A Boy Is 6 years Younger Than His Sister.The Product Of Their Ages Is 135.Find The Ages

x=9years and x=15yers

Let's assume that the age of the boy is represented by the variable "x" years.

According to the given information, we know that the sister's age is 6 years older, so we can represent her age as "x + 6" years.

The product of their ages is 135, so we can set up the equation:

x * (x + 6) = 135

Expanding and rearranging the equation:

x^2 + 6x = 135

Now, let's solve this quadratic equation.

To factorize it, we can look for two numbers that multiply to give 135 and add up to 6. The numbers are 15 and 9.

So, we have:

(x + 15)(x - 9) = 0

Setting each factor equal to zero:

x + 15 = 0 or x - 9 = 0

Solving these equations:

x = -15 or x = 9

Since we are looking for the age of a boy, the age cannot be negative. Therefore, x = 9.

Thus, the age of the boy is 9 years and the age of the sister is x + 6 = 9 + 6 = 15 years.

To find the ages of the boy and his sister, let's set up some equations based on the information given.

Let's assume the sister's age is represented by S, and the boy's age is represented by B.

We are given that the boy is 6 years younger than his sister. So, we can write the equation:
B = S - 6

The product of their ages is 135. So, we can write the equation:
B * S = 135

Now, we have a system of equations:
B = S - 6
B * S = 135

Let's solve this system of equations.

By substituting the first equation into the second equation, we get:
(S - 6) * S = 135

Expanding the equation, we have:
S^2 - 6S = 135

Rearranging the equation to bring all terms to one side, we have:
S^2 - 6S - 135 = 0

Now, we can solve this quadratic equation for S using factoring, completing the square, or the quadratic formula. In this case, factoring would be the simplest approach.

The factors of -135 that sum up to -6 are -15 and 9. So, we can rewrite the equation as:
(S - 15)(S + 9) = 0

Setting each factor equal to zero, we get two possible values for S:
S - 15 = 0 or S + 9 = 0

Solving these equations, we find:
S = 15 or S = -9

Since age cannot be negative, we discard the solution S = -9.

So, the sister's age is 15 (S = 15).

Substituting this value back into the equation B = S - 6, we get:
B = 15 - 6
B = 9

Therefore, the ages of the boy and his sister are 9 years and 15 years, respectively.

sister's age --- x

the boy's age -- x-6

x(x-6) = 135
x^2 - 6x - 135 = 0
solve for x, Hint: it factors nicely