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question: A Businessman advertised two job openings for peons in his firm. Two men applied and the businessman decided to engage both of them. He offered them salary of $2000$ rupees per year. $1000$ rupees to be paid every half year,with a promise that their salary would be raised if their work proved satisfactory.They could have a raise of $300$ rupees per year ,(or) if they preferred,$100$ rupees each half year .The two men thought for few moments and then one of them expressed his wish to take the raise at $300$ rupees per year ,while the other man said he would accept the half yearly increase of $100$ rupees .between the two men,who was gainer .

$(a)$ First person

$(b)$ second person

$(c)$ both are equal

$(d)$none of these

my attempt :i thought answer should be first person because after one year he will get total sum as $1000+1000+300$(yearly raise)$=2300$

and second person ,after one year,end up getting $1000+100+1000+100=2200 $

so, definitely $300$ yearly offer is more lucrative .

but answer is given option $(b)$

please explain

The way I interpret this problem, the salaries go as follows:

Person 1:

He gets paid $2000$ for the first year, $2300$ for the second year, $2600$ for the third year, etc. So for the $n$th year he gets paid $2000+300(n-1)$.

Person 2:

He gets paid $1000+1100$ for the first year, $1200+1300$ for the second year, $1400+1500$ for the third year, etc. So he does end up getting paid more, year-by-year, because his raises occur more frequently and so can build up more.

Suppose if they get $100$ each half year. Then the following are the possible outcomes
$$1^{st}mbox{ year } 1000+1100=2100$$
$$2^{nd}mbox{ year } 1200+1300=2500$$
$$3^{rd}mbox{ year } 1400+1500=2900$$
$$4^{th}mbox{ year } 1600+1700=3300$$

Suppose if they get $300$ each per year. Then the following are the possible outcomes
$$1^{st}mbox{ year } 1000+1000=2000$$
$$2^{nd}mbox{ year } 1150+1150=2300$$
$$3^{rd}mbox{ year } 1300+1300=2600$$
$$4^{th}mbox{ year } 1450+1450=2900$$

Now can you see which one is profitable.

You are paid at the end of the work period, so in the first year the first man gets $1000$ twice for $2000$, then is raised to $2300/$year for the second year. The second gets $1000$ for the first six months and $1100$ for the second, giving a total of $2100$. He is ahead by $100$ after the first year.

The second year the first man gets $1150$ each time for a total of $2300$. The second gets another raise to $1200$ for the first six months and one to $1300$ for the second six months, giving a total of $2500$. He is ahead by $200$ in the second year.

In general, in year $k$, the first man gets $2000+300(k-1)$. The second gets $2000+2cdot 100cdot (k-1)+2cdot 100 cdot (k-1)+100=2000+400(k-1)+100$ and his advantage grows by $100$ each year.

You have calculated the year end salary of the first person correctly as he's getting his salary annually with an annual raise of $300$. So

$$N_1 = 2000+300=2300/year$$

Now the second person is getting his salary and his raises half yearly

$$N_2 = 1000+100 =1100/half year$$
during the first half. Now during the second half he again gets a raise of $100$ making his salary

$$N_3 = 1100 +100 =1200/halfyearly =2400/year$$

Some of the other questions assume one person gets a raise starting six months before the other, but it turns out it doesn't matter.

Consider they both work for one year before any raises go into effect: $2000$ and $2000$. The second year, one gets a bonus of $300$ total ($150$ per pay), while the other gets a bonus of $300$ total ($100$ and then $200$ per pay). At the end of the second year, they are still both equal ($2300$ and $2300$ annual pay). The third year they start to diverge.

In the third year, one gets a bonus of $600$ total (increase from $300$ to $600$ total, for $300$ per pay), while the other gets a bonus of $700$ total (increase from $200$ to $300$, and then again from $300$ to $400$). In this way the total sums continue to diverge by $100$ per year.

You can see that eventually the $100$ per pay increase is more beneficial.