One hundred people were surveyed. Of these, $87$ indicated they liked Mozart and $70$ indicated they liked Bach. What is the minimum number of people surveyed who could have said they liked both Mozart and Bach?
The minimum number of people who like both Mozart and Bach is achieved when the number of people who like Mozart but not Bach is maximized. There are $100-70=30$ people who do not like Bach. If all these people like Mozart, then the number of people who like Bach and Mozart is $87-30=\boxed{57}$.
57
Level 2
Prealgebra
Problem: One hundred people were surveyed. Of these, $87$ indicated they liked Mozart and $70$ indicated they liked Bach. What is the minimum number of people surveyed who could have said they liked both Mozart and Bach?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the minimum number of people who liked both Mozart and Bach, we can use the principle of inclusion-exclusion. Let A be the set of people who like Mozart and B be the set of people who like Bach. The sizes of these sets are given as |A| = 87 and |B| = 70. 

The total number of people surveyed is 100. The formula for the union of two sets is given by:

|A ∪ B| = |A| + |B| - |A ∩ B|

Where |A ∩ B| represents the number of people who like both Mozart and Bach. We want to find the minimum value of |A ∩ B|.

Since the total number of surveyed people is 100, we have:

|A ∪ B| ≤ 100

Substituting the values into the inclusion-exclusion formula gives:

|A ∪ B| = 87 + 70 - |A ∩ B| ≤ 100

This simplifies to:

157 - |A ∩ B| ≤ 100

Rearranging this inequality leads to:

|A ∩ B| ≥ 157 - 100

Thus, we find:

|A ∩ B| ≥ 57

This means that at least 57 people must have liked both Mozart and Bach to satisfy the conditions of the survey. Therefore, the final answer is 57.<|END_OF_LATENT|> The final answer is $57$.