which is the largest angle.

If the sides of a triangle are equal to the sides of anothertriangle, then the angles of the triangles are also equal.

So, ∠A=∠R

The smallest side of ABC is AB, and the angle opposite thisside is ∠C, which is the smallest angle.

The smallest side of PQR is PR, and the angleopposite this side is ∠Q, which is the smallest angle.

So, ∠C=∠Q

The medium sized side of ABC is AC, and the angleopposite this side is ∠B, which is the medium sized angle.

Like this, the medium sized side of PQR is QR, andthe angle opposite this side is ∠P, which is the medium sizedangle.

So, ∠B=∠P

If the sides of a triangle are equal to the sides of an-other triangle, then the angles opposite to the equal sides ofthese triangles are equal.

Draw triangle ABC with sides 4 cnetimetres, 5 centimetres, 6centimetres.

Draw the same triangle below AB, with right and left flipped.

The sides AC and BC of ABC are equal to the sides BD andAD of ABD.

The third side of both triangle is AB.

Since the legths of all three sides are equal, the angles mustalso be equal.

That is, CAB =DBA, CBA=DAB

CAB and DBA are alternate angles made by the line ABmeeting the pair AC, BD of lines. Since the angles are equal,the lines AC and BD are parallel.

Similarly, CBA and DAB are alternate angles made bythe line AB meeting the pair BC, AD of lines. Since the anglesare equal, the lines BC and AD are parallel.

That is, ACBD is a parallelogram.

Q : Draw a parallelogram of sides 5 centimetres, 6centimetres and one diogonal 8 cnetimetres.

Draw a triangle of sides 5 centimetres, 6 centimetres and8 centimetres. (Take 8 centimetres side as base.)

ACBD is a parallelogram.

So, ∠A=∠R

The smallest side of ABC is AB, and the angle opposite thisside is ∠C, which is the smallest angle.

The smallest side of PQR is PR, and the angleopposite this side is ∠Q, which is the smallest angle.

So, ∠C=∠Q

The medium sized side of ABC is AC, and the angleopposite this side is ∠B, which is the medium sized angle.

Like this, the medium sized side of PQR is QR, andthe angle opposite this side is ∠P, which is the medium sizedangle.

So, ∠B=∠P

If the sides of a triangle are equal to the sides of an-other triangle, then the angles opposite to the equal sides ofthese triangles are equal.

Draw triangle ABC with sides 4 cnetimetres, 5 centimetres, 6centimetres.

The sides AC and BC of ABC are equal to the sides BD andAD of ABD.

The third side of both triangle is AB.

Since the legths of all three sides are equal, the angles mustalso be equal.

That is, CAB =DBA, CBA=DAB

CAB and DBA are alternate angles made by the line ABmeeting the pair AC, BD of lines. Since the angles are equal,the lines AC and BD are parallel.

Similarly, CBA and DAB are alternate angles made bythe line AB meeting the pair BC, AD of lines. Since the anglesare equal, the lines BC and AD are parallel.

That is, ACBD is a parallelogram.

Q : Draw a parallelogram of sides 5 centimetres, 6centimetres and one diogonal 8 cnetimetres.

Draw a triangle of sides 5 centimetres, 6 centimetres and8 centimetres. (Take 8 centimetres side as base.)

ACBD is a parallelogram.

One Sides and Two Angle

We can draw triangle in various methods with the lengthof one sides 8 centimetres and the angles at its ends are 400

and 600.

Third angle of all these triangles is 600.

(Sum of angles in a triangle is 1800.

Since the sum of all three angles in a triangle is 1800, if anytwo angles of a tringle are equal to any two angles of anothertriangle, then the third angles are also equal.

In the figure, ABCD is a parallelogram

Drawing the diagonal AC, we can split it into two triangles.

In both ABC and ADC, one side is AC.

∠CAB and ∠DCA are alternated angles made by the line ACmeeting the pair AB, CD of parallel lines.

So ∠CAB=∠DCA

∠ACB and ∠DAC are alternate angles made by the line ACmeeting the pair AD, BC of parallel lines.

So ∠ACB =∠DAC

Thus in ABC and ADC, the side AC and the angles at itsends are equal. So, the sides opposite to equal angles are alsoequal.

That is, AB=CD, AD=BC.Equal triangles 5 Equal triangles 6

If one side of a triangle and the angles at its ends areequal to one side of another triangle and the angles at itsends, then the third angles are also equal and the sidesopposite equal angles are equal.

One Sides and Two Angle

We can draw triangle in various methods with the lengthof one sides 8 centimetres and the angles at its ends are 400

and 600.

Third angle of all these triangles is 600.

(Sum of angles in a triangle is 1800.

Since the sum of all three angles in a triangle is 1800, if anytwo angles of a tringle are equal to any two angles of anothertriangle, then the third angles are also equal.

In the figure, ABCD is a parallelogram

Drawing the diagonal AC, we can split it into two triangles.

In both ABC and ADC, one side is AC.

∠CAB and ∠DCA are alternated angles made by the line ACmeeting the pair AB, CD of parallel lines.

So ∠CAB=∠DCA

∠ACB and ∠DAC are alternate angles made by the line ACmeeting the pair AD, BC of parallel lines.

So ∠ACB =∠DAC

Thus in ABC and ADC, the side AC and the angles at itsends are equal. So, the sides opposite to equal angles are alsoequal.

That is, AB=CD, AD=BC.Equal triangles 5 Equal triangles 6

If one side of a triangle and the angles at its ends areequal to one side of another triangle and the angles at itsends, then the third angles are also equal and the sidesopposite equal angles are equal.

In any parallelogram, opposite sides are equal.

Draw the second diagonal BD and the dioganalsintersecting at P.

The sides AB and CD of APB and CPD are equal. Also∠CAB=∠DCA

That is, ∠PAB=∠PCD

∠PBA and ∠PDC are alternate angles made by the line BDmeeting the pair AB, CD of parallel lines.

So, ∠PBA =∠PDC

Thus in APB and CPD, the sides AB and CD are equal;and the angles at their ends are also equal. So the sides oppo-site equal angles are also equal.

That is, AP=CP, BP=DP

In other words, P is the mid point of both the diagonals AC andBD.

In any parallelogram, the point of intersection of thediagonals is the midpoint of both.

In other words,

In any parallelogram, the diagonals bisect each other.

i. AB=RQ, BC=QP, AC=RP

ii. LM=YZ, MN=XZ, LN=XY

(∠N=1800-(300+800)=700

∠Z=1800-(700+300)=800)

Q : In the figure, AP and BQ equal and parallel lines aredrawn at the ends of the line AB. The point of intersectionof PQ and AB is marked as M.

i. Are the sides of AMP equal to the sides of BMQ?Why?

In AMP and BMQ,

∠PAM=∠QBM

(Alternate angles made by the line AB meeting the pair andBQ of parallel lines.)

∠APM=∠BQM

(Alternate angles made by the line PQ meeting the pair AP andBQ of parallel lines.)

If one side of AMP and the angles at its ends are equal toone side of BMQ and the angles at its ends, then the threesides of AMP are equal to three sides of BMQ.

In any parallelogram, opposite sides are equal.

Draw the second diagonal BD and the dioganalsintersecting at P.

The sides AB and CD of APB and CPD are equal. Also∠CAB=∠DCA

That is, ∠PAB=∠PCD

∠PBA and ∠PDC are alternate angles made by the line BDmeeting the pair AB, CD of parallel lines.

So, ∠PBA =∠PDC

Thus in APB and CPD, the sides AB and CD are equal;and the angles at their ends are also equal. So the sides oppo-site equal angles are also equal.

That is, AP=CP, BP=DP

In other words, P is the mid point of both the diagonals AC andBD.

In any parallelogram, the point of intersection of thediagonals is the midpoint of both.

In other words,

In any parallelogram, the diagonals bisect each other.

i. AB=RQ, BC=QP, AC=RP

ii. LM=YZ, MN=XZ, LN=XY

(∠N=1800-(300+800)=700

∠Z=1800-(700+300)=800)

Q : In the figure, AP and BQ equal and parallel lines aredrawn at the ends of the line AB. The point of intersectionof PQ and AB is marked as M.

i. Are the sides of AMP equal to the sides of BMQ?Why?

In AMP and BMQ,

∠PAM=∠QBM

(Alternate angles made by the line AB meeting the pair andBQ of parallel lines.)

∠APM=∠BQM

(Alternate angles made by the line PQ meeting the pair AP andBQ of parallel lines.)

If one side of AMP and the angles at its ends are equal toone side of BMQ and the angles at its ends, then the threesides of AMP are equal to three sides of BMQ.

ii. What is the special about the position of M on AB?

Since AM=BM, M is the midpoint of AB.

Isosceles Triangles

In this triangle two sies are equal.

If two sides of a triangle are equal, the angles oppositethese sides are also equal.

Eg: Let’s check whether, if two angles of a triangle areequal, then the sides opposite to the equal angles are equal.

Draw ABC and draw a perpendicular from C to AB.

In APC and BPC,

∠A=∠B, ∠APC=∠BPC=900

Since the sum of all three angles in a triangle is 1800, if anytwo angles of a triangle are equal to any two angles of anothertriangle, then the third angles are also equal.

So, ∠ACP=∠BCP

If two angles of a triangle are equal, then the sidesopposite to the equal angles are equal.

A triangle with two sides equal, is called an isosceles triangle.

Triangles with two angles equal are also isosceles.

The triangle with all three sides equal, is called an equilateraltrianle.

In ABC, since AC=BC, ∠B=∠A.

Q : One angle of an isosceles triangle is 900. What are theother two angles?

Two angles of an isosceles triangle are equal. 900 cannotbe the measure of equal angles. Therefore, sum of the mea-sures of two equal angles is 900(i.e., 1800-900)

i.e. each angle is equal to 450 (i.e., 900+2)

Therefore, other angles are 450, 450.

Bisector of an Angle

First draw anb isosceles triangle including this angle likethis.

ii. What is the special about the position of M on AB?

Since AM=BM, M is the midpoint of AB.

Isosceles Triangles

In this triangle two sies are equal.

If two sides of a triangle are equal, the angles oppositethese sides are also equal.

Eg: Let’s check whether, if two angles of a triangle areequal, then the sides opposite to the equal angles are equal.

Draw ABC and draw a perpendicular from C to AB.

In APC and BPC,

∠A=∠B, ∠APC=∠BPC=900

Since the sum of all three angles in a triangle is 1800, if anytwo angles of a triangle are equal to any two angles of anothertriangle, then the third angles are also equal.

So, ∠ACP=∠BCP

If two angles of a triangle are equal, then the sidesopposite to the equal angles are equal.

A triangle with two sides equal, is called an isosceles triangle.

Triangles with two angles equal are also isosceles.

The triangle with all three sides equal, is called an equilateraltrianle.

In ABC, since AC=BC, ∠B=∠A.

Q : One angle of an isosceles triangle is 900. What are theother two angles?

Two angles of an isosceles triangle are equal. 900 cannotbe the measure of equal angles. Therefore, sum of the mea-sures of two equal angles is 900(i.e., 1800-900)

i.e. each angle is equal to 450 (i.e., 900+2)

Therefore, other angles are 450, 450.

Bisector of an Angle

First draw anb isosceles triangle including this angle likethis.

Now we need only draw the perpendicular bisector of the sidePQ of PBQ.

Q : Draw an angle of 750 and draw its bisector.

* * * * * * * * * * * * * * * * * * *

Chapter 2

Equations

Addition and Subtraction

Q : “Six more marks and I would’ve got full hundredmarks in the math test”, Rajan was sad. How muchmark did he actually get?

Actual mark for Rajan = 100 - 6 = 94

Q : Gopalan bought a bunch of bananas. 7 of them wererotten which he threw away. Now there are 46. Howmany bananas were there in the bunch?

Number of bananas in the bunch = 46 + 7 = 53

Multiplication and Division

Q : A number multiplied by 12 gives 756. What is the

number?

Number x 12 = 756

Number = 756 ÷ 12 = 63

Q : A number divided by 21 gives 756. What is the

number?

Number ÷ 21 = 756

Number = 756 x 21 = 15876

Equal triangles 11 Equations 12

Now we need only draw the perpendicular bisector of the sidePQ of PBQ.

Q : Draw an angle of 750 and draw its bisector.

* * * * * * * * * * * * * * * * * * *

Chapter 2

Equations

Addition and Subtraction

Q : “Six more marks and I would’ve got full hundredmarks in the math test”, Rajan was sad. How muchmark did he actually get?

Actual mark for Rajan = 100 - 6 = 94

Q : Gopalan bought a bunch of bananas. 7 of them wererotten which he threw away. Now there are 46. Howmany bananas were there in the bunch?

Number of bananas in the bunch = 46 + 7 = 53

Q : A number multiplied by 12 gives 756. What is the

number?

Number x 12 = 756

Number = 756 ÷ 12 = 63

Q : A number divided by 21 gives 756. What is the

number?

Number ÷ 21 = 756

Number = 756 x 21 = 15876

Equal triangles 11 Equations 12

Different changes

Q : When a number is tripled and then two added, itbecame 50. What is the number?

Three times of number = 50 - 2=48

Number = 48 ÷ 3=16

Q : Anita and her friends bought pens. For five pensbought together, they got a discount of three rupeesand it cost them 32 rupees. Had they bought the pensseparately, how much would each have to spend?

Actual cost of five pen = 32 + 3 = 35 rupees

Cost of one pen when they bought the pens separately

= 35 ÷ 5 = 7 rupees

Algebraic Method

Q : The price of a chair and a table together is 4500rupees. The price of the table is 1000 rupees more thanthat of the chair. What is the price of each?

Let x be the price of a chair

Since price of the table is 1000 rupees more than that ofthe chair, the price of table = x + 1000 rupees

If x+(x+1000) = 4500, then find x.

2x+1000 = 4500

That is, 1000 is added to two times of a number, it

becomes 4500.

2x=4500 -1000 = 3500

Then, x = 3500 ÷ 2 = 1750

Price of a chair = 1750 rupees

Price of a table = x + 1000 = 1750 + 1000

= 2750 rupees

Q : i. The sum of three consecutive natural numbers is 36. What are the numbers?

ii. The sum of three consecutive even numbers is 36. What are the numbers?

iii. Can the sum of three consecutive odd numbers be 36? Why?

iv. The sun if three consecutive odd numbers is 33. What are the numbers?

Addition and subtraction

If x + a = b, then x = b - a

If x - a = b, then x = b + a

If ax = b(a ≠ 0, then x = ba

= b, then x = ab

Equations 13 Equations 14

Different changes

Q : When a number is tripled and then two added, itbecame 50. What is the number?

Three times of number = 50 - 2=48

Number = 48 ÷ 3=16

Q : Anita and her friends bought pens. For five pensbought together, they got a discount of three rupeesand it cost them 32 rupees. Had they bought the pensseparately, how much would each have to spend?

Actual cost of five pen = 32 + 3 = 35 rupees

Cost of one pen when they bought the pens separately

= 35 ÷ 5 = 7 rupees

Algebraic Method

Q : The price of a chair and a table together is 4500rupees. The price of the table is 1000 rupees more thanthat of the chair. What is the price of each?

Let x be the price of a chair

Since price of the table is 1000 rupees more than that ofthe chair, the price of table = x + 1000 rupees

If x+(x+1000) = 4500, then find x.

2x+1000 = 4500

That is, 1000 is added to two times of a number, it

becomes 4500.

2x=4500 -1000 = 3500

Then, x = 3500 ÷ 2 = 1750

Price of a chair = 1750 rupees

Price of a table = x + 1000 = 1750 + 1000

= 2750 rupees

Q : i. The sum of three consecutive natural numbers is 36. What are the numbers?

ii. The sum of three consecutive even numbers is 36. What are the numbers?

iii. Can the sum of three consecutive odd numbers be 36? Why?

iv. The sun if three consecutive odd numbers is 33. What are the numbers?

Addition and subtraction

If x + a = b, then x = b - a

If x - a = b, then x = b + a

If ax = b(a ≠ 0, then x = ba

= b, then x = ab

v. The sum of three consecutive natural numbers is 33. What are the numbers?

i. Let three consecutive natural numbers are x, x+1, x+2

Sum of three consecutive natural numbers = 36

That is, x + x + 1 + x + 2 = 36

3x + 3 = 36

3x = 36 - 3 = 33

Numbers = 11,12,13

ii. Let three consecutive even numbers are x, x + 2, x + 4.

Sum of three consecutive even numbers = 36

That is, x + x + 2 + x + 4 = 36

3x + 6 = 36

3x = 36 - 6 = 30

Even Numbers = 10,12,14

iii. Sum of three consecutive odd numbers is an odd number.There fore, sum of three consecutive odd numbers cannot be 36.

iv. Let three consecutive odd numbers are x,x+2, x+4.

Sum of three consecutive odd numbers = 33

That is, x + x + 2 + x + 4 = 33

3x + 6=33

3x = 33 - 6 = 27

Odd numbers = 9,11,13

v. Let three consecutive natural numbers are x, x + 1, x + 2

x + x + 1 + x + 2 = 33

3x + 3=33

3x = 33 - 3 = 30

x = 303

Natural numbers = 10,11,12

Q : i. In a calendar, a square of four numbers is marked. The sum of the numbers is 80. What are the numbers?

ii. A square of nine numbers is marked in a calendar. The sum of all these numbers is 90. What are the numbers?

i. Let the numbers in a square of four numbers marked inthe calendar are x, x + 1, x + 7, x + 8

x + x + 1 + x + 7 + x + 8 = 80

4x + 16 = 80

4x = 80 - 16 = 64

x+7 x+8

v. The sum of three consecutive natural numbers is 33. What are the numbers?

i. Let three consecutive natural numbers are x, x+1, x+2

That is, x + x + 1 + x + 2 = 36

3x + 3 = 36

3x = 36 - 3 = 33

Numbers = 11,12,13

ii. Let three consecutive even numbers are x, x + 2, x + 4.

Sum of three consecutive even numbers = 36

That is, x + x + 2 + x + 4 = 36

3x + 6 = 36

3x = 36 - 6 = 30

Even Numbers = 10,12,14

iii. Sum of three consecutive odd numbers is an odd number.There fore, sum of three consecutive odd numbers cannot be 36.

iv. Let three consecutive odd numbers are x,x+2, x+4.

Sum of three consecutive odd numbers = 33

That is, x + x + 2 + x + 4 = 33

3x + 6=33

3x = 33 - 6 = 27

Odd numbers = 9,11,13

v. Let three consecutive natural numbers are x, x + 1, x + 2

x + x + 1 + x + 2 = 33

3x + 3=33

3x = 33 - 3 = 30

x = 303

Natural numbers = 10,11,12

Q : i. In a calendar, a square of four numbers is marked. The sum of the numbers is 80. What are the numbers?

ii. A square of nine numbers is marked in a calendar. The sum of all these numbers is 90. What are the numbers?

i. Let the numbers in a square of four numbers marked inthe calendar are x, x + 1, x + 7, x + 8

x + x + 1 + x + 7 + x + 8 = 80

4x + 16 = 80

4x = 80 - 16 = 64

x+7 x+8

Numbers in the calandar = 16,17, 23, 24

ii. Let the numbers in a square of nine numbers marked inthe calandar are x, x + 1, x + 2, x + 7, x + 8, x + 9, x + 14,

x + 15, x+16

x + x + 1 + x + 2 + x + 7 + x + 8 + x + 9 + x + 14 + x + 15 +

x+16 = 90

That is, 9x + 72 = 90

9x = 90 - 72 = 18

Numbers in the calandar = 2, 3, 4, 9, 10, 11, 16, 17, 18

Q : A class has the same number of girls and boys. Onlyeight boys were absent on a particular day and thenthe number of girls was double the number of boys.What is the number of boys and girls?

Let the number of boys and girls each be x.

If eight boys were absent,

number of boys = x - 8

number of girls = xEquations 17 Equations 18

Number of girls = 2 x number of boys

∴ x = 2(x - 8)

That is, x = 2x - 16

x - 2x = 2x - 16 - 2x

Subtract 2x from both sides

-x = -16

∴ x = 16

Number of boys = 16

Number of girls = 16

Q : Ajayan is ten years older than Vijayan. Next year,Ajayan’s age would be double that of Vijayan. Whatare their age now?

Let x be the present age of Vijayan. Then presentage of Ajayan = x+10

After one year

x + 11 = 2(x+1)

x + 11 = 2x+2

x - 2x = 2 - 11

-x = -9

Age of Vijayan = 9, Age of Ajayan = 19.

x x+1 x+2

x+7 x+8 x+9

x+14 x+15 x+16

Present age Age in next yearVijayan x x + 1

Ajayan x + 10 x + 11

Numbers in the calandar = 16,17, 23, 24

ii. Let the numbers in a square of nine numbers marked inthe calandar are x, x + 1, x + 2, x + 7, x + 8, x + 9, x + 14,

x + 15, x+16

x + x + 1 + x + 2 + x + 7 + x + 8 + x + 9 + x + 14 + x + 15 +

x+16 = 90

That is, 9x + 72 = 90

9x = 90 - 72 = 18

Numbers in the calandar = 2, 3, 4, 9, 10, 11, 16, 17, 18

Q : A class has the same number of girls and boys. Onlyeight boys were absent on a particular day and thenthe number of girls was double the number of boys.What is the number of boys and girls?

Let the number of boys and girls each be x.

If eight boys were absent,

number of boys = x - 8

number of girls = xEquations 17 Equations 18

Number of girls = 2 x number of boys

∴ x = 2(x - 8)

That is, x = 2x - 16

x - 2x = 2x - 16 - 2x

Subtract 2x from both sides

-x = -16

∴ x = 16

Number of boys = 16

Number of girls = 16

Q : Ajayan is ten years older than Vijayan. Next year,Ajayan’s age would be double that of Vijayan. Whatare their age now?

Let x be the present age of Vijayan. Then presentage of Ajayan = x+10

After one year

x + 11 = 2(x+1)

x + 11 = 2x+2

x - 2x = 2 - 11

-x = -9

Age of Vijayan = 9, Age of Ajayan = 19.

x x+1 x+2

x+7 x+8 x+9

x+14 x+15 x+16

Present age Age in next yearVijayan x x + 1

Ajayan x + 10 x + 11

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