The ages of X and Y are in the ratio 4 : 5 After 6 years, the ration will become 6 : 7. What is the current age of Y (in years)?
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Understanding the Age Ratio Problem
This question asks us to find the current age of Y, given the current ratio of ages of X and Y, and the ratio of their ages after 6 years. We are given two pieces of information related to the ages of X and Y:
- The current ages of X and Y are in the ratio 4 : 5.
- After 6 years, their ages will be in the ratio 6 : 7.
We need to use these ratios to set up equations and solve for the ages.
Setting up Equations Based on Age Ratios
Let the current age of X be \(4k\) years and the current age of Y be \(5k\) years, where \(k\) is a common factor. This representation satisfies the first condition that their current ages are in the ratio 4:5.
Now, consider their ages after 6 years:
- Age of X after 6 years will be: Current age of X + 6 years = \(4k + 6\) years.
- Age of Y after 6 years will be: Current age of Y + 6 years = \(5k + 6\) years.
According to the problem, the ratio of their ages after 6 years is 6 : 7. So, we can write the equation:
\[\frac{\text{Age of X after 6 years}}{\text{Age of Y after 6 years}} = \frac{6}{7}\]
Substituting the expressions for their ages after 6 years, we get:
\[\frac{4k + 6}{5k + 6} = \frac{6}{7}\]
Solving the Equation for the Common Factor
To find the value of \(k\), we need to solve the equation \(\frac{4k + 6}{5k + 6} = \frac{6}{7}\). We can do this by cross-multiplication.
Multiply the numerator of the left side by the denominator of the right side, and vice versa:
\[7 \times (4k + 6) = 6 \times (5k + 6)\]
Now, distribute the numbers on both sides of the equation:
\[7 \times 4k + 7 \times 6 = 6 \times 5k + 6 \times 6\]
\[28k + 42 = 30k + 36\]
To isolate the variable \(k\), rearrange the equation. Subtract \(28k\) from both sides:
\[42 = 30k - 28k + 36\]
\[42 = 2k + 36\]
Now, subtract 36 from both sides:
\[42 - 36 = 2k\]
\[6 = 2k\]
Finally, divide both sides by 2 to find the value of \(k\):
\[\frac{6}{2} = k\]
\[k = 3\]
Calculating the Current Age of Y
We represented the current age of Y as \(5k\). Now that we have found the value of \(k\) is 3, we can calculate the current age of Y.
Current age of Y = \(5k = 5 \times 3 = 15\) years.
Let's also quickly check the age of X. Current age of X = \(4k = 4 \times 3 = 12\) years.
After 6 years, X's age would be \(12 + 6 = 18\) years, and Y's age would be \(15 + 6 = 21\) years. The ratio of their ages after 6 years would be \(18 : 21\), which simplifies to \(6 : 7\) (dividing both by 3). This matches the condition given in the question, confirming our value of \(k\) is correct.
Therefore, the current age of Y is 15 years.
| Item | Current Representation | Value |
|---|---|---|
| Ratio of current ages (X:Y) | 4:5 | N/A |
| Current age of X | \(4k\) | \(4 \times 3 = 12\) years |
| Current age of Y | \(5k\) | \(5 \times 3 = 15\) years |
| Ages after 6 years (X:Y) | \((4k+6) : (5k+6)\) | \((12+6) : (15+6) = 18 : 21\) |
| Ratio after 6 years | 6:7 | \(18 : 21 = 6 : 7\) |
| Value of k | \(k\) | 3 |
Revision Table: Key Concepts
| Concept | Description | Application in Problem |
|---|---|---|
| Ratio | A comparison of two quantities by division. Written as a:b or a/b. | Used to represent the relationship between X and Y's ages at two different times. |
| Algebraic Representation | Using variables (like \(k\)) to represent unknown quantities or common factors in ratios. | Representing ages as \(4k\) and \(5k\) allows us to set up equations. |
| Setting up Equations | Translating word problems into mathematical equations. | Forming the equation \(\frac{4k + 6}{5k + 6} = \frac{6}{7}\) from the given information. |
| Solving Linear Equations | Using algebraic operations (like cross-multiplication, addition, subtraction) to find the value of the variable. | Solving for \(k\) in the equation \(28k + 42 = 30k + 36\). |
Additional Information: Age Problems and Ratios
Age problems often involve ratios or differences between ages at different points in time. A common technique is to represent the current ages using a common multiplier (like \(k\)) for the given ratio. For example, if the ratio of ages is A:B, the ages can be written as Ak and Bk.
When the problem mentions a change in time (e.g., "after X years" or "X years ago"), you add or subtract that amount from the current age expressions. So, after 6 years, age Ak becomes Ak + 6, and age Bk becomes Bk + 6.
The new ratio or difference given for the ages at the future or past time provides the second piece of information needed to form an equation. Solving this equation allows you to find the value of the multiplier \(k\), which can then be used to find the actual current ages.
It's always a good practice to check your answer by plugging the calculated ages back into the conditions given in the problem to ensure they hold true.
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