A brother is elder to his sister by 7 years. Five years ago, the product of their ages was 30. What is the current age of the brother?

Let the current age of the brother be x years.
Since the brother is elder to his sister by 7 years, the current age of the sister is (x−7) years.

Five years ago:

Age of the brother = x−5

Age of the sister = (x−7)−5=x−12

According to the problem, the product of their ages 5 years ago was 30:

(x−5)(x−12)=30
x
2
−12x−5x+60=30
x
2
−17x+30=0
Solving the quadratic equation by splitting the middle term:

x
2
−15x−2x+30=0
x(x−15)−2(x−15)=0
(x−2)(x−15)=0
This gives two possible values for x:
x=2 or x=15

If x=2, the brother's current age is 2 years, which makes the sister's age negative (2−7=−5), which is impossible.
Thus, x=15.

The current age of the brother is 15 years.