David expressed the ratio of the number of books that he owns to the number of DVDs that he owns as 27:11. Which statement could be true based upon this relationship?

Question

David has 11 books and 27 DVDs.
David has 11 DVDs and 27 books.
David has 11 DVDs and 27 items total.
David has 11 books and 27 items total.

Answer 1

To understand the relationship expressed in the ratio of the number of books to the number of DVDs that David owns, which is given as 27:11, we can break down the meaning of that ratio.

This means that for every 27 books, David has 11 DVDs. It also implies that the number of books (let's denote it as \( b \)) and DVDs (let's denote it as \( d \)) can be expressed as:

\[ \frac{b}{d} = \frac{27}{11} \]

This means:

1. \( b = 27k \)
2. \( d = 11k \)

where \( k \) is some positive integer.

Now, let's evaluate the provided statements based on this understanding:

1. David has 11 books and 27 DVDs.

   • This would mean the ratio is \( \frac{11}{27} \), which is not equal to \( \frac{27}{11} \). So, this statement is not true.

2. David has 11 DVDs and 27 books.

   • This matches the ratio perfectly as \( b = 27 \) and \( d = 11 \). The ratio \( \frac{27}{11} \) is correct. This statement could be true.

3. David has 11 DVDs and 27 items total.

   • If he has 11 DVDs, and there are 27 items total, that would mean he has \( 27 - 11 = 16 \) books. The ratio of books to DVDs would then be \( \frac{16}{11} \), which is not equal to \( \frac{27}{11} \). So, this statement is not true.

4. David has 11 books and 27 items total.

   • If he has 11 books and a total of 27 items, then he must have \( 27 - 11 = 16 \) DVDs. The ratio of books to DVDs would then be \( \frac{11}{16} \), which is not equal to \( \frac{27}{11} \). So, this statement is not true.

In conclusion, the statement that could be true based on the ratio given is:

David has 11 DVDs and 27 books.