Equations

6 marks more would make 100 marks in Math test, which means 6 less than 100, that is, 100 – 6 = 94.

13 rupees more will make the total to 60 rupees, which means 13 less than 60, that is, 60 – 13 = 47.

If we add 7 more bananas to the 46 bananas, we will get the total number of bananas Gopalan bought, which means 7 more than 46, that is, 46 + 7 = 53.

If we add remaining 217 rupees to 163 rupees which were spent, we will get total money she had at first, which means 163 more than 217, that is, 163 + 217 = 380.

264 more will make the total to 452, which means 264 less than 452, that is, 452 – 264 = 188.

If we subtract 198 from the number we will get 163, which means 163 more than 198, that is, 198 + 163 = 361.

40000 rupees is 5 times the peon’s salary. Hence peon’s salary is of the manager’s salary, that is,

1300 rupees is given by each, which means number of travelers are 1300 of 5200, that is,

756 is 12 times of the number, which means number is 12 of 756, that is,

.

756 is of the number, which means number is 21 times of 756, that is,

756 × 21 = 15876.

Five pens cost 32 rupees after a discount of 3 rupees.

∴ their cost without discount = 32 + 3 = 35 rupees.

∴ cost of five pens (without discount) = 35 rupees

Hence, cost of each pen = 35/5 = 7rupees

Given:

Perimeter = 25 metres

Let the two sides of the rectangle be ‘l’ & ‘b’, where l is the length & b is the breadth.

Then, l = 5 m (given)

b = ? (to find)

Now,

Perimeter of rectangle = 2 × (length + breadth)

⇒ 25 = 2 × (l + b)

⇒ 25 = 2 × (5 + b)

Simplifying the bracket, we get,

⇒ 25 = 2 × 5 + 2b

⇒ 25 = 10 + 2b

⇒ 25 –10 = 2b

⇒ 15 = 2b

Or

Perimeter of rectangle = 2 × (length + breadth)

= 25 metres (given)

∴ Perimeter is obtained by multiplying the sum of length and breadth by 2

∴ sum of length can be obtained by dividing the perimeter by 2

∴ sum of length and breadth of the given rectangle = perimeter/2

⇒ 25/2

= 12.5metres

∴ sum of 2 sides is 12.5 metres

Since one side is 5 metres

∴ Other side = 12.5-5 = 7.5 metres

After adding 3 to the double, the number becomes 101,

Therefore, the double of the number is 101-3 = 98

∴ the required number is = = 49

After adding 3 to the triple, the number becomes 101,

Therefore, the triple of the number is 101-2 = 99

∴ the required number is = = 33

After subtracting 3 from the double, the number becomes 101,

Therefore, the double of the number is 101 + 3 = 104

∴ the required number is = = 52

After subtracting 2 from the triple, the number becomes 101,

Therefore, the triple of the number is 101 + 2 = 103

∴ the required number is = 34.33

∴ Half a number added to the number will give a number which will be times the original number.

∴ times the original number = 111

Hence, the original number =

⇒ 37 × 2

⇒74

Let the number of birds be ‘b’

We and us again = b + b = 2b

With half of us =

And half of that =

With one more =

Would make hundred =

⇒ 11b = 99 × 4

⇒ b = 36

Hence, number of birds = 36.

Let the breadth of the rectangle be x metres.

Twice the breadth = 2x metres

One more than twice the breadth = 2x + 1 metres

∴ Length of the given rectangle = 2x + 1 metres

Perimeter of the rectangle = 80 metres (given)

⇒ 2(length + breadth) = 80

⇒ 2(2x + 1 + x) = 80

⇒ 2(3x + 1) = 80 (Transporting 2 to RHS)

⇒

⇒ 3x + 1 = 40

⇒ 3x = 40-1

⇒ 3x = 39

⇒

⇒x = 13

Hence,

Length of the rectangle = 2x + 1 = 2 × 13 + 1 = 27 metres

Breadth of the rectangle = x = 13 metres

Let smaller angle be x°

Angle on the other side is 50° more than the smaller angle.

∴ Angle on the other side = (x + 50)°

Since, both the angles together form a line,

Their sum = 180°

⇒ x + 50 + x = 180

⇒ 2x + 50 = 180

⇒ 2x = 180-50

⇒ 2x = 130

⇒

⇒ x = 65

Hence,the smaller angle = x° = 65°

Let price of a pen be x rupees

Then, price of 2 pens = 2x rupees

Since, Price of book is 4 rupees more than price of a pen,

∴ Price of a book = x + 4rupees

∴ price of 5 books = 5(x + 4) = 5x + 5 × 4 rupees = 5x + 20 rupees

Price of a pencil is two rupees less than that of pen

∴ price of a pencil = x-2 rupees

∴ price of 3 pencils = 3(x-2) = 3x-3 × 2 rupees = 3x-6 rupees

Total price of 5 books, 2 pens and 3 pencils = 74 rupees

⇒ 5x + 20 + 2x + 3x-6 = 74

⇒ 10x + 14 = 74

⇒ 10x = 74-14

⇒ 10x = 60

⇒ x = 6

Hence ,

Price of a pen = x = 6 rupees

Price of a book = x + 4 = 10 rupees

Price of a pencil = x-2 = 4 rupees

Let the three consecutive natural numbers be x,x + 1,x + 2.

Their sum = 36

⇒ x + x + 1 + x + 2 = 36

⇒ 3x + 3 = 36

⇒ 3x = 36-3

⇒ 3x = 33

⇒

⇒ x = 11

Hence, the required three consecutive numbers are 11 ,11 + 1,11 + 2 = 11,12 and 13.

Let the first even number be 2x.(We are taking 2x to make sure that the number is even i.e. divisible by 2)

Then, the next even numbers = 2x + 2 and 2x + 2 + 2 = 2x + 4

Their sum = 36

⇒ 2x + 2x + 2 + 2x + 4 = 36

⇒ 6x + 6 = 36

⇒ 6x = 36-6

⇒ 6x = 30

⇒

⇒x = 5

⇒ 2x = 10

2x + 2 = 2 × 5 + 2 = 12

2x + 4 = 2 × 5 + 4 = 14

Hence,the required three consecutive even numbers are 10,12,and 14.

If possible, let the three consecutive odd numbers be 2x + 1,2x + 3,2x + 5(we take 2x + 1 to make sure that it is not divisible by 2,i.e. it is a odd number(here x is a natural number))

Their sum = 36

⇒ 2x + 1 + 2x + 3 + 2x + 5 = 36

⇒ 6x + 9 = 36

⇒ 6x = 27

⇒ x = 4.5

2x + 1 = 2 × 4.5 + 1 = 10

According to our assumption x must be a natural number, but here it is not.

As a result of which 2x + 1 is also not an odd number.

Hence, sum of three consecutive odd numbers cannot be 36.

Let the three consecutive odd numbers be 2x + 1,2x + 3,2x + 5(we take 2x + 1 to make sure that it is not divisible by 2,i.e. it is a odd number(here x is a natural number))

Their sum = 33

⇒ 2x + 1 + 2x + 3 + 2x + 5 = 33

⇒ 6x + 9 = 33

⇒ 6x = 33-9

⇒ 6x = 24

⇒

⇒ x = 4

⇒ 2x + 1 = 2 × 4 + 1 = 9

2x + 3 = 2 × 4 + 3 = 11

2x + 5 = 2 × 4 + 5 = 13

Hence, the required three consecutive odd numbers are 9,11 and 13.

Let the three consecutive natural numbers be x,x + 1,x + 2.

Their sum = 33

⇒ x + x + 1 + x + 2 = 33

⇒ 3x + 3 = 33

⇒ 3x = 33-3

⇒ 3x = 30

⇒

⇒ x = 10

⇒ x + 1 = 11

x + 2 = 12

Hence, the three required consecutive natural numbers are 10,11 and 12.

Since, the square is composed of 4 numbers,

2 numbers will mark its length, and 2 numbers will mark its breadth.

(as both are equal in square)

Two numbers will be consecutive and the other two numbers will be the same days of the next week (∴ they will also be consecutive)

Let the smallest number be x.

∴ next number = x + 1

Next week, same day

Number will be x + 7

And next number to it = x + 7 + 1 = x + 8

Sum of all the four numbers = 80(given)

⇒ x + x + 1 + x + 7 + x + 8 = 80

⇒ 4x + 16 = 80

⇒ 4x = 80-16

⇒ 4x = 64

⇒

⇒x = 16

x + 1 = 17

x + 7 = 23

x + 8 = 24

Hence, the required numbers are 16,17,23 and 24.

Since, the square is composed of 9 numbers,

3 numbers will mark its length, and 3 numbers will mark its breadth.

(as both are equal in square)

three numbers will be consecutive and the next three numbers will be the same days of the next week (∴ they will also be consecutive)

and the next three numbers will be the same days of the next week (∴ they will also be consecutive)

Let the smallest number be x.

∴ next number = x + 1

∴next number = x + 3

Next week, same day

Number will be x + 7

And next number to it = x + 7 + 1 = x + 8

Next number = x + 8 + 1 = x + 9

Next week, same day

Number will be x + 7 + 7 = x + 14

Next number = x + 14 + 1 = x + 15

Next number = x + 15 + 1 = x + 16

Sum of all the nine numbers = 90(given)

⇒ x + x + 1 + x + 2 + x + 7 + x + 8 + x + 9 + x + 14 + x + 15 + x + 16 = 90

⇒ 9x + 72 = 90

⇒ 9x = 90-72

⇒ 9x = 18

⇒

⇒x = 2

x + 1 = 3

x + 2 = 4

x + 7 = 9

x + 8 = 10

x + 9 = 11

x + 14 = 17

x + 15 = 18

x + 16 = 19

Hence, the required numbers are 2,3,4,9,10,11,17,18and 19.

Let number of adults be x

Since, total number of persons is 50

Number of children = 50-x

Ticket rate for a child is 10 rupees

∴ ticket rate of 50-x children = 10(50-x) = 500-10x rupees

Ticket rate of an adult is 26 rupees

∴ ticket rate of x adults = 26x rupees

Total amount collected = 740 rupees

⇒ 26x + 500-10x = 740

⇒ 16x + 500 = 740

⇒ 16x = 740-500

⇒ 16x = 240

⇒

⇒x = 15

Hence, Number of children = 50-x = 50-15 = 35

Let the number of girls and boys in the class be x each.

On a particular day

8 boys were absent

∴ number of boys present = x-8

Since, number of girls is twice the number of boys present on that day

x = 2(x-8)

Solving the brackets, we get

X = 2x-2 × 8

⇒ x = 2x-16

⇒ 16 = 2x-x

⇒16 = x

Hence, number of boys and girls in the class = 16 each

Let Vijayan’s present age be x years.

Since, Ajayan is 10 years older than Vijayan

Ajayan’s present age = x + 10 years

Next year,

Vijayan’s age will be = x + 1 years

And Ajayan’s age will be = x + 10 + 1 = x + 11 years

Since it is given that Next year, Ajayan’s age would be double that of Vijayan

⇒ x + 11 = 2(x + 1)

Solving the brackets,we get

x + 11 = 2x + 2 × 1

⇒ x + 11 = 2x + 2

⇒11-2 = 2x-x

⇒9 = x

Hence, Present age of

Ajayan = x + 10 = 9 + 10 = 19 years.

Vijayan = x = 9 years.

Let the required number be x.

∴ Five times the number = 5x

Sum of the number and 4 = x + 4

∴ 3 times the sum of the number and four = 3(x + 4) = 3x + 3 × 4 = 3x + 12

Since, five times the number is equal to three times the sum of the number and 4

∴ 5x = 3x + 12

⇒ 5x-3x = 12

⇒ 2x = 12

⇒

⇒x = 6

Hence, the required number = 6

Lethe original number of women in the society be x.

Since number of men is thrice the number of women,

Number of men = 3x

Number of women after joining of 29 more women = x + 29

Number of men after joining of 16 more men = 3x + 16

Since, the new number of men is twice the number of women

∴ 3x + 16 = 2(x + 29)

Solving the brackets, we get

3x + 16 = 2x + 2 × 29

⇒ 3x + 16 = 2x + 58

⇒ 3x = 2x + 58-16

⇒ 3x = 2x + 42

⇒ 3x-2x = 42

⇒x = 42

Hence, number of women in the society at first = x = 42