Question 10 of 11intermediate🔧 ApplyLong Answer3 marks

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.

Correct Answer

No, this situation is not possible.

Exercise: EXERCISE 4.3 | Q: 4 | (Page 10)
For More Understanding

Explanation

This is Exercise 4.3, Question 4 from the NCERT textbook. The question tests whether students can formulate a quadratic equation from a word problem and determine if real solutions exist using the discriminant. When the discriminant is negative, the quadratic equation has no real roots, making the situation impossible.

Solution Steps

  1. Step 1: Let the present ages of the two friends be xx years and yy years.

    Given: x+y=20x + y = 20, so y=20−xy = 20 - x.

  2. Step 2: Four years ago, their ages were (x−4)(x - 4) years and (y−4)(y - 4) years.

    Given:

    (x−4)(y−4)=48(x - 4)(y - 4) = 48

  3. Step 3: Substituting y=20−xy = 20 - x:

    (x−4)(20−x−4)=48(x - 4)(20 - x - 4) = 48

    which gives

    (x−4)(16−x)=48(x - 4)(16 - x) = 48

  4. Step 4: Expanding:

    −x2+20x−64=48-x^2 + 20x - 64 = 48

    which simplifies to

    x2−20x+112=0x^2 - 20x + 112 = 0

  5. Step 5: Calculating discriminant:

    D=b2−4ac=(−20)2−4(1)(112)=400−448=−48D = b^2 - 4ac = (-20)^2 - 4(1)(112) = 400 - 448 = -48

  6. Step 6: Since D<0D < 0, the quadratic equation has no real roots.

    ∴ Therefore, this situation is not possible.

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