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I am looking for some expert advice in answering this finance question. I am having trouble figuring out how to compare these put and strikes. Is there a formula to approach this type of problem?

Let $p_K$ be the current price of a European put expiring at time $T$ with strike price $K$.

Let $S_0$ be the spot price of the underlying asset. Compare the following quantities if possible.

(a) Compare $p_{50}$, $p_{55}$, $p_{60}$.

(b) Compare $p_{50}$ and 50

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    Well, $p_K$ has to be worth less than $K$ since it would be discounted by the interest rate, i.e., $p_K \leq Ke^{-rT} < K$ for a continuously compounded rate $r>0$. Additionally, we see that if $K_1 < K_2$, then $$p_{K_1} \leq p_{K_2}$$ since being able to sell the stock for $K_1$ at time $T$ is worth less than selling the same stock for $K_2$ at the same time. – 2017-02-28
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    Ok. So for (a), can we compare? The way I read it is that we can't since we don't have any info on S. is this correct? – 2017-02-28
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    No, as a matter of fact, it's (mostly) independent of stock price -- a 50-strike put is worth less than a 55-strike put, no matter the current stock price, at expiry, the 55 strike put is worth at least as much as the 50 strike put. – 2017-02-28

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You can use the Black and Scholes formula to answer this questions rigorously: https://en.wikipedia.org/wiki/Black–Scholes_model

This should help you confirm that $p_{50}\leq p_{55}\leq p_{60}$. This is because a put is like an insurance, it allows you to sell something at no less than $K$. Therefore, the higher is $K$ the more valuable the insurance. To see this, note that if at time $T$ the price is higher than $K$, then selling the good at the market price is better than exercising your put and selling it at $K$. In that case, the put is worthless. It is clear that the higher the strike price, the less likely it is for your put to be worthless.

As for the second question, you can use the formula of Black and Scholes to see that $P\leq K$ for any positive interest rate $r\geq 0$.