Is the following situation possible? If so, determine their present ages.

The sum of the ages of two friends is $20$ years. Four years ago, the product of their ages in years was $48$ .

Question

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

Homi · Accepted Answer

Understand the Question

Let the present age of the first friend be $x$ years. Since the sum of their ages is $20$ years, the present age of the other friend is $(20 - x)$ years.
Express their ages $4$ years ago and set their product equal to $48$ to form a quadratic equation $ax^2 + bx + c = 0$ .
Evaluate the discriminant $D = b^2 - 4ac$ . If $D < 0$ , the equation has no real roots, meaning the situation is not mathematically possible.

Step 1 · Formulate the Quadratic Equation

Let the present age of the first friend be $x$ years. Present age of the second friend $= (20 - x) \text{ years}$ .

Four years ago:

Age of first friend $= (x - 4) \text{ years}$
Age of second friend $= (20 - x - 4) = (16 - x) \text{ years}$

Given that the product of their ages four years ago was $48$ :

\begin{aligned} (x - 4)(16 - x) &= 48 \\ 16x - x^2 - 64 + 4x &= 48 \\ -x^2 + 20x - 64 &= 48 \\ -x^2 + 20x - 64 - 48 &= 0 \\ -x^2 + 20x - 112 &= 0 \\ x^2 - 20x + 112 &= 0 \end{aligned}

Step 2 · Check the Discriminant for Real Roots

Comparing $x^2 - 20x + 112 = 0$ with $ax^2 + bx + c = 0$ : $a = 1, \quad b = -20, \quad c = 112$

Calculate the discriminant $D = b^2 - 4ac$ :

\begin{aligned} D &= (-20)^2 - 4(1)(112) \\ &= 400 - 448 \\ &= -48 \end{aligned}

Since $D < 0$ , the quadratic equation has no real roots. Therefore, the given situation is not possible.

Answer

The given situation is not possible.

Common Mistakes

Past Age Formulation: Forgetting to subtract $4$ from both friends' ages, or writing $(20 - x + 4)$ instead of $(20 - x - 4) = 16 - x$ .
Sign Errors in Squaring: Incorrectly calculating $(-20)^2$ as $-400$ instead of $+400$ .
Interpreting the Discriminant: Overlooking that a negative discriminant ( $D < 0$ ) means no real roots exist, indicating that the described real-world scenario is impossible.

Question 4

More questions in Exercise 4.3