Question

(i) If the difference of the squares of two consecutive odd numbers is 40, find the larger number.
(ii) If two consecutive even numbers are such that half of the larger exceeds one fourth of the smaller by 5. Find the numbers.

Answer 1

Answer :-

i) 11

ii) 16 and 18

Solution :-

i)

Let the two consecutive odd numbers be x, (x + 2)

Difference of their squares = 40

⇒ (x + 2)² - x² = 40

⇒ x² + 2² + 2(x)(2) - x² = 40

⇒ 4 + 4x = 40

⇒ 4x = 40 - 4

⇒ 4x = 36

⇒ x = 36/4 = 9

Larger no. = (x + 2) = 9 + 2 = 11

Therefore the larger number is 11

ii)

Let the two consecutive even numbers be x, (x + 2)

Given

1/2 of larger = 1/4 of smaller + 5

⇒ 1/2 * (x + 2) = 1/4 * x + 5

⇒ (x + 2)/2 = x/4 + 5

⇒ (x + 2)/2 - x/4 = 5

Taking LCM

⇒ { 2(x + 2) - x } / 4 = 5

⇒ (2x + 4 - x)/4 = 5

⇒ (x + 4)/4 = 5

⇒ x + 4 = 5 * 4

⇒ x + 4 = 20

⇒ x = 20 - 4 = 16

Therefore the number are 16 and 18.

Answer 2

Answer:

Solution:-

i) Given:-

• Difference between squares of two consecutive odd numbers = 40

To find:-

• Larger number = ?

Let's take the first consecutive odd number as y.

$\therefore$ 2nd consecutive number = (y + 2)

A/Q,

$\Rightarrow\rm (y + 2)^2 - y^2 = 40 \\\\\Rightarrow\rm \cancel{y^2} + 2y * 2 + 2^2 -\cancel{ y^2 }= 40 \\\\\Rightarrow\rm 4y + 4 = 40 \\\\\Rightarrow\rm 4y = 40 - 4 \\\\\Rightarrow\rm 4y = 36 \\\\\Rightarrow\rm y = 36/4 \\\\\Rightarrow\rm y = {\boxed{\rm{9}}}$

$\because\rm Larger \:number = y + 2$

$\therefore\rm Larger \: number = y + 2 = 9 + 2$

${\underline{\therefore{\text{Larger \: number = 11 }}}}$

$\rule{200}{1}$

ii) Given:-

• 1/2 of larger consecutive number = 1/4 of smaller number + 5

To find:-

• Consecutive even numbers = ?

Solution:-

Let's take smaller even number be y .

$\therefore$ Larger even number = (y + 2)

A/Q,

$\Rightarrow\rm \dfrac{1}{2} (y + 2) = \dfrac{1}{4}(y) + 5 \\\\\Rightarrow\rm \dfrac{y + 2}{2} = \dfrac{y}{4} + 5 \\\\\Rightarrow\rm \dfrac{y + 2}{2} = \dfrac{y + 20}{4} \\\\\Rightarrow\rm 4(y + 2) = 2(y + 20) \\\\\Rightarrow\rm 4y + 8 = 2y + 40 \\\\\Rightarrow\rm 4y - 2y = 40 - 8 \\\\\Rightarrow\rm 2y = 32 \\\\\Rightarrow\rm y = 32/2 \\\\\Rightarrow\rm y = {\boxed{\rm{16}}}$

$\therefore\rm Larger \: number = y + 2 = 16 + 2$

$\therefore\rm Larger \: number = 18$

${\underline{\underline{\therefore{\text{Numbers \: are \:16 \: \& \: 18 .}}}}}$