Comparing Reading Ratios: Dexter Vs. Lena - Math Problem

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Hey guys! Let's dive into a fun math problem involving ratios. We've got Dexter and Lena, two bookworms who've been hitting the books, but they have different tastes in genres. Dexter is really into science fiction and history, and so is Lena, but they've read a different amount of books in each category. Our mission? To figure out who has a greater ratio of science fiction books to history books read. Let's break it down and make it super easy to understand.

Understanding the Problem

To kick things off, let's make sure we understand what we're dealing with. Ratios are basically a way of comparing two quantities. In this case, we're comparing the number of science fiction books read to the number of history books read. Think of it like a fraction – the number of science fiction books will be our numerator (the top number), and the number of history books will be our denominator (the bottom number). This comparison allows us to see the proportion of science fiction books to history books for both Dexter and Lena. When we compare these ratios, we can then see who leans more towards science fiction in their reading habits relative to history.

  • Dexter's Reading: Dexter has read 11 science fiction books and 5 history books. So, his ratio of science fiction to history books is 11 to 5, which we can write as 11/5.
  • Lena's Reading: Lena, on the other hand, has read 9 science fiction books and 4 history books. Her ratio is 9 to 4, or 9/4.

Now, the question is, how do we compare these fractions? Simply looking at them might not give us a clear picture, so we need to find a way to make them comparable.

Calculating the Ratios

Okay, so we've got Dexter's ratio (11/5) and Lena's ratio (9/4). To compare these ratios effectively, we need to find a common denominator. Remember those fraction lessons from school? Finding a common denominator means we're looking for a number that both 5 and 4 can divide into evenly. The least common multiple of 5 and 4 is 20. This is our magic number!

Let's convert these fractions:

  • For Dexter's ratio (11/5), we need to multiply both the numerator and the denominator by 4 to get a denominator of 20. So, (11 * 4) / (5 * 4) = 44/20.
  • For Lena's ratio (9/4), we need to multiply both the numerator and the denominator by 5 to get a denominator of 20. So, (9 * 5) / (4 * 5) = 45/20.

Now we have two fractions with the same denominator: 44/20 and 45/20. This makes it super easy to see which ratio is larger. It’s like comparing apples to apples, or in this case, twentieths to twentieths. We can now directly compare the numerators (the top numbers) to determine which ratio is greater.

Comparing the Ratios

Alright, we've done the heavy lifting and now we're at the juicy part – comparing the ratios! We've converted Dexter's ratio to 44/20 and Lena's ratio to 45/20. Now, it's a simple matter of looking at the numerators. Which is bigger, 44 or 45? Obviously, it's 45!

This means that 45/20 is greater than 44/20. Translating this back to our original problem, it tells us that Lena's ratio of science fiction books to history books is greater than Dexter's ratio. In other words, for every history book Lena read, she read a slightly larger proportion of science fiction books compared to Dexter. This comparison is crucial because it answers the core question of the problem: who has the greater ratio of science fiction books to history books read?

So, to put it simply, by finding a common denominator and comparing the resulting fractions, we've successfully determined which ratio is larger. This method is super useful for comparing any kind of ratios, whether we're talking about books, ingredients in a recipe, or anything else where we want to see how two quantities relate to each other. Now let's get to the bottom of the solution!

Identifying the Correct Comparison

Now, let's circle back to the given options and see which one correctly compares the ratios. We've already figured out that Lena's ratio (45/20) is greater than Dexter's ratio (44/20). Remember the option we were given:

  • A. 5/20 > 4/20

This looks like a bit of a trick! It's trying to throw us off by using the original numbers of history books read (5 and 4) and a denominator of 20. However, this comparison doesn't accurately reflect the ratios of science fiction to history books. Remember, we need to compare the fractions we calculated after finding the common denominator: 44/20 and 45/20.

To correctly represent the comparison, we need to show that Lena's ratio is greater than Dexter's. That means we need a statement that looks something like this: 45/20 > 44/20. But hold on! The option given to us (5/20 > 4/20) is actually comparing the number of history books read relative to a total (implied to be 20). It doesn't reflect the science fiction to history ratio at all. Therefore, we need to be super careful to ensure we're comparing the correct values and interpreting the results in the right context.

This step is crucial because it highlights the importance of understanding what we're comparing. It's not just about doing the math; it's about knowing what the numbers represent and making sure our comparison is meaningful. In this case, we needed to focus on the ratio of science fiction to history books, not just the raw number of history books read.

Final Answer

Alright guys, we've gone through the whole problem step-by-step, and it's time to nail down the final answer! We started by understanding the problem, then we calculated the ratios of science fiction to history books for both Dexter and Lena. We found a common denominator to make the ratios comparable, and finally, we compared the numerators to see whose ratio was greater.

We determined that Lena's ratio (45/20) is greater than Dexter's ratio (44/20). However, the given option A (5/20 > 4/20) does not correctly show this comparison. It's comparing something else entirely – the number of history books read relative to a total, not the ratio of science fiction to history.

Therefore, the correct answer is that option A does not show the correct comparison of the ratios for this problem. Option A compares the number of history books read by each person with a common denominator, which does not represent the ratios of science fiction to history books.

This problem highlights the importance of carefully reading and understanding what a question is asking. It's not enough just to do the calculations; we need to make sure we're comparing the right things and interpreting the results correctly. So, awesome job working through this with me! Keep practicing, and you'll become a ratio master in no time!