The present ages of Kavitha, Rajitha and Haritha are in the ratio of 4 : 7 : 9. Eight years ago, the sum of their ages was 56. Find the present ages (in years).
16, 28, 36
Solving the Age Ratio Problem: Finding Present Ages
This problem involves finding the present ages of three people, Kavitha, Rajitha, and Haritha, based on their age ratio and the sum of their ages at a past time.
Setting Up the Problem with Ratios
The present ages of Kavitha, Rajitha, and Haritha are in the ratio of 4 : 7 : 9.
Let the common ratio be \(x\). Therefore, we can represent their present ages as:
- Kavitha's present age = \(4x\) years
- Rajitha's present age = \(7x\) years
- Haritha's present age = \(9x\) years
Calculating Ages Eight Years Ago
The problem gives us information about their ages eight years ago. To find their ages eight years ago, we subtract 8 from their present ages:
- Kavitha's age 8 years ago = \(4x - 8\) years
- Rajitha's age 8 years ago = \(7x - 8\) years
- Haritha's age 8 years ago = \(9x - 8\) years
Forming the Equation
We are told that the sum of their ages eight years ago was 56. We can write this as an equation:
\((4x - 8) + (7x - 8) + (9x - 8) = 56\)
Solving the Equation for x
Now, let's solve this linear equation to find the value of \(x\):
Combine the terms with \(x\) and the constant terms:
\(4x + 7x + 9x - 8 - 8 - 8 = 56\)
\((4x + 7x + 9x) + (-8 - 8 - 8) = 56\)
\(20x - 24 = 56\)
Add 24 to both sides of the equation:
\(20x = 56 + 24\)
\(20x = 80\)
Divide both sides by 20 to find \(x\):
\(x = \frac{80}{20}\)
\(x = 4\)
Finding the Present Ages
Now that we have the value of \(x\), we can find the present ages of Kavitha, Rajitha, and Haritha:
- Kavitha's present age = \(4x = 4 \times 4 = 16\) years
- Rajitha's present age = \(7x = 7 \times 4 = 28\) years
- Haritha's present age = \(9x = 9 \times 4 = 36\) years
So, the present ages are 16, 28, and 36 years.
Verification
Let's check if these ages satisfy the condition that the sum of their ages eight years ago was 56.
- Kavitha's age 8 years ago = \(16 - 8 = 8\) years
- Rajitha's age 8 years ago = \(28 - 8 = 20\) years
- Haritha's age 8 years ago = \(36 - 8 = 28\) years
Sum of ages 8 years ago = \(8 + 20 + 28 = 56\). This matches the given information.
Summary of Present Ages
| Person | Present Age (in years) |
|---|---|
| Kavitha | 16 |
| Rajitha | 28 |
| Haritha | 36 |
The present ages are 16, 28, and 36 years, which corresponds to one of the options provided.
Analysis of Options
Let's quickly look at why other options might be incorrect:
- Option 1: 16, 36, 28 - The ages are 16, 28, 36, but the order does not match the ratio 4:7:9 for Kavitha, Rajitha, Haritha. The ratio 16:36:28 simplifies to 4:9:7, not 4:7:9.
- Option 3: 20, 35, 45 - The ratio 20:35:45 simplifies to 4:7:9 (\(20/5=4\), \(35/5=7\), \(45/5=9\)). Let's check the sum 8 years ago: \((20-8) + (35-8) + (45-8) = 12 + 27 + 37 = 76\). This is not 56.
- Option 4: 12, 21, 27 - The ratio 12:21:27 simplifies to 4:7:9 (\(12/3=4\), \(21/3=7\), \(27/3=9\)). Let's check the sum 8 years ago: \((12-8) + (21-8) + (27-8) = 4 + 13 + 19 = 36\). This is not 56.
Only the ages 16, 28, 36 satisfy both the ratio and the sum of ages condition.
Revision Table: Key Concepts for Age Problems
| Concept | Explanation | How it applies here |
|---|---|---|
| Ratio | A comparison of two or more quantities. Used to represent proportional relationships between ages. | Ages are in the ratio 4:7:9, represented as 4x, 7x, 9x. |
| Representing Ages | Using a variable (like x) to represent unknown values based on the given ratio. | Present ages are 4x, 7x, 9x. |
| Ages in Past/Future | Subtracting years for the past, adding years for the future. | Ages 8 years ago are present age minus 8. |
| Forming Equation | Translating the word problem's conditions into a mathematical equation. | Sum of ages 8 years ago is 56, leading to (4x-8) + (7x-8) + (9x-8) = 56. |
| Solving Linear Equation | Using algebraic techniques (combining like terms, isolating the variable) to find the unknown value. | Solving 20x - 24 = 56 for x. |
| Substitution | Plugging the found value of the variable back into the expressions for the unknown quantities. | Substituting x=4 into 4x, 7x, 9x to find present ages. |
| Verification | Checking if the calculated values satisfy all conditions given in the problem. | Checking if the sum of calculated ages 8 years ago equals 56. |
Additional Information on Age and Ratio Problems
Age and ratio problems are common in quantitative aptitude sections of various exams. They usually involve setting up linear equations based on the relationships between ages at different points in time.
- Key steps: Identify the unknowns, assign variables based on ratios or relationships, write expressions for ages at different times (past or future), form equations based on the given conditions (sum, difference, product, etc.), solve the equations, and finally calculate the required ages.
- Common variations: Problems might involve the sum or difference of ages, the age of one person being a multiple of another's, or conditions based on ages after or before a certain number of years.
- Importance of verification: Always check your final answers against the original problem statement to ensure they meet all conditions. This helps catch calculation errors.
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