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:

✰ The sum of ages of two friends is 30 years.

✰ Four years ago, the product of their ages in year was 112.

To find:

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

Solution:

Let the present age of first friend be x.

Then the present age of second friend = 30 - x

4 years ago,

⤳ Age of first friend = x - 4

⤳ Age of second friend = 30 - x - 4

⤳ Age of second friend = 26 - x

Now, we know that the product of their ages is 112. Thus, we will multiply both their ages and after doing the required calculations we will get a quadratic equation.

According to the question,

➛ ( x - 4 ) ( 26 - x ) = 112

➛ 26x - x² - 104 + 4x = 112

➛ - x² + 30x - 216 = 0

➛ x² - 30x + 216 = 0

Now,

✧ D = b² - 4ac

Here,

• b = - 30
• a = 1
• c = 216

Putting the values, we have:

➛ D = (- 30)² - 4 × 1 × 216

➛ D = 900 - 864

➛ D = 36

∴ If can determine the ages of both the friends as the roots are real.

So, let's continue...

➛ x² - 12x - 18x + 216 = 0

➛ x ( x - 12 ) - 18 ( x - 12 ) = 0

➛ ( x - 12 ) ( x - 18 ) = 0

Take

➛ x - 12 = 0

➛ x = 12

Now, take

➛ x - 18 = 0

➛ x = 18

∴ The present ages of both the friends are 12 years and 18 years respectively.

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