Question

★ Math Problem !!

Is the following situation $\sf{\red{possible}}$ ? 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 i years was 48.

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Answer 1

First solution

Let the ages of two people be $a,b$.

The sum of the ages of two friends is 20 years.

→ $a+b=20$ ...[Eqn. 1]

Four years ago, the product of their ages was 48.

→ $(a-4)(b-4)=48$ ...[Eqn. 2]

Let's find the product of their ages at now. Using both equations,

→ $ab-4a-4b+16=48$

→ $ab-4(a+b)+16=48$

→ $ab-80+16=48$

→ $ab=112$ ...[Eqn. 3]

Suppose a quadratic equation with their ages as zeros. By factor theorem,

→ $(x-a)(x-b)=0$

→ $x^2-(a+b)x+ab=0$

We can use the information given in [Eqn. 2, 3].

→ $x^2-20x+112=0$

→ $D=20^2-4\times 112$

→ $D=400-448<0$

Their ages are both imaginary numbers. So, this situation is not possible.

Second solution

Let the age of two people 4 years ago be $a,b$.

→ $a+b=12$

→ $ab=48$

Using the same method as Sol. 1, we find two numbers $a,b$.

→ $x^2-12x+48=0$

→ $D=12^2-4\times48$

→ $D=144-192<0$

This situation is not possible as their ages are both imaginary.

Answer 2

Given

• Sum of their ages = 20
• The product of their ages (4 years ago) = 48

To Find

• Thier present ages

Solution

☯ x + y = 20, so

• Four years ago their ages would have been (x-4) & (20-x-4) or (16-x)

━━━━━━━━━━━━━━━━━━━━━━━━━

✭ According to the Question :

→ (x-4)(16-x) = 48

→ 16x + 4x - 64 - x² = 48

→ -x² + 20x - 64 = 48

→ x² - 20x + 64 = -48 [Div the whole eq by "-"]

→ x² - 20x + 64 + 48 = 0

→ x² - 20x + 112 = 0

• So now we shall check if the equation has any real roots & if it does then it means the situation is possible and if it doesn't then it's impossible

→ b² - 4ac

→ (-20)² - 4(1)(112)

→ 400 - 448

→ -48 < 0

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Therefore,

The given situation is impossible as the quadratic equations has no real roots.