Five years ago, the ratio of the ages of Tarun and Saurabh was 4 ∶ 1. After five years, the ratio of their ages will be 2 ∶ 1. What is the present age (in years) of Saurabh?
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Understanding the Age Ratio Problem
This question involves solving a problem based on the ratio of ages of two individuals, Tarun and Saurabh, at different points in time. We are given their age ratio five years ago and their projected age ratio five years from now. Our goal is to find Saurabh's current age.
Setting Up Variables for Past Ages
Let's represent the ages of Tarun and Saurabh five years ago using a variable. The ratio of their ages five years ago was given as 4 ∶ 1.
- Let Tarun's age five years ago be $4x$ years.
- Let Saurabh's age five years ago be $1x$ or $x$ years.
Here, $x$ is a common factor for their ages at that time.
Calculating Present Ages
To find their present ages, we need to add 5 years to their ages from five years ago.
- Tarun's present age = (Tarun's age five years ago) + 5 = $4x + 5$ years.
- Saurabh's present age = (Saurabh's age five years ago) + 5 = $x + 5$ years.
Determining Ages After Five Years
Next, let's find their ages five years from the present time. We add another 5 years to their present ages.
- Tarun's age after five years = (Tarun's present age) + 5 = $(4x + 5) + 5 = 4x + 10$ years.
- Saurabh's age after five years = (Saurabh's present age) + 5 = $(x + 5) + 5 = x + 10$ years.
Formulating the Equation Using Future Age Ratio
We are given that the ratio of their ages after five years will be 2 ∶ 1. We can set up an equation using the ages we just calculated.
Ratio of Tarun's age to Saurabh's age after five years:
$\frac{\text{Tarun's age after 5 years}}{\text{Saurabh's age after 5 years}} = \frac{2}{1}$
Substituting the expressions for their ages:
$\frac{4x + 10}{x + 10} = \frac{2}{1}$
Solving the Age Ratio Equation
Now, we solve this equation to find the value of $x$. We can cross-multiply:
$1 \times (4x + 10) = 2 \times (x + 10)$
$4x + 10 = 2x + 20$
To isolate the $x$ terms, subtract $2x$ from both sides:
$4x - 2x + 10 = 2x - 2x + 20$
$2x + 10 = 20$
To isolate the $x$ term further, subtract 10 from both sides:
$2x + 10 - 10 = 20 - 10$
$2x = 10$
Finally, divide both sides by 2 to find the value of $x$:
$\frac{2x}{2} = \frac{10}{2}$
$x = 5$
Calculating Saurabh's Present Age
We found that $x = 5$. Saurabh's present age was represented by the expression $x + 5$.
Saurabh's present age = $x + 5 = 5 + 5 = 10$ years.
Thus, Saurabh's present age is 10 years.
| Time Period | Tarun's Age | Saurabh's Age | Ratio |
|---|---|---|---|
| Five years ago | $4x = 4 \times 5 = 20$ | $x = 1 \times 5 = 5$ | 20 : 5 = 4 : 1 (Matches given) |
| Present Age | $4x + 5 = 20 + 5 = 25$ | $x + 5 = 5 + 5 = 10$ | 25 : 10 = 5 : 2 |
| After five years | $4x + 10 = 20 + 10 = 30$ | $x + 10 = 5 + 10 = 15$ | 30 : 15 = 2 : 1 (Matches given) |
Revision Table for Age Ratio Problems
| Concept | Description | How to Apply |
|---|---|---|
| Representing Ages with Variables | Use a common variable (e.g., $x$) to represent a part of the ratio. | If ratio is A:B, ages can be $Ax$, $Bx$. |
| Calculating Past/Future Ages | Add or subtract years from the current or assumed age. | Past age = Present age - Years; Future age = Present age + Years. |
| Setting up Equation | Use the given ratio at a specific time to form an algebraic equation. | $\frac{\text{Age 1}}{\text{Age 2}} = \frac{\text{Ratio Part 1}}{\text{Ratio Part 2}}$ |
| Solving Linear Equations | Use algebraic techniques (cross-multiplication, isolating variable) to find the value of the variable. | Combine like terms, perform inverse operations. |
Additional Information on Age Ratio Calculations
Age ratio problems are a common type in quantitative aptitude tests. They typically involve using ratios to represent the proportional relationship between the ages of two or more people at different points in time. The key is to set up a consistent algebraic representation of the ages across these time periods and then use the given ratio at one of those times to form an equation. Solving this equation allows you to find the value of the variable, which can then be used to determine the actual ages at any specified time (past, present, or future).
Always read the question carefully to identify which time period each ratio refers to (e.g., 'five years ago', 'present', 'after five years') and what age you are ultimately asked to find (e.g., Tarun's present age, Saurabh's age five years from now).
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