Marble Equations Setup

Let A represent the number of marbles Anjali has and S represent the number of marbles Sanjana has.

Translate the conditions given in the problem into algebraic equations:

1. Condition 1: Sanjana gives 15 marbles to Anjali.

   Anjali's marbles become $A + 15$. Sanjana's marbles become $S - 15$.

   The resulting equation is: $A + 15 = 2 \times (S - 15) - 10$

2. Condition 2: Sanjana gives 40 marbles to Anjali.

   Anjali's marbles become $A + 40$. Sanjana's marbles become $S - 40$.

   The resulting equation is: $A + 40 = 3 \times (S - 40) + 20$

Simplify Marble Equations

Simplify the two equations derived from the conditions:

• Equation 1: $A + 15 = 2S - 30 - 10 \implies A + 15 = 2S - 40 \implies A - 2S = -55$
• Equation 2: $A + 40 = 3S - 120 + 20 \implies A + 40 = 3S - 100 \implies A - 3S = -140$

Solving Marble Equations

Solve the system of linear equations formed:

1) $A - 2S = -55$

2) $A - 3S = -140$

Subtract Equation 2 from Equation 1 to eliminate A:

$ (A - 2S) - (A - 3S) = -55 - (-140) $

$ A - 2S - A + 3S = -55 + 140 $

$ S = 85 $

Substitute the value $S = 85$ back into Equation 1:

$ A - 2(85) = -55 $

$ A - 170 = -55 $

$ A = 170 - 55 $

$ A = 115 $

Final Marble Count for Anjali

The number of marbles with Anjali is 120.