Question

Kanika was given her pocket money on 1st Jan 2016. She puts Rs. 1 on Day 1, Rs. 2 on Day 2, Rs. 3 on Day 3, and continues doing so till the end of the month, from her pocket money into her piggy bank. She also spent Rs. 204 of her pocket money, and was found that at the end of the month she will have Rs. 100 with her. How much money was her pocket money for the month?

Answer 1

Hint: Use the sum of n terms of an AP formula given as \[{S_n} = \dfrac{n}{2}(a + l)\] to calculate the amount stored in the piggy bank. Then, use the spendings and the amount left with Kanika to calculate the total pocket money.

It is given that Kanika puts Rs. 1 on Day 1, Rs. 2 on Day 2, and so on until the end of the month. January has 31 days, hence, on the last day, she saves Rs. 31.

Observe that the savings form an arithmetic progression (AP) with the first term as 1 and the common difference being 1. Hence, we need to find the sum below:

\[1 + 2 + 3 + ......... + 31\]

We know that the sum of n terms of an AP is given as follows:

\[{S_n} = \dfrac{n}{2}(a + l)...........(1)\]

The given AP has 31 terms with 1 as the first term and 31 as the last term. Then using equation (1), we have:

\[{S_{31}} = \dfrac{{31}}{2}(1 + 31)\]

Simplifying, we have:

\[{S_{31}} = \dfrac{{31}}{2}(32)\]

\[{S_{31}} = 31 \times 16\]

\[{S_{31}} = 496............(2)\]

Hence, Kanika saved Rs. 496 in the piggy bank out of her pocket money.

She also spends Rs.204 from her pocket money and at the end of the month, she has Rs. 100.

Hence, the total pocket money she received at the beginning of the month is the sum of the money stored in the piggy bank, money spent, and the money she has. Hence, we have:

\[S = 496 + 204 + 100\]

Simplifying, we have:

\[S = 800\]

Hence, the correct answer is Rs. 800.


Note: You may make a mistake by thinking that Rs. 204 was spent from her piggy bank but it is wrong. In the question, it is given that Rs. 204 was spent from her pocket money, so you need to account for it separately.