Lesson Spending money according to dates

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<H2>Spending money according to dates</H2>

<H3>Problem 1</H3>Mr. &nbsp;<B>X</B>&nbsp; had a habit of spending money according to dates. &nbsp;For example, &nbsp;if date was &nbsp;19&nbsp; he was spending &nbsp;$19, &nbsp;and if date was &nbsp;15&nbsp; he was spending &nbsp;$15. 
One night he calculated total spending of &nbsp;5&nbsp; consecutive days - Monday to Friday, &nbsp;and he found that he spent &nbsp;$61 in &nbsp;5&nbsp; days. 
So, &nbsp;identify the dates. &nbsp;(Assume that everyday spending is integer number of dollars).

<B>Answer</B>. &nbsp;There is only one solution:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;27 and 28 &nbsp;of February and &nbsp;1, 2, &nbsp;and &nbsp;3 &nbsp;of March.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(27 + 28 + 1 + 2 + 3 = 61).

<B>Solution</B>

If these days would be inside one month, &nbsp;then the dates are &nbsp;(x-2), &nbsp;(x-1), &nbsp;x, &nbsp;(x+1) &nbsp;and &nbsp;(x+2), &nbsp;where x is the date in the middle of &nbsp;5 days.

Then the sum must be multiple of &nbsp;5, &nbsp;since

(x-2) + (x-1) + x + (x+1) + (x+2) = 5x.

But the integer &nbsp;61&nbsp; is not multiple of &nbsp;5. &nbsp;Contradiction.

Hence, &nbsp;the dates are partly the end of some month and the beginning of the next month.

Then, &nbsp;it is easy to check that the dates &nbsp;27, &nbsp;28, &nbsp;1, &nbsp;2 &nbsp;and 3 &nbsp;satisfy the condition &nbsp;27 + 28 + 1 + 2 + 3 = 61.

Next, &nbsp;it is easy to see that there is no other solution.

<H3>Problem 2</H3>Same problem if &nbsp;Mr. &nbsp;<B>X</B>&nbsp; spent &nbsp;$63 &nbsp;in &nbsp;5&nbsp; consecutive days. &nbsp;Identify the dates. &nbsp;Solve the problem yourself.

<H3>Problem 3</H3>Next time &nbsp;Mr. &nbsp;<B>X</B>'s &nbsp;spending was &nbsp;$90 &nbsp;of &nbsp;6&nbsp; consecutive days - Monday to Saturday.
Identify the dates. &nbsp;Solve the problem yourself.

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