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Three consecutive integers are as such when they are taken in increasing order and multiplied by 2, 3, and 4 respectively, they add up to 56. Find these numbers.

Let x,x+1 ,x+2 are three consecutive integers in increasing order

Then according to question

$2(x) + 3(x+1) + 4(x+2) =56$

$2x + 3x +3 + 4x + 8 =56$

$9x + 11 =56$

$9x = 56 -11$

$9x = 45$

x=5

So,numbers are 5,6,7

The perimeter of a rectangular swimming pool is 154 meters. Its length is 2 m more than twice its breadth. What are the length and breadth of the pool?

Let breadth be x

Then length =2x + 2

According to Question

Perimeter =154 m

$2(L + B) = 154$

$2( 2x+2 + x)=154$

$2(3x + 2)=154$

$6x + 4 =154$

$6x= 150$

x= 25 m

So Length =52 m and breadth =25 m

Sum of two numbers is 95. If one exceeds the other by 15 find the numbers.

Let one number be x, then other number will be x +15

According to question

Sum of numbers=95

$x + x + 15=95$

$2x+15=95$

$2x=95-15$

$2x=80$

$x=40$

So, the two numbers are 40 and 55

Two numbers are in the ration 4:3. If they differ by 18, find these numbers

When the numbers are in ratio, we assume numbers as the value in ratio multiplied by variable

So, Let numbers be 4x and 3x

Now Difference of numbers =18

$4x -3x=18$

$x=18$

So,numbers are 72 and 54

Three consecutive integers add up to 57. What are these integers?

Let x,x+1 ,x+2 are three consecutive integers

Now sum of these=57

$x + x + 1 + x +2=57$

$3x + 3=57$

$3x=54$

$x=18$

So, numbers are are 18,19,20

There is a narrow rectangular plot. The length and breadth of the plot are in the ratio of 11:4. At the rate of Rs. 100 per meter it will cost village panchayat Rs.75000 to fence the plot. What are the dimensions of the plot?

Let lenght and breadth are 11x and 4x

Perimeter of plot = 2(L+ B) = 2(11x + 4x)=30x

Now given

$100 \times 30x = 75000$

$3000x=75000$

$x= \frac {75000}{3000}$

$x=25 m$

So, lenght and breadth are $11 \times 25= 225 m$ and $4 \times 25= 100 m$

Convert the following statements into equations.

(a) 3 added to a number is 11

(b) 2 subtracted from a number is equal to 15.

(c) 3 times a number decreased by 2 is 4.

(d) 2 times the sum of the number x and 7 is 13.

a. $x +3 =11$

b. $x- 2=15$

c. $3x -2=4$

d. 2(x+ 7) =13

Amina thinks of a number and subtracts 5/2 from it. She multiplies the result by 8. The final result is 3 times her original number. Find the number

Let x be the number,then

$ 8(x - \frac {5}{2}) = 3x$

$ 8x - 20 =3x$

$ 8x -3x = 20$

$5x = 20$

$x=4$

A number is 12 more than the other. Find the numbers if their sum is 48.

Let x be the number , then other number will be x + 12

According to question

$x+ x +12=48$

$2x + 12=48$

$x=18$

So, the numbers are 18 and 30

The sum of three consecutive odd numbers is 51. Find the numbers.

Let x,x+1 ,x+2 are three consecutive integers

Now sum of these=51

$x + x + 1 + x +2=51$

$3x + 3=51$

$3x=48$

$x=16$

So, numbers are are 16,17,18

Jane is 6 years older than her younger sister. After 10 years, the sum of their ages will be 50 years. Find their present ages.

Let her younger sister age is x, the Jane age is x + 6

After 10 years, Jane age will be $x+6 + 10=x+16$ and sister age will be $x +10$

Now ,

$x+16 + x +10=50$

$2x + 26=50$

$2x=24$

$x=12$

So, Jane age is 18 years and her sister age is 12 years

The denominator of a fraction is greater than the numerator by 8. If the numerator is increased by 17 and denominator is decreased by 1, the number obtained is 3/2, find the fraction.

Let numerator be x ,then denominator is x + 8 and fraction is $\frac {x}{x+8}$

If the numerator is increased by 17, New numerator becomes $x+17$

If denominator is decreased by 1, New denominator becomes $x +8 -1 =x+7$

Now

$\frac {x + 17}{x+7} = \frac {3}{2}$

By cross multiplication

$2(x+17) = 3(x+7)$

$2x + 34 = 3x +31$

$ 3x +31=2x + 34 $

$ 3x -2x = 34 -31$

$x =3$

So fraction is $\frac {3}{11}$

A sum of Rs 2700 is to be given in the form of 63 prizes. If the prize is of either Rs 100 or Rs 25, find the number of prizes of each type.

Let x be the type of Prize Rs 100.

Since the total number of prize is 63, Rs 25 type will be 63 -x

Now according to Question,

The value of these total prizes = Rs 2700

$100 \times x + (63 -x) \times 25 = 2700$

$ 100x + 1575 -25 x = 2700$

$ 75x = 2700 -1575$

$75x = 1125$

$x=15$

So 15 Rs 100 type prize and 47 Rs 25 type prizes were present

In an isosceles triangle, the base angles are equal and the vertex angle is 80°. Find the measure of the base angles.

Let x be the base angle

Now

$ x +x + 80 = 180$

$2x =100$

$x=50$

True and False statement

a. The three consecutive positive integer can be written as x, x+1, x+2 where x is any positive integer

b. The cost of a pencil is 5 Rs more than the cost of an eraser. If the cost of 8 pencils and 10 erasers is Rs 130, then the cost of pencil is 10 Rs

c. if $2(x-13) = 14$, then $x=20$

d. The shifting of one number from one side of linear equation to another side is called transposition

e. The three consecutive multiple of 7 would 7x,7x+7, 7x+21

a. True

b. True

c. False , x=25

d. True

e. False . Correct numbers are 7x,7x+7, 7x+14

Fifteen years from now Ravi's age will be 4 times his current age. What is his current age?

a. 4 year

b. 5 years

c. 6 years

d. 3 years

Let x be the current age

Age after 15 years =x + 15

Now

$x+ 15 = 4x$

$ 4x= x + 15$

$ 4x -x =15$

$ 3x=15$

$x=3 years$

Ramesh is a cashier in a Canara bank. he has notes of denominations of Rs. 100, 50 and 10 respectively. The ratio of number of these notes is 2:3:5 respectively. The total cash with Ramesh is 4,00,000. How many notes of each denomination does he have?

a. 2000 100’s notes,3000 50’s notes and 5000 50’s notes

c.4000 100’s notes,6000 50’s notes and 10000 50’s notes

c.1000 100’s notes,1500 50’s notes and 2500 50’s notes

d.None of these

Let the number of notes are 2x,3x and 5x

$ 100 \times 2x + 50 \times 3x + 10 \times 5x = 400000$

$200x + 150 x + 50x =400000$

$400x = 400000$

$x= 1000$

So ,Banks as 2000 100’s notes,3000 50’s notes and 5000 50’s notes

If the perimeter of the triangle is 30 cm, find the length of each side

$2x+3 +5x +1 + 5x+2=30$

$12x=30-6=24$

$x=2$

So sides are 7,11,12

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**Notes****Worksheets****Ncert Solutions**- NCERT Solutions for Class 8 Maths Chapter 2 Exercise 2.1
- NCERT Solutions for Linear equations Class 8 maths Chapter 2 Exercise 2.2
- class 8 maths chapter 2 exercise 2.3 solutions
- class 8 maths chapter 2 exercise 2.4 solutions
- class 8 maths chapter 2 exercise 2.5 solutions
- class 8 maths chapter 2 exercise 2.6 solutions

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