Question

Kanika was given her pocket money on 1st January, 2008. She puts Rs 1. On day1, Rs 2 on day2, Rs 3 on day3, and continued on doing so till the end of the month, from this money into her piggy bank. She also spent Rs 204 of her pocket money, and was found at the end of month she still has Rs 100 with her. How much money was her pocket money for the month?

Answer 1

Hint: Consider that her pocket money is x rupees.
Sum of first `n' natural numbers is \[\dfrac{{n\left( {n + 1} \right)}}{2}\]
Total pocket money=money spent + money left at the end + money saved in piggy bank.

Complete step-by-step answer:
Given she spent 204 rupees and 100 rupees is left with her at the end of month.
Money in piggy bank:
January has 31 days and given she puts 1rupee on day1, 2rupees on day2, 3rupees on day3 and continues on doing till the end month.
Therefore, money in piggy bank
=money put in day1 + money put in day2 + money put in day3 +......................+ money put in day 31.
=1+2+3+............+31
We know they sum of first n natural numbers is \[\dfrac{{n\left( {n + 1} \right)}}{2}\]
\[ \Rightarrow \]Money in piggy bank=\[\dfrac{{31\left( {31 + 1} \right)}}{2}\]
\[ = \dfrac{{31\left( {32} \right)}}{2}\]
  =496.
Therefore, money in the piggy bank is 496 rupees.

Now total pocket money is equal to
=money in piggy bank + money spent + money left at the end.
=496+100+204
=800.
Therefore, her pocket money is 800 rupees.

Note: Money in the piggy bank can be calculated by using the sum of terms in arithmetic progression i.e,
\[\dfrac{{n\left( {2a + \left( {n - 1} \right)d} \right)}}{2}\]
Where, n=number of terms
             a=first term
             d=common difference
Here, n=31; d=1; a=1.
 Therefore, money in the piggy bank is \[ = \dfrac{{31\left( {32} \right)}}{2}\]= 496.