Question

(i) $\square$ABCD is a parallelogram. The ratio of $\angle$A and $\angle$B of this parallelogram is 5 : 4. Find the measure of $\angle$B.

(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.

(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.

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

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

Answer 1

(i)
Quadrilateral ABCD is a parallelogram.

Let the measure of $\angle$A and $\angle$B be 5x and 4x, respectively.

Now,

$\angle$A + $\angle$B = 180º (Adjacent angles of a parallelogram are supplementary)

∴ 5x + 4x = 180º

⇒ 9x = 180º

⇒ x = 20º

∴ Measure of $\angle$B = 4x = 4 × 20º = 80º

Thus, the measure of $\angle$B is 80º.

(ii)
Let the present ages of Albert and Salim be 5x years and 9x years, respectively.

5 years hence,

Age of Albert = (5x + 5) years

Age of Salim = (9x + 5) years

It is given that five year hence, the ratio of their ages will be 3 : 5.

$$\begin{gathered} \therefore \frac{5x+5}{9x+5}=\frac{3}{5} \\ \Rightarrow 25x+25=27x+15 \\ \Rightarrow 27x-25x=25-15 \\ \Rightarrow 2x=10 \\ \Rightarrow x=5 \end{gathered}$$
∴ Present age of Albert = 5x = 5 × 5 = 25 years

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

Thus, the present age of Albert is 25 years and the present age of Salim is 45 years.

(iii)
Let the length and breadth of the rectangle be 3x cm and x cm, respectively.

Perimeter of the rectangle = 36 cm

∴ 2(Length + Breadth) = 36 cm

⇒ 2(3x + x) = 36

⇒ 2 × 4x = 36

⇒ 8x = 36

⇒ x = 4.5

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

Breadth of the rectangle = x = 4.5 cm

Thus, the length and breadth of the rectangle is 13.5 cm and 4.5 cm, respectively.

(iv)
Let the two numbers be 31x and 23x.

Sum of the two numbers = 216

∴ 31x + 23x = 216

⇒ 54x = 216

⇒ x = $\frac{216}{54}$ = 4

∴ One number = 31x = 31 × 4 = 124

Other number = 23x = 23 × 4 = 92

Thus, the two numbers are 92 and 124.

(v)
Let the two numbers be 10x and 9x.

Product of the two numbers = 360

∴ 10x × 9x = 360

⇒ 90x² = 360

⇒ x² = 4

⇒ x = 2

∴ One number = 10x = 10 × 2 = 20

Other number = 9x = 9 × 2 = 18

Thus, the two numbers are 18 and 20.