At present, the ratio between the ages of Anu and Dimpy is 5 : 8. After 2 years, the ratio of their ages will become 2 : 3. What was Dimpy’s age before 2 years?
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Solving Anu and Dimpy's Age Ratio Problem
This problem involves understanding and solving age-based word problems using ratios and simple algebraic equations. We are given the current ratio of ages and the ratio after a certain number of years and asked to find the age before a certain period.
Step-by-Step Solution for Age Calculation
Let's break down the problem to find Dimpy's age before 2 years.
- Represent the current ages using the given ratio: The current ratio of Anu's age to Dimpy's age is 5 : 8. Let their current ages be \(5x\) years and \(8x\) years, respectively, where \(x\) is a common factor.
- Express their ages after 2 years: After 2 years, Anu's age will be \(5x + 2\) years, and Dimpy's age will be \(8x + 2\) years.
- Set up an equation based on the ratio after 2 years: The problem states that after 2 years, the ratio of their ages will become 2 : 3. So, we can write the equation: \[\frac{5x + 2}{8x + 2} = \frac{2}{3}\]
- Solve the equation for \(x\): To solve for \(x\), we will cross-multiply: \[3 \times (5x + 2) = 2 \times (8x + 2)\] \[15x + 6 = 16x + 4\] Now, rearrange the terms to isolate \(x\): \[6 - 4 = 16x - 15x\] \[2 = x\] So, the value of \(x\) is 2.
- Calculate the current ages:
- Anu's current age = \(5x = 5 \times 2 = 10\) years.
- Dimpy's current age = \(8x = 8 \times 2 = 16\) years.
- Calculate Dimpy's age before 2 years: The question asks for Dimpy's age before 2 years. \[\text{Dimpy's age before 2 years} = \text{Dimpy's current age} - 2\] \[\text{Dimpy's age before 2 years} = 16 - 2 = 14 \text{ years}\]
Therefore, Dimpy's age before 2 years was 14 years.
Verification of Age Ratio
Let's verify the ages and ratios based on our calculation:
- Current ages: Anu = 10, Dimpy = 16. Ratio = 10 : 16 = 5 : 8 (Matches the given current ratio).
- Ages after 2 years: Anu = 10 + 2 = 12, Dimpy = 16 + 2 = 18. Ratio = 12 : 18 = 2 : 3 (Matches the given future ratio).
The calculated ages satisfy both conditions in the problem statement.
Understanding Age Ratio Problems
Age ratio problems are common in quantitative aptitude. They typically involve setting up equations based on given ratios at different points in time (present, past, or future).
- Key Idea: The difference between the ages of two people remains constant over time. However, the ratio of their ages changes over time.
- Method: Use a variable (like \(x\)) to represent the common multiple in the ratio. Formulate expressions for ages at different times and set up an equation based on the ratio given for one of those times. Solve the equation to find the variable, and then calculate the required ages.
Revision Table: Key Information
| Description | Value/Expression |
|---|---|
| Current Anu:Dimpy Ratio | 5 : 8 |
| Current Ages (Anu, Dimpy) | \(5x\), \(8x\) |
| Ages After 2 Years (Anu, Dimpy) | \(5x + 2\), \(8x + 2\) |
| Ratio After 2 Years | 2 : 3 |
| Equation Solved | \(\frac{5x + 2}{8x + 2} = \frac{2}{3}\) |
| Value of \(x\) | 2 |
| Current Dimpy's Age | 16 years |
| Dimpy's Age Before 2 Years | 14 years |
Additional Information on Ratio Problems
Ratios provide a way to compare quantities. In age problems, ratios change as time passes, but the difference in age remains constant.
For example, if person A is 10 and person B is 15, the ratio is 10:15 = 2:3. The difference is 5 years. After 10 years, A is 20 and B is 25. The ratio is 20:25 = 4:5. The ratio changed, but the age difference is still 5 years (25 - 20).
Setting up proportions (like the fraction equation we used) is a standard method to solve problems involving ratios changing over time.
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Important Questions from Quant Based Puzzle
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- A player holds 13 cards of four suits, of which seven are black and six are red. There are twice as many diamonds as spades and twice as many hearts as diamonds. How many clubs does he hold?
5 years ago, father’s age was 8 times Rohan’s age. 5 years hence, the ratio of father’s age to Rohan’s age will be 10 : 3. What is Rohan’s present age?
