# How to compute the cardinality of these two sets?

$A$ and $B$ are two sets of consecutive integers, $|A|=m \text{ and } |B|=2m$ and sum of the elements of $A$ and $B$ are $2m$ and $m$ respectively. If the difference between the largest numbers of $A$ and $B$ is $99$, how could we find the value of $m$ ?

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It'll be nice if you include what you tried and where you are stuck. – Srivatsan Nov 19 '11 at 3:06

Let $A=\{k_a, k_a +1,\dots,k_a+m-1\}$ and $B=\{k_b, k_b +1,\dots,k_b+2m-1\}$. Then we have:

$$S_a = m k_a + \frac{(m-1) m}{2}$$

$$S_b = 2m k_b + \frac{(2m-1) 2m}{2}$$

We are given that $S_a = 2m$ and $S_b = m$. Thus,

$$m k_a + \frac{(m-1) m}{2} = 2m$$

$$2m k_b + \frac{(2m-1) 2m}{2} = m$$

We also know that:

$$(k_b+2m-1) -(k_a+m-1) = 99$$ or

$$(k_a+m-1) - (k_b+2m-1) = 99$$

Does the above help?

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