# Larry is 2 years younger than Mary. The difference between the squares of their ages is 28. How old is each?

Jan 28, 2016

Mary is $8$; Larry is $6$

#### Explanation:

Let
$\textcolor{w h i t e}{\text{XXX}} L$ represent Larry's age, and
$\textcolor{w h i t e}{\text{XXX}} M$ represent Mary's age.

We are told:
[equation 1]$\textcolor{w h i t e}{\text{XXX}} L = M - 2$
and
[equation 2]$\textcolor{w h i t e}{\text{XXX}} {M}^{2} - {L}^{2} = 28$

Substituting $M - 2$ from equation [1] for $L$ in equation [2]
$\textcolor{w h i t e}{\text{XXX}} {M}^{2} - {\left(M - 2\right)}^{2} = 28$

$\textcolor{w h i t e}{\text{XXX}} {M}^{2} - \left({M}^{2} - 4 M + 4\right) = 28$

$\textcolor{w h i t e}{\text{XXX}} 4 M - 4 = 28$

$\textcolor{w h i t e}{\text{XXX}} 4 M = 32$

$\textcolor{w h i t e}{\text{XXX}} M = 8$

Substituting $8$ for $M$ in equation [1]
$\textcolor{w h i t e}{\text{XXX}} L = 8 - 2 = 6$

Jan 28, 2016

$6 \mathmr{and} 8$

#### Explanation:

Let the age of $L a r r y = x$

Age of $M a r y = x + 2$ (Difference of their ages are 2)

Given that the difference between the squares of their ages is 28

So,${\left(2 + x\right)}^{2} - {x}^{2} = 28$

Use the formula ${\left(a + b\right)}^{2} = {a}^{2} + 2 a b + {b}^{2}$

$\rightarrow \left(4 + 4 x + {x}^{2}\right) - {x}^{2} = 28$

$\rightarrow 4 + 4 x + {x}^{2} - {x}^{2} = 28$

$\rightarrow 4 + 4 x = 28$

$\rightarrow 4 x = 28 - 4$

$4 x = 24$

$x = \frac{24}{4} = 6$

We know now that the Age of

$L a r r y = 6$

So, Age of $M a r y = \left(x + 2\right) = 6 + 2 = 8$