Understanding Age Ratio Problems

Age ratio problems are common in quantitative aptitude tests. They involve setting up equations based on the given ratios of ages at different points in time (present, past, or future).

Setting Up the Problem

We are given the ratio of the present ages of two people, A and B, and the ratio of their ages after a certain number of years. We need to find the present age of B.

Let the present age of A be represented by $2x$ and the present age of B be represented by $3x$, based on the given ratio of $2:3$. Here, $x$ is a common factor.

Ages After 5 Years

After 5 years, the age of A will be their present age plus 5 years, which is $2x + 5$.

Similarly, after 5 years, the age of B will be their present age plus 5 years, which is $3x + 5$.

We are told that the ratio of their ages after 5 years will be $3:4$. So, we can write this as an equation:

$$ \frac{\text{Age of A after 5 years}}{\text{Age of B after 5 years}} = \frac{3}{4} $$

Substituting the expressions for their ages, we get:

$$ \frac{2x + 5}{3x + 5} = \frac{3}{4} $$

Solving the Equation for x

To solve for $x$, we can cross-multiply the equation:

$$ 4 \times (2x + 5) = 3 \times (3x + 5) $$

Distribute the numbers on both sides:

$$ 8x + 20 = 9x + 15 $$

Now, we need to isolate $x$. Let's move the terms involving $x$ to one side and the constant terms to the other side. Subtract $8x$ from both sides:

$$ 20 = 9x - 8x + 15 $$

$$ 20 = x + 15 $$

Subtract 15 from both sides:

$$ 20 - 15 = x $$

$$ x = 5 $$

Calculating B's Present Age

We defined B's present age as $3x$. Now that we have found the value of $x$, we can calculate B's present age:

$$ \text{B's present age} = 3 \times x = 3 \times 5 = 15 \text{ years} $$

So, B's present age is 15 years.

Let's check the ratio after 5 years: Age of A after 5 years is 15, and age of B after 5 years is 20. The ratio is $15:20$, which simplifies to $3:4$. This matches the information given in the problem, confirming our value of $x$ is correct.

Revision Table: Age Ratio Calculations

Additional Information: Ratio and Proportion Basics

A ratio is a comparison of two quantities. For example, the ratio $2:3$ means that for every 2 units of the first quantity, there are 3 units of the second quantity.

A proportion is an equation that states that two ratios are equal. For example, $\frac{a}{b} = \frac{c}{d}$ is a proportion. In a proportion, the product of the means equals the product of the extremes (cross-multiplication), i.e., $ad = bc$. This principle was used to solve the equation in this problem.

When dealing with ratios of ages, we use a common multiplier (like $x$) to represent the actual ages, as the ratio only gives the relative sizes. For instance, a ratio of $2:3$ could mean ages 2 and 3, 4 and 6, 10 and 15, and so on. By using $2x$ and $3x$, we cover all these possibilities until we find the specific value of $x$ that satisfies the given conditions.