The sum of the current ages of Vishal and Armaan is 70 years. 5 years ago, Vishal was twice as old as Armaan. What is Armaan’s current age?
25 years
Solving Age Word Problems: Finding Vishal and Armaan's Current Ages
This problem involves finding the current ages of two people, Vishal and Armaan, based on given information about the sum of their current ages and a relationship between their ages in the past. These are classic age word problems often solved using algebraic equations.
Setting Up the Equations for Vishal and Armaan's Ages
Let's use variables to represent the current ages:
- Let \(V\) represent Vishal's current age in years.
- Let \(A\) represent Armaan's current age in years.
We are given two pieces of information:
- The sum of their current ages is 70 years.
- 5 years ago, Vishal was twice as old as Armaan.
Let's translate these statements into algebraic equations.
Equation 1: Sum of Current Ages
The first statement says the sum of their current ages is 70 years. So, we have:
\(V + A = 70 \quad \cdots (1)\)
Equation 2: Ages 5 Years Ago
The second statement talks about their ages 5 years ago.
- Vishal's age 5 years ago was \(V - 5\).
- Armaan's age 5 years ago was \(A - 5\).
According to the problem, 5 years ago, Vishal was twice as old as Armaan. This gives us the equation:
\(V - 5 = 2 \times (A - 5)\)
Let's simplify this second equation:
\(V - 5 = 2A - 10\)
\(V = 2A - 10 + 5\)
\(V = 2A - 5 \quad \cdots (2)\)
Solving the System of Equations
Now we have a system of two linear equations with two variables:
| Equation (1): | \(V + A = 70\) |
| Equation (2): | \(V = 2A - 5\) |
We can solve this system using the substitution method. Since Equation (2) already gives us an expression for \(V\), we can substitute this expression into Equation (1).
Substitute \(V = 2A - 5\) into Equation (1):
\((2A - 5) + A = 70\)
Now, let's solve for \(A\):
\(3A - 5 = 70\)
\(3A = 70 + 5\)
\(3A = 75\)
\(A = \frac{75}{3}\)
\(A = 25\)
So, Armaan's current age is 25 years.
If needed, we could also find Vishal's current age by substituting the value of \(A\) back into either equation. Using Equation (1):
\(V + 25 = 70\)
\(V = 70 - 25\)
\(V = 45\)
Vishal's current age is 45 years. Let's quickly check this with the second condition: 5 years ago, Vishal was \(45-5=40\) and Armaan was \(25-5=20\). Indeed, \(40 = 2 \times 20\).
Conclusion
Based on the calculations, Armaan's current age is 25 years.
Revision Table: Key Concepts in Age Problems
| Concept | Explanation | Example |
|---|---|---|
| Representing Current Age | Use variables (e.g., \(x\), \(y\)) for current ages. | Let John's current age be \(J\). |
| Representing Age in the Past | Subtract the number of years from the current age variable. | John's age 10 years ago was \(J - 10\). |
| Representing Age in the Future | Add the number of years to the current age variable. | John's age in 5 years will be \(J + 5\). |
| Setting up Equations | Translate the relationships given in the problem into algebraic equations using the age expressions. | "Sum of ages is 50": \(J + M = 50\). "John is twice Mary's age": \(J = 2M\). |
| Solving Systems | Use substitution or elimination to find the values of the variables (the ages). | Solve \(J+M=50\) and \(J=2M\). |
Additional Information: Solving Simultaneous Equations
Many age word problems, like the one about Vishal and Armaan, lead to a system of simultaneous linear equations. Here are common methods to solve them:
1. Substitution Method:
- Solve one equation for one variable in terms of the other variable.
- Substitute this expression into the other equation.
- Solve the resulting single-variable equation.
- Substitute the value found back into one of the original equations to find the value of the other variable.
This method was used in the solution for Vishal and Armaan's ages.
2. Elimination Method:
- Multiply one or both equations by constants so that the coefficients of one variable are opposites (e.g., \(+2x\) and \(-2x\)).
- Add the equations together to eliminate one variable.
- Solve the resulting single-variable equation.
- Substitute the value found back into one of the original equations to find the value of the other variable.
Both methods are effective, and the choice often depends on the specific structure of the equations.
Important Questions from Quant Based Puzzle
Three years ago, the difference between the age of Ravish and the age of Kailash was 18 years. Three years from today, Ravish will be three times as old as Kailash. What is the present age of Ravish (in years)?
Seven years from now, Anamika will be as old as Malini was 4 years ago. Srinidhi was born 2 years ago. The average age of Anamika, Malini and Srinidhi 10 years from now will be 33 years. What is the present age of Anamika?
An amount of ₹1,003 is to be distributed among A, B and C in the ratio of 11 : 23 : 25. How many rupees would B get more than A?
In an exam of 80 questions, a correct answer gives 1 marks but a wrong answer deducts 1 marks, and if a question in not attempted there is no deduction in marks. If a student attempted only 80% of the question and got 32 marks, then how many questions did he answer correctly?
The ratio of the present ages of Asha and Lata is 5 : 6. If the difference between their ages is 6 years, then what will be Lata’s age after 5 years?
