Linear Equations in One Variable - 8th math

Question (1): If you subtract from a number and multiply the result by , you get . What is the number?

Solution:

Let the required number is n.

Now, according to question, we have

After dividing both sides by 1/2 we get,

After transposing – 1/2 to RHS we get,

Thus, required number Answer

Alternate method

Let the required number is n.

Now, according to question, we have

After transposing 1/2 to RHS, we get

Now, after transposing 1/2 to RHS, we get

Thus, required number Answer

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

Solution:

Given, perimetre of the rectangular swimming pool = 154 m

And Length = 2 m more than twice of the breadth.

Let the breadth of the swimming pool = b metre

∴ Length of the swimming pool = 2 b m + 2 metre

Thus, Length and breadth of the given swimming pool = ?

We know that, Perimetre of a Rectangle = 2 (length + breadth)

∴ 154 = 2 [ (2b + 2) + b ]

⇒ 154 = 2(2 b + 2 + b )

⇒ 154 = 2 (3 b + 2 )

⇒ 154 = 6 b + 4

After subtracting 4 from both sides, we get

⇒ 154 – 4 = 6 b + 4 – 4

⇒ 150 m = 6 b

After dividing both sides by 6, we get

⇒ 25 = b

Therefore, breaadth (b) = 25 m

Now, since length = 2 b + 2

Therefore, by substituting the value of breadth (b), we get

2 × 25 + 2 m= 50 + 2 = 52

Thus, length of the given pool = 52 m And breadth = 25 m Answer

Question (3) The base of an isosceles triangle is 4/3 cm The perimeter of the triangle is cm. What is the length of either of the remaining equal sides?

Solution:

Given,

Base of the isosceles triangle = cm

And, Perimeter cm

Thus, length of the remaining equal sides = ?

We know that, Isosceles triangles have two sides equal.

Now, We know that,

Perimeter of an isosceles triangle = Sum of two equal sides + third side

Let the length of equal sides of the given isosceles triangle = a

And length of unequal side = b

Therefore, Perimeter = 2a + b

Therefore,

After transposing 4/3 to RHS, we get

After transposing 2 to RHS, we get

cm

Thus, length of the one of the equal sides = 7/5 cm Answer

Question (4) Sum of two numbers is 95. If one exceeds the other by 15, find the numbers.

Solution:

Given, sum of two numbers = 95

One of number exceeds other by 15

Therefore, numbers = ?

Let one number is a

Therefore, according to question second number = a + 15

Now, as given in the question, sum of two numbers = 95

Therefore,

a + a + 15 = 95

⇒ 2a + 15 = 95

By transposing 15 to RHS, we get

⇒ 2a = 95 – 15

⇒ 2 a = 80

After dividing both sides by 2, we get

⇒ a=40

Now, since second number = a + 15

Therefore, by substituting the value of a we get

The second number = 40 + 15 = 55

Thus, first number = 40 and second number = 55 Answer

Question (5) Two numbers are in the ratio 5:3. If they differ by 18, what are the numbers?

Solution:

Given, ratio of the two numbers = 5:3

And their difference = 18

Therefore, numbers = ?

Let first number = 5x

And second number = 3x

As per question, 5x – 3x = 18

⇒ 2x = 18

After dividing both sides by 2, we get

⇒ x = 9

Now, we have first number = 5 x = 5 × 9

Thus, first number = 45

And, we have the second number = 3 x = 3 × 9

Thus, second number = 27

Thus, required numbers are 45 and 27 Answer

Question (6) Three consecutive integers add up to 51. What are these integers?

Solution:

Given, the sum of three consecutive integers = 51

Therefore, integers = ?

Let the first integer = a

Therefore, second consecutive integer = a + 1

And, third consecutive integer = a + 2

Since, according to question sum of the given three consecutive number = 51

Therefore,

a + (a +1) + (a + 2) = 51

⇒ a + a + 1 + a + 2 = 51

After rearranging the above expression

⇒ a + a + a + 1 + 2 = 51

⇒ 3a + 1 + 2 = 51

⇒ 3a + 3 = 51

After transposing 3 to the RHS, we get

3a = 51 – 3

⇒ 3a = 48

After dividing both sides by 3, we get

⇒ a =16

Now, the first integer = 16

Therefore, next two consecutive integers

= 16 + 1 = 17 and 17 + 1 = 18

Thus, required three consecutive integers are 16, 17 and 18 Answer