Question

Jaideep was given some money by his mother on his birthday. Jaideep spent all of it in five stores. In each store he spent one rupee more than half of what he had when he came in. How much did he get from his mother?

• A. Rs. 25
• B. Rs. 30
• C. Rs. 50
• D. Rs. 62

Hint:

Work backwards from the fact that nothing is left after the fifth store. If he leaves each store with \(\frac{y}{2}-1\), then what he had entering that store is \(2(\text{amount left}+1)\).

Answer 1

The Correct Option is D

Step 1: Set up the relationship between what he has and what remains.
Let \(y_i\) be the amount of money Jaideep has when he walks into store \(i\).
In each store he spends "one rupee more than half of what he had", so he spends \(\frac{y_i}{2} + 1\).
What remains after that store is
\[ y_i - \left(\frac{y_i}{2}+1\right) = \frac{y_i}{2} - 1 \]
and this remaining amount becomes \(y_{i+1}\), the money he carries into the next store.

Step 2: Use the fact that he has exactly Rs 0 left after the fifth store.
After store 5, nothing remains, so
\[ \frac{y_5}{2} - 1 = 0 \implies y_5 = 2 \]

Step 3: Work backwards, store by store, using \(y_i = 2(y_{i+1}+1)\).
This backward formula comes directly from rearranging \(y_{i+1} = \frac{y_i}{2}-1\).
\[ y_4 = 2(y_5+1) = 2(2+1) = 6 \]
\[ y_3 = 2(y_4+1) = 2(6+1) = 14 \]
\[ y_2 = 2(y_3+1) = 2(14+1) = 30 \]
\[ y_1 = 2(y_2+1) = 2(30+1) = 62 \]

Step 4: Check the answer by simulating forward from Rs 62.
Store 1: has 62, spends \(31+1=32\), left with 30.
Store 2: has 30, spends \(15+1=16\), left with 14.
Store 3: has 14, spends \(7+1=8\), left with 6.
Store 4: has 6, spends \(3+1=4\), left with 2.
Store 5: has 2, spends \(1+1=2\), left with 0.
Every rupee is spent in exactly five stores, ending at precisely Rs 0, confirming the value is correct.

Final Answer:
Jaideep's mother gave him Rs 62. Testing Rs 30 directly shows it cannot satisfy the question: starting with Rs 30, he already reaches Rs 0 after only the fourth store, one store short of the five stores the question describes, so Rs 30 fails the problem's own condition. Rs 62 is the value consistent with all five stores.
\[ \boxed{62} \]