Question

The difference of the squares of two consecutive even natural numbers is 92. Taking x as the smaller of the two numbers, form an equation in x, and hence find the larger of the two numbers.

• A. $18$
• B. $24$
• C. $23$
• D. $21$

Answer 1

The correct option is B $24$

Given that,

The difference of squares of two consecutive even natural numbers is $92$.

The smaller number is $x$.

To find out,

$\left(i\right)$ The equation in $x$ representing the given information.

$\left(ii\right)$ The larger of the two numbers.

$\left(i\right)$ If the smaller number is $x$, then the larger number will be $x+2$

According to the question,

$(x+2{)}^2-x^2=92$

Hence, $(x+2{)}^2-x^2=92$ is the equation that represents the given information.

$\left(ii\right)$ Solving $(x+2{)}^2-x^2=92$

We know that, $(a+b{)}^2=a^2+2ab+b^2$

Applying the same identity, we get:

$(x^2+4x+4)-x^2=92$

$\Rightarrow x^2+4x+4-x^2=92$

$\Rightarrow 4x+4=92$

Dividing both sides by $4$, we get:

$x+1=23$

$x=23-1$

$=22$

Hence, $x=22$

$\therefore x+2=22+2$

$=24$

Hence, the larger of the two numbers is $24$.