Explanation

Reading this question, we will call the number of $0.10 d for "dimes" and the number of $0.50 coins h for "half dollars." Then, the total number of coins Pat has is . The total value of currency Pat has is , in dollars. And, we're told also that . We're asked for the number of coins, . That could come in the form of solving for d and h individually, or otherwise in algebra that fortuitously yields the exact expression . Let's examine the statements, separately first.

Statement (1) tells us that . This information alone is fairly worthless to our aim. Analyzing by cases, we can see that h could still be anything from 10 to 10 trillion. Insufficient.

Statement (2) is more promising. It indicates that , or , switching from dollars to cents. Here the fact that may play a role. Analyzing by cases, we see, if , then $3.00 comes from the half dollars and the rest from dimes. But the data permit the case that , and then $4.50 comes from the half dollars, with only two dimes filling the gap, and a much smaller total number of coins. Statement (2) is therefore insufficient to tell us the total number of coins.

We must combine the statements. Often when you combine statements in Data Sufficiency, and you have done analysis by cases in either or both of the statements, some of the statements that were allowed in one statement are not allowed jointly. That's precisely what happens here. When we were looking at Statement (2), we had a case in which , but that's not allowed when we combine statements, because that would lead to way more dimes than 10. If the total is $4.70 and the total number of dimes is fewer than 10, then dimes are contributing less than $1.00 and half-dollars must contribute at least $3.70. But half-dollars come in multiples of $0.50. That would appear to leave two possibilities: the half-dollars add up to $4.00, or they add up to $4.50. Both are allowed and lead to a different total number of coins, so we still don't have sufficient information to answer the question.

The correct answer is (E).