Question: The difference of the present ages of two sisters is 5 years. If the product of their ages is 204, find the age of two sisters.

Solution:

Let the present ages of one sister be x years and that of the other be y years.

According to the question,

Condition I,
The difference of their present ages is 5 years.
$or, x - y = 5$ - (i)

Condition II,
Product of their ages is 204.
$or, xy = 204$
$or, x = \frac{204}{y}$ - (ii)

Put value of x from equation (ii) in equation (i), we get,

$or, \dfrac{204}{y} - y = 5$

$or, \dfrac{204 - y²}{y} = 5$

$or, 204 - y² = 5y$

$or, y² +5y -204 = 0$

$or, y² + (17-12)y -204 = 0$

$or, y² + 17y -12y -204 = 0$

$or, y(y +17) -12(y +17) = 0$

$or, (y -12)(y +17) = 0$

Either,
$y -12 = 0$
$\therefore y = 12$

Or,
$y +17 = 0$
$\therefore y = -17$

Since, age in years can never be negative, put value of y = 12 in equation (ii), we get,
$x = \frac{204}{12}$
$\therefore x = 17$

So, (x,y) = (17,12)

Therefore, the required present ages of two sisters in years are 17 years and 12 years, respectively.

#SciPiPupil