Raman's age is twice to the age of his daughter Shruti. If ten years ago Raman's age was three times to the age of Shruti, then what is the present age of Shruti?
20 years
Solving Raman and Shruti's Age Problem
This problem involves setting up equations based on the given information about Raman's and his daughter Shruti's ages at two different points in time: the present and ten years ago. We can use algebra to solve for their current ages.
Setting Up the Age Equations
Let's define the variables:
- Let \(S\) represent Shruti's present age in years.
- Let \(R\) represent Raman's present age in years.
Based on the problem statement, we can form two equations:
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Condition 1: Present Age Relationship
Raman's age is twice the age of his daughter Shruti.
\(R = 2S\)
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Condition 2: Age Relationship Ten Years Ago
Ten years ago, Raman's age was \(R - 10\) and Shruti's age was \(S - 10\).
Ten years ago, Raman's age was three times Shruti's age.
\(R - 10 = 3(S - 10)\)
Step-by-Step Solution
Now we have a system of two linear equations:
1. \(R = 2S\)
2. \(R - 10 = 3(S - 10)\)
We can solve this system using substitution. Substitute the expression for \(R\) from equation (1) into equation (2).
Substitute \(2S\) for \(R\) in the second equation:
\(2S - 10 = 3(S - 10)\)
Now, we solve for \(S\):
Distribute the 3 on the right side of the equation:
\(2S - 10 = 3S - 30\)
To isolate the variable \(S\), subtract \(2S\) from both sides of the equation:
\(-10 = 3S - 2S - 30\)
\(-10 = S - 30\)
Add 30 to both sides of the equation to find the value of \(S\):
\(-10 + 30 = S\)
\(20 = S\)
So, Shruti's present age (\(S\)) is 20 years.
Verifying the Answer
If Shruti's present age is 20 years, then Raman's present age is \(R = 2S = 2 \times 20 = 40\) years.
Ten years ago:
- Shruti's age was \(20 - 10 = 10\) years.
- Raman's age was \(40 - 10 = 30\) years.
Is Raman's age ten years ago (30) three times Shruti's age ten years ago (10)?
\(30 = 3 \times 10\)
Yes, the condition holds true. The calculated present age for Shruti is correct.
The present age of Shruti is 20 years.
Revision Table: Key Age Problem Concepts
| Concept | Description | How Used Here |
|---|---|---|
| Defining Variables | Assigning letters (like S, R) to unknown quantities (ages). | Used S for Shruti's present age, R for Raman's present age. |
| Translating Words to Equations | Converting sentences about relationships into mathematical equations. | "Raman's age is twice Shruti's" becomes \(R=2S\). "Ten years ago..." involves subtracting 10. |
| Solving System of Equations | Finding the values of variables that satisfy multiple equations simultaneously. | Used substitution method to solve for S and R. |
| Verification | Plugging the solution back into the original problem statement to check if it works. | Checked if ages 10 years ago satisfied the given condition. |
Additional Information: Solving Age Word Problems
Age word problems are common in algebra and test your ability to translate real-world situations into mathematical equations. Here are some tips:
- Always define your variables clearly, specifying what age (present, past, or future) they represent.
- Pay close attention to time references like "ago," "hence," "in X years," etc., as they require adding or subtracting from the present age.
- "Is," "was," and "will be" usually correspond to the equals sign (=).
- "Twice," "three times," "half of," etc., indicate multiplication or division.
- Set up as many equations as there are unknown variables.
- Use methods like substitution or elimination to solve the system of equations.
- Always verify your solution by plugging the values back into the original word problem to ensure all conditions are met.
Practice with different types of age problems will help you become more comfortable with setting up the equations correctly.
Important Questions from Quant Based Puzzle
- There are deers and peacocks in a zoo. By counting heads they are 80. The number of their legs is 200. How many peacocks are there?
- A certain number of horses and an equal number of men are going somewhere. Half of the owners are on their horses' back while the remaining ones are walking along leading their horses. If the number of legs walking on the ground is 70, how many horses are there?
- A, B, C, D and E play a game of cards. A says to B, "If you give me three cards, you will have as many as E has and if I give you three cards, you will have as many as D has". A and B together have 10 cards more than what D and E together have. If B has two cards more than what C has and the total number of cards be 133, how many cards does B have?
- 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?
There are fourteen teams playing in a tournament. If every team plays one match with every other team, how many matches will be played in the tournament?
