This solution explains how to find the initial amount a woman started shopping with, given the amount spent and the remaining amount in terms of variables.

Algebraic Representation of Amounts

Let the starting amount be Rs. $X$ and $Y$ paise.

• Total starting amount in paise: $100X + Y$
• Amount spent: Rs. 3.50 = 350 paise
• Remaining amount: Rs. $2Y$ and $2X$ paise = $100(2Y) + 2X$ paise

Formulating the Equation

The relationship between these amounts is:

Starting Amount - Amount Spent = Remaining Amount

Substituting the expressions in paise:

$(100X + Y) - 350 = 100(2Y) + 2X$

Simplifying the equation:

$100X + Y - 350 = 200Y + 2X$

$100X - 2X + Y - 200Y = 350$

$98X - 199Y = 350$

Verifying the Correct Option

The correct answer is Rs. 32.14. This implies that for the starting amount, $X = 32$ (Rupees part) and $Y = 14$ (Paise part).

We test these values in the derived equation:

Left Hand Side (LHS): $98X - 199Y = 98(32) - 199(14)$

$LHS = 3136 - 2786 = 350$

Right Hand Side (RHS): $350$

Since $LHS = RHS$, the values $X=32$ and $Y=14$ are correct.

Confirming Spending and Remaining Amount

Let's confirm the transaction details:

• Starting Amount = Rs. 32.14
• Amount Spent = Rs. 3.50
• Calculated Remaining Amount = Rs. 32.14 - Rs. 3.50 = Rs. 28.64

Now, let's check the remaining amount based on the formula Rs. $2Y$ and $2X$ paise, using $Y=14$ and $X=32$:

• Rs. $2Y$ = Rs. $2 \times 14$ = Rs. 28
• $2X$ paise = $2 \times 32$ paise = 64 paise
• Remaining amount as per formula = Rs. 28 and 64 paise = Rs. 28.64

The calculated remaining amount matches the amount derived from the formula, confirming the starting amount.

Conclusion

The starting amount is Rs. 32.14.