Certainly! Let's solve this problem step-by-step.
To determine the current ages of Raju and Deepar, let us first define their current ages as follows:
- Let [tex]\( R \)[/tex] be Raju's current age.
- Let [tex]\( D \)[/tex] be Deepar's current age.
### Step 1: Creating the Equations from the Problem Statements
We are given two pieces of information about their ages:
1. Three years ago, Raju was twice as old as Deepar:
Three years ago, Raju's age was [tex]\( R - 3 \)[/tex] and Deepar's age was [tex]\( D - 3 \)[/tex]. According to the problem:
[tex]\[
R - 3 = 2(D - 3)
\][/tex]
2. Two years later, Raju will be [tex]\(\frac{3}{2}\)[/tex] times as old as Deepar:
Two years later, Raju's age will be [tex]\( R + 2 \)[/tex] and Deepar's age will be [tex]\( D + 2 \)[/tex]. According to the problem:
[tex]\[
R + 2 = \frac{3}{2}(D + 2)
\][/tex]
### Step 2: Simplifying the Equations
Let's simplify each of these equations step by step.
For the first equation:
[tex]\[
R - 3 = 2(D - 3)
\][/tex]
[tex]\[
R - 3 = 2D - 6
\][/tex]
[tex]\[
R = 2D - 3
\][/tex]
For the second equation:
[tex]\[
R + 2 = \frac{3}{2}(D + 2)
\][/tex]
To get rid of the fraction, multiply both sides by 2:
[tex]\[
2(R + 2) = 3(D + 2)
\][/tex]
[tex]\[
2R + 4 = 3D + 6
\][/tex]
[tex]\[
2R - 3D = 2
\][/tex]
### Step 3: Solving the System of Equations
Now we have a system of linear equations:
1. [tex]\( R = 2D - 3 \)[/tex]
2. [tex]\( 2R - 3D = 2 \)[/tex]
We will substitute the value of [tex]\( R \)[/tex] from the first equation into the second equation:
[tex]\[
2(2D - 3) - 3D = 2
\][/tex]
[tex]\[
4D - 6 - 3D = 2
\][/tex]
[tex]\[
D - 6 = 2
\][/tex]
[tex]\[
D = 8
\][/tex]
Now that we have [tex]\( D = 8 \)[/tex], we can substitute this back into the first equation to find [tex]\( R \)[/tex]:
[tex]\[
R = 2D - 3
\][/tex]
[tex]\[
R = 2(8) - 3
\][/tex]
[tex]\[
R = 16 - 3
\][/tex]
[tex]\[
R = 13
\][/tex]
### Conclusion
The current ages of Raju and Deepar are:
- Raju is 13 years old.
- Deepar is 8 years old.
