Solving Age Ratio Problems: Finding the Difference

This question involves finding the difference between the current ages of two individuals, X and Y, given ratios of their ages at different points in time. We are provided with two key pieces of information:

• The ratio of their current ages is 4:7.
• The ratio of their ages three years ago was 1:2.

Setting up the Age Equations

Let the current age of X be \(4k\) years and the current age of Y be \(7k\) years, where \(k\) is a common multiple. This represents the given ratio of their current ages (4:7).

Three years ago, the ages of X and Y would have been:

• Age of X three years ago: \(4k - 3\) years
• Age of Y three years ago: \(7k - 3\) years

We are told that the ratio of their ages three years ago was 1:2. We can write this as an equation:

\[\frac{4k - 3}{7k - 3} = \frac{1}{2}\]

Solving for the Unknown Variable

To find the value of \(k\), we can cross-multiply the equation:

\[2 \times (4k - 3) = 1 \times (7k - 3)\]

Now, distribute on both sides of the equation:

\[8k - 6 = 7k - 3\]

To solve for \(k\), we need to isolate the \(k\) terms on one side and the constant terms on the other side. Subtract \(7k\) from both sides:

\[8k - 7k - 6 = 7k - 7k - 3\]

\[k - 6 = -3\]

Now, add 6 to both sides:

\[k - 6 + 6 = -3 + 6\]

\[k = 3\]

So, the value of the common multiple \(k\) is 3.

Calculating Current Ages

Now that we have the value of \(k\), we can find the current ages of X and Y:

• Current age of X = \(4k = 4 \times 3 = 12\) years
• Current age of Y = \(7k = 7 \times 3 = 21\) years

Finding the Difference in Current Ages

The question asks for the difference between their current ages (Y - X). We calculate this difference:

\[\text{Difference} = \text{Current age of Y} - \text{Current age of X}\]

\[\text{Difference} = 21 - 12\]

\[\text{Difference} = 9\]

The difference between the current ages of Y and X is 9 years.

Verification of Age Ratios

Let's check if our calculated ages satisfy the conditions given in the problem:

• Current ages: X = 12, Y = 21. Ratio = \(12 : 21\). Dividing both by 3, we get \(4 : 7\). This matches the given current ratio.
• Ages three years ago: X = \(12 - 3 = 9\), Y = \(21 - 3 = 18\). Ratio = \(9 : 18\). Dividing both by 9, we get \(1 : 2\). This matches the given ratio from three years ago.

The calculated ages satisfy both conditions, confirming our solution is correct.

Revision Table: Key Steps in Age Ratio Problems

Additional Information: Understanding Ratios and Age Word Problems

Ratios are used to compare quantities. A ratio of 4:7 means that for every 4 units of X's age, there are 7 units of Y's age. When solving age word problems involving ratios over time, we typically use a variable (like \(k\)) to represent the common multiple in the ratio. This allows us to set up algebraic equations that can be solved.

It's crucial to correctly represent the ages at different points in time. If the problem refers to ages 'n' years ago, you subtract 'n' from the current ages. If it refers to ages 'n' years hence (in the future), you add 'n' to the current ages.

Always verify your calculated ages by plugging them back into the original problem statements to ensure they satisfy all given conditions, especially the age ratios at different times.