Question

Problem 1. Every day $20.27 gets added to an account (that starts empty). How many days does it take for the account total to first end in .99? What is the account total at this point?

Answer 1

So, the problem simply asks us to find the time when adding $20.27 daily to an empty account will make the account total end in .99. Now, the fractional part, 0.27 is the only part that is important to us here.

In fact, we can even discard the decimals at this point.

We have to find the number of times we can add 27 until the last two digits become 99 for the first time. Now, the fastest way to find the last two digits on a number is to take the number modulo 100, that is, the remainder of the number when divided by 100.

That is, for any number n, $\small n(\mod 100)$ gives us the last two digits of the number.

Now, if we add 27 each day, then the total after x days is 27x, so our equation modelling the whole problem reduces to the linear congruence

$\small 27x\equiv 99\mod 100$

Now, solving this linear congruence we get the first solution as x = 37, in fact, if we check we can see that

27 x 37 = 999 = 99 mod 100

Thus, we have a solution.

Therefore, the account total will first end in .99 after 37 days, and the corresponding total will be:

$\small 20.27\times 37=\$ \;749.99$