# Problem of the Day

Out of a group of 100 people, how many speak both Spanish and French?

(1) 70 people speak Spanish.

(2) 40 people speak French.

### Solution

In this problem, each statement is obviously not sufficient on its own. Statement (1) tells us nothing about the people who speak French, and Statement (2) tells us nothing about the people who speak Spanish. Therefore, the answer must be ‘C’ or ‘E’.

Taking both statements together, we know that 70 people speak Spanish and 40 people speak French. Adding those together, we get 110 people. This number exceeds the total number of people in the group because there is some overlap between the groups. Some people must be getting counted twice. To find the minimum overlap, subtract 100 from 110. At least 10 people must speak both Spanish and French. However, this is only the minimum number of people who speak both Spanish and French. There are many other possible answers. For instance, it could be that all 40 people who speak French also speak Spanish. In other words, the entire set of people who speak French is a subset of the group who speak Spanish. This would leave 30 people who speak neither French nor Spanish. This implies that the *maximum*Â number of people who speak both French and Spanish is 40. So, the given information allows us to conclude only that the number of people who speak both languages is somewhere between 10 and 40, inclusive.

Note that the answer could be shifted to ‘C’ with a simple modification of the problem. If the problem had guaranteed that everyone in the group spoke at least one of the two languages, then the overlap would have to be exactly 10. These types of problems are common on the GMAT and it is important that you are familiar with how the different phrasings of the question can impact the possibilities.

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