Solving the Age Ratio Problem

This problem involves calculating the future age ratio of two individuals, A and B, based on their past age ratio and the sum of their current ages.

Representing Ages 6 Years Ago

We are given that the ratio between the ages of A and B 6 years ago was 5 : 6. We can represent their ages at that time using a variable, let's say $x$.

• Age of A 6 years ago = $5x$ years
• Age of B 6 years ago = $6x$ years

Calculating Present Ages

To find their present ages, we need to add 6 years to their ages from 6 years ago.

• A's present age = Age 6 years ago + 6 years = $5x + 6$ years
• B's present age = Age 6 years ago + 6 years = $6x + 6$ years

We are also given that the sum of their present ages is 45 years. We can set up an equation:

A's present age + B's present age = 45

$(5x + 6) + (6x + 6) = 45$

Now, we solve for $x$:

$11x + 12 = 45$

$11x = 45 - 12$

$11x = 33$

$x = \frac{33}{11}$

$x = 3$

Now we can find their present ages:

• A's present age = $5x + 6 = 5(3) + 6 = 15 + 6 = 21$ years
• B's present age = $6x + 6 = 6(3) + 6 = 18 + 6 = 24$ years

Let's check if the sum is correct: $21 + 24 = 45$. Yes, it matches the given information.

Determining Future Ages

The question asks for the ratio of their ages 3 years hence. This means we need to calculate their ages after 3 years from the present.

• A's age after 3 years = A's present age + 3 = $21 + 3 = 24$ years
• B's age after 3 years = B's present age + 3 = $24 + 3 = 27$ years

Finding the Final Ratio

Finally, we find the ratio of their ages 3 years from now.

Ratio = A's age after 3 years : B's age after 3 years

Ratio = $24 : 27$

To simplify the ratio, we find the greatest common divisor (GCD) of 24 and 27, which is 3.

Divide both numbers by 3:

$\frac{24}{3} = 8$

$\frac{27}{3} = 9$

So, the ratio of their ages 3 years hence will be $8 : 9$.