**Question 1.**

Solution:

**Question 2.Find the following ratios.i. The ratio of radius to circumference of the circle.ii. The ratio of circumference of circle with radius r to its area.iii. The ratio of diagonal of a square to its side, if the length of side is 7 cm.iv. The lengths of sides of a rectangle are 5 cm and 3.5 cm. Find the ratio of numbers denoting its perimeter to area.Solution:**

i. Let the radius of circle be r.

then, its circumference = 2πr

Ratio of radius to circumference of the circle

The ratio of radius to circumference of the circle is 1 : 2π.

ii. Let the radius of the circle is r.

∴ circumference = 2πr and area = πr^{2}

Ratio of circumference to the area of circle

∴ The ratio of circumference of circle with radius r to its area is 2 : r.

iii. Length of side of square = 7 cm

∴ Diagonal of square = √2 x side

= √2 x 7

= 7 √2 cm

Ratio of diagonal of a square to its side

∴ The ratio of diagonal of a square to its side is √2 : 1.

iv. Length of rectangle = (l) = 5 cm,

Breadth of rectangle = (b) = 3.5 cm

Perimeter of the rectangle = 2(l + b)

= 2(5 + 3.5)

= 2 x 8.5

= 17 cm

Area of the rectangle = l x b

= 5 x 3.5

= 17.5 cm^{2}

Ratio of numbers denoting perimeter to the area of rectangle

∴ Ratio of numbers denoting perimeter to the area of rectangle is 34 : 35.

**Question 3.Compare the followingSolution:**

**Question 4.Solve.ABCD is a parallelogram. The ratio of ∠A and ∠B of this parallelogram is 5 : 4. FInd the measure of ∠B. [2 MarkslSolution:**

Ratio of ∠A and ∠B for given parallelogram is 5 : 4

Let the common multiple be x.

m∠A = 5x°and m∠B=4x°

Now, m∠A + m∠B = 180° …[Adjacent angles of a parallelogram arc supplementary]

∴ 5x° + 4x°= 180°

∴ 9x° = 180°

∴ x° = 20°

∴ m∠B=4x°= 4 x 20° = 80°

∴ The measure of ∠B is 800.

**ii. The ratio of present ages of Albert and Salim is 5 : 9. Five years hence ratio of their ages will be 3 : 5. Find their present ages.Solution:**

The ratio of present ages of Albert and Salim is 5 : 9

Let the common multiple be x.

∴ Present age of Albert = 5x years and

Present age of Salim = 9x years

After 5 years,

Albert’s age = (5x + 5) years and

Salim’s age = (9x + 5) years

According to the given condition,

Five years hence ratio of their ages will be 3 : 5

∴ 5(5x + 5) = 3(9x + 5)

∴ 25x + 25 = 27x + 15

∴ 25 – 15 = 27 x – 25 x

∴ 10 = 2x

∴ x = 5

∴ Present age of Albert = 5x = 5 x 5 = 25 years

Present age of Salim = 9x = 9 x 5 = 45 years

∴ The present ages of Albert and Salim are 25 years and 45 years respectively.

**iii. The ratio of length and breadth of a rectangle is 3 : 1, and its perimeter is 36 cm. Find the length and breadth of the rectangle.Solution:**

The ratio of length and breadth of a rectangle is 3 : 1

Let the common multiple be x.

Length of the rectangle (l) = 3x cm

and Breadth of the rectangle (b) = x cm

Given, perimeter of the rectangle = 36 cm

Since, Perimeter of the rectangle = 2(l + b)

∴ 36 = 2(3x + x)

∴ 36 = 2(4x)

∴ 36 = 8x

Length of the rectangle = 3x = 3 x 4.5 = 13.5 cm

∴ The length of the rectangle is 13.5 cm and its breadth is 4.5 cm.

**iv. The ratio of two numbers is 31 : 23 and their sum is 216. Find these numbers.Solution:**

The ratio of two numbers is 31 : 23

Let the common multiple be x.

∴ First number = 31x and

Second number = 23x

According to the given condition,

Sum of the numbers is 216

∴ 31x + 23x = 216

∴ 54x = 216

∴ x = 4

∴ First number = 31x = 31 x 4 = 124

Second number = 23x = 23 x 4 = 92

∴ The two numbers are 124 and 92.

**v. If the product of two numbers is 360 and their ratio is 10 : 9, then find the numbers.Solution:**

Ratio of two numbers is 10 : 9

Let the common multiple be x.

∴ First number = 10x and

Second number = 9x

According to the given condition,

Product of two numbers is 360

∴ (10x) (9x) = 360

∴ 90x

^{2}= 360

∴ x

^{2}= 4

∴ x = 2 …. [Taking positive square root on both sides]

∴ First number = 10x = 10x

^{2}= 20

Second number = 9x = 9x

^{2}= 18

∴ The two numbers are 20 and 18.

**Question 5.If a : b = 3 : 1 and b : c = 5 : 1, then find the value of [3 Marks each]Solution:**

Given, a : b = 3 : 1

Substituting (ii) in (i),

we get a = 3(5c)

∴ a = 15c …(iii)

∴ 0.04 x 0.4 x a = (0.4)^{2} x (0.04)^{2} x b … [Squaring both sides]

**Question 7. (x + 3) : (x + 11) = (x – 2) : (x + 1), then find the value of x.Solution:**(x + 3) : (x + 11) = (x- 2) : (x+ 1)

∴ (x + 3)(x +1) = (x – 2)(x + 11)

∴ x(x +1) + 3(x + 1) = x(x + 11) – 2(x + 11)

∴ x

^{2}+ x + 3x + 3 = x

^{2}+ 1 lx – 2x – 22

∴ x

^{2}+ 4x + 3 = x

^{2}+ 9x – 22

∴ 4x + 3 = 9x – 22

∴ 3 + 22 = 9x – 4x

∴ 25 = 5x

∴ x = 5