Astha and Saumya each have certain number of oranges Astha says to Saumya if you give me 10 of your oranges I will have twice the number of oranges left with you,"Saumya reples,"if you give me 10 of your oranges ,I will have the same number of oranges as left with you,"find the number of oranges with Astha and Saumya respectively.

Options

A. 60,40
B. 70,50
C. 60,80
D. 70,90

Answer

To solve the problem, we will set up equations based on the statements made by Astha and Saumya. Let's denote the number of oranges Astha has as \( X \) and the number of oranges Saumya has as \( Y \).

### Step 1: Set up the first equation based on Astha's statement.
Astha says, "If you give me 10 of your oranges, I will have twice the number of oranges left with you."

This can be translated into the equation:
\[ X + 10 = 2(Y - 10) \]
Expanding this gives:
\[ X + 10 = 2Y - 20 \]
Rearranging it, we get:
\[ X - 2Y = -30 \quad \text{(Equation 1)} \]

### Step 2: Set up the second equation based on Saumya's statement.
Saumya replies, "If you give me 10 of your oranges, I will have the same number of oranges as you have left."

This translates to:
\[ Y + 10 = X - 10 \]
Rearranging this gives:
\[ X - Y = 20 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations.
Now we have two equations:
1. \( X - 2Y = -30 \)
2. \( X - Y = 20 \)

We can solve these equations simultaneously. From Equation 2, we can express \( X \) in terms of \( Y \):
\[ X = Y + 20 \]

### Step 4: Substitute \( X \) in Equation 1.
Now, substitute \( X \) in Equation 1:
\[ (Y + 20) - 2Y = -30 \]
This simplifies to:
\[ 20 - Y = -30 \]
Rearranging gives:
\[ -Y = -30 - 20 \]
\[ -Y = -50 \]
Thus:
\[ Y = 50 \]

### Step 5: Find \( X \) using the value of \( Y \).
Now substitute \( Y = 50 \) back into Equation 2:
\[ X - 50 = 20 \]
So:
\[ X = 70 \]

### Conclusion:
Astha has \( 70 \) oranges and Saumya has \( 50 \) oranges.

### Summary of the Solution:
- Astha's oranges \( X = 70 \)
- Saumya's oranges \( Y = 50 \)