Word problem about a diminishing military contract?

Question

Here's a question I've had a bit of trouble answering:

A military member is assigned $5$ years to serve abroad. For each month they serve in which they have exemplary behavior though, $10$ days is taken off their $5$ year contract. Assuming they have exemplary behavior every month, how much time do they have to serve altogether? How many days get taken off their $5$ year contract?

You can't just take $60\text{ months} \times 10 \text{ days}$ and then deduct it from the $5$ years because they won't be serving all $5$ years.

Answer 1

To do this with equations, measure the time in days, and ignore leap years for now. There are 365 days in a year and 12 months, so every $\frac {365}{12}$ days, they earn $10$ days credit. Let $d$ be the number of days that they actually serve. Then they earn $$\frac {10}{\frac {365}{12}}d = \frac{120}{365}d$$days of credit total.

The actual days on duty plus the credited days will add up to the full 5 years of service:

$$d + \frac{120}{365}d = 5\times 365\\d= \frac{5\cdot 365}{\frac{485}{365}} = 1373.5 \text{ days}$$ which rounds up to 1374 days, or 3 years, 9 months, 5 days.

But of course, somewhere in the 5 year term, we are going to have a leap year day. Possibly two, but likely just one. So the calculation is one or two days short. Thus the actual duty would be 3 years, 9 months, 6 days (or 7 days).

Note that this "solution by equation" makes a number of simplifying assumptions. The credit days accrue steadily, instead of in 10-day lumps at the beginning of each month. The months are all of even length, just under 30.5 days.

The equation method only estimates the complexity of the real situation, and thus, its result is less accurate. In order to model the situation more accuratebrly, you need to break it down into details, as Aniruddha Deshmukh has done. Even that solution is only approximate, as an exact calculation would require more information than is given in the question:

• On what year, month and day does the tour of duty start? (The year is only needed to figure out where the leap year days fall.)
• How exactly are the credit days awarded? (For example, on the first day of the following month? Or maybe a month later, or possibly continuously as my equation calculates.)

Answer 2

Observe that after 36 months, that is 3 years, 360 days are taken off the $5$ year contract. Now, after 3 years, the contract remains of $4$ years and 5 days, which means $12$ months and $5$ days more to go. But for every month, $10$ days get cut off from the contract. So, after $1$st month, the contractual time remaining is $10$ months $25$ days. After $2$nd month, we are left with $9$ months $15$ days. After the $3$rd, we are left with $8$ months $5$ days. After the $4$th, we are left with $6$ months $25$ days. After the $5$th month, time left would be $5$ months $15$ days. After $6$th month, we have $4$ months $5$ days. After $7$th month, we are left with $2$ months $25$ days which will be served completely.

Hence, the total time of service would be $3$ years, $9$ months and $5$ days.

If the leap year concept is taken into consideration, the calculation might change a bit. But the process is the same as described above.