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.
Important Questions from Algebra (Notes)
- What is the remainder when 2023²⁰²⁴ + 2025²⁰²⁴ is divided by 2024?
In an examination, a student scores 4 marks for every correct answer and loses 1 mark for every wrong answer. If she/he attempts all 60 questions and secures 130 marks, the number of questions she/he attempts wrongly, are?
Match List-I with List-II
List-1 List-II (A) If $\begin{bmatrix}\lambda-1 & 0 \\ 0 & \lambda-1 \end{bmatrix} $, then $\lambda$ is (I) 0 (B) If A=$ \begin{bmatrix}1 & 2 \\2 & 4 \end{bmatrix} $, then $\Delta$ is (II) 1 (C) If A = $ \begin{bmatrix}1 & 0 \\0 & \frac{1}{2} \end{bmatrix} $, then $|A^{-1}|$ is (III) -2 (D) If $ \begin{bmatrix}a+1 & 1 \\1 & 2 \end{bmatrix} = \begin{bmatrix}-1 & 1 \\1 & 2 \end{bmatrix} $, then a is (IV) 2 Choose the correct answer from the options given below:
- If (x - 1) is a factor of $2x^2 - 5x + k = 0$, then the value of k is:
- If $x = (2+\sqrt{3})^{\frac{1}{3}} + (2+\sqrt{3})^{-\frac{1}{3}}$ and $x^3-3x + k = 0$, then the value of k is:
