Question

Is the following situation possible ? If so, determine their present age. The sum of ages of two friends is 30 years. Four years ago, the product of their ages in year was 112.​

Answer 1

Given,

• Sum of ages of two friends = 30 years
• Product of ages of two friends = 112 years

Let the ages of two friends be x and y respectively.

Then,

• x + y = 30 ––– (i)
• xy = 112 ––– (ii)

$\boxed{ \tt{y = \frac{112}{x}} } \qquad \qquad \{ \text{from } \rm {eq}^{n} \: (i) \}$

And,

$\to \tt{}x + \frac{112}{x} = 30 \qquad\qquad \\ \\ \to \tt{} {x}^{2} + 112 = 30x \\ \\ \to \tt{} {x}^{2} - 30x + 112 = 0$

• On comparing with ax² + bx + c, we have

a = 1, b = - 30, c = 112

Now, finding the discriminant to check the nature of roots of the obtained equation,

D = b² – 4ac

→ D = (-30)² - 4(1)(112)

→ D = 900 - 448 = 452

As, the discriminant is positive integer, the roots of this equation will be distinct & real.

Thus, Getting back to the equation,

$\to \tt {x}^{2} - 30 x + 112 = 0 \\ \\ \to \tt {(x)}^{2} - 2(x)(15) + {(15)}^{2} - {(15)}^{2} + 112 = 0 \\ \\ \to \tt {(x - 15)}^{2} - 225 + 112 = 0 \\ \\ \to \tt {(x - 15)}^{2} - 113 = 0 \\ \\ \to \tt {(x - 15)}^{2} = 113 \\ \\ \to \tt x - 15 = \sqrt{113} \\ \\ \sf\therefore \: \: x = \red{\sqrt{113} - 5}$

But, The ages can't have values which is irrational in nature.

Thus, our approach is incomplete.