Enrique's Coffee Consumption How Much Per Hour

In this mathematical exploration, we delve into a practical problem involving Enrique's coffee consumption. This scenario allows us to apply fundamental arithmetic operations, particularly division and fractions, to determine his hourly coffee intake. It's a relatable problem that highlights the relevance of mathematics in everyday situations. By carefully dissecting the information provided, we can arrive at the correct solution and gain a better understanding of rate problems.

Enrique, a dedicated coffee enthusiast, consumed a specific amount of coffee over a defined period. The problem states that Enrique drank $2^3 / 16$ cups of coffee in $2^1 / 2$ hours. The central question we aim to answer is: How much coffee did Enrique drink each hour? This is a classic rate problem, where we need to find the quantity of coffee consumed per unit of time (in this case, per hour). To solve this, we'll need to divide the total coffee consumption by the total time taken.

Before we dive into the calculations, let's break down the given information into simpler terms:

• Total Coffee Consumed: Enrique drank $2^3 / 16$ cups of coffee. We need to simplify this expression. $2^3$ means 2 multiplied by itself three times, which equals 8. So, the total coffee consumed is $8 / 16$ cups.
• Total Time: Enrique drank the coffee over $2^1 / 2$ hours. This is a mixed number, and it's easier to work with improper fractions. $2^1 / 2$ can be converted to an improper fraction by multiplying the whole number (2) by the denominator (2) and adding the numerator (1), then placing the result over the original denominator. So, $2^1 / 2$ becomes $(2 * 2 + 1) / 2 = 5 / 2$ hours.

Now we have the total coffee consumption as $8 / 16$ cups and the total time as $5 / 2$ hours. The next step is to divide the total coffee by the total time.

To find out how much coffee Enrique drank each hour, we need to divide the total amount of coffee he drank by the total time he spent drinking it. This can be represented as:

Coffee per hour = (Total coffee) / (Total time)

Substituting the values we have:

Coffee per hour = $(8 / 16) / (5 / 2)$

Dividing fractions involves multiplying by the reciprocal of the divisor. The reciprocal of $5 / 2$ is $2 / 5$. So, we have:

Coffee per hour = $(8 / 16) * (2 / 5)$

Now, we can multiply the numerators and the denominators:

Coffee per hour = $(8 * 2) / (16 * 5) = 16 / 80$

We can simplify this fraction by finding the greatest common divisor (GCD) of 16 and 80, which is 16. Dividing both the numerator and the denominator by 16, we get:

Coffee per hour = $(16 / 16) / (80 / 16) = 1 / 5$ cups

So, Enrique drank $1 / 5$ of a cup of coffee each hour.

Now that we've calculated that Enrique drank $1 / 5$ of a cup of coffee per hour, let's analyze the answer choices provided to see which one matches our result. The answer choices are:

A) $7 / 8$ of a cup B) $4 / 5$ of a cup C) $1 \frac{1}{8}$ cups D) $515 / 32$ cups

Comparing our answer, $1 / 5$ of a cup, with the given options, we can see that none of the options directly match our answer. However, it's important to recognize that the initial simplification of $8/16$ to $1/2$ could have been done earlier, potentially leading to a slightly different calculation path. Let's revisit the calculation using the simplified fraction for total coffee consumed:

Coffee per hour = $(1 / 2) / (5 / 2)$

Coffee per hour = $(1 / 2) * (2 / 5)$

Coffee per hour = $(1 * 2) / (2 * 5) = 2 / 10$

Simplifying $2 / 10$ by dividing both numerator and denominator by 2, we get:

Coffee per hour = $1 / 5$ cups

Again, our result is $1 / 5$ cups per hour. This indicates a potential issue with the provided answer choices, as none of them directly correspond to the calculated value. It's possible that there was an error in the options provided or in the initial problem setup. To ensure clarity, it would be beneficial to double-check the original problem statement and the answer choices for any discrepancies.

It’s always a good practice to double-check our work and also consider if there might be alternative ways to interpret the problem. We have meticulously followed the steps of converting mixed numbers to improper fractions, performing division of fractions, and simplifying the results. Our calculations consistently point to $1/5$ cups of coffee per hour.

Given the discrepancy between our calculated answer and the provided options, it is crucial to acknowledge the possibility of an error in the answer choices or a misunderstanding of the problem statement. However, based on the information provided and the standard interpretation of rate problems, our solution of $1/5$ cups per hour is mathematically sound.

In conclusion, by carefully analyzing the problem and applying the principles of fraction division, we determined that Enrique drank 1/5 of a cup of coffee each hour. While this result doesn't directly align with the provided answer choices (A, B, C, and D), our step-by-step calculations demonstrate the accuracy of our solution. This exercise underscores the importance of not only solving mathematical problems but also critically evaluating the results in the context of the given information and potential answer options.

It also highlights the need for clear and accurate problem statements and answer choices in educational settings. When discrepancies arise, it encourages a deeper investigation into the problem's context and the mathematical processes involved, fostering a more thorough understanding of the concepts.

Based on our calculations, Enrique drank $\bf{1/5}$ cups of coffee each hour. This answer does not match any of the provided options, suggesting a potential error in the choices given.