Question

paige walks to the park $\frac{2}{3}$ mile away. it takes her 16 minutes to get there. paige wants to know her speed in miles per minute. which fraction represents paiges speed in miles per minute? $\frac{\frac{2}{3}}{16}$, $\frac{2}{16}$, $\frac{2}{3}$, $\frac{16}{\frac{2}{3}}$

Explanation:

Step1: Recall the speed formula

Speed is calculated as distance divided by time, so the formula is \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \).

Step2: Identify distance and time

The distance Paige walks is \( \frac{2}{3} \) mile, and the time taken is 16 minutes.

Step3: Calculate the speed

Substitute the values into the speed formula: \( \text{Speed} = \frac{\frac{2}{3}}{16} \), which is equivalent to \( \frac{2}{3} \div 16=\frac{2}{3}\times\frac{1}{16}=\frac{2}{48}=\frac{1}{24} \), but in terms of the fraction representing the speed (before simplifying the division), it is \( \frac{\frac{2}{3}}{16}=\frac{2}{3\times16}=\frac{2}{48} \), and the fraction form from the formula \( \frac{\text{Distance}}{\text{Time}} \) is \( \frac{\frac{2}{3}}{16} \) which can also be written as \( \frac{2}{3}\div16 = \frac{2}{3\times16}=\frac{2}{48} \), and the option that matches the \( \frac{\text{Distance}}{\text{Time}} \) format (with distance \( \frac{2}{3} \) and time 16) is \( \frac{\frac{2}{3}}{16} \) which is the same as \( \frac{2}{3\times16}=\frac{2}{48} \), and looking at the options, the first option is \( \frac{\frac{2}{3}}{16} \) (written as \( \frac{\frac{2}{3}}{16} \)) and the second option is \( \frac{2}{16} \) (wait, no, let's re - examine the options:

Option 1: \( \frac{\frac{2}{3}}{16} \) (which is \( \frac{2}{3}\div16 \))

Option 2: \( \frac{2}{16} \) (this is incorrect as it ignores the fraction in the distance)

Option 3: \( \frac{2}{3} \) (this is the distance, not speed)

Option 4: \( \frac{16}{\frac{2}{3}} \) (this is time divided by distance, which is not speed)

So the correct fraction representing the speed is \( \frac{\frac{2}{3}}{16} \) (the first option) or when we write division of a fraction by a whole number, \( \frac{2}{3}\div16=\frac{2}{3\times16}=\frac{2}{48}=\frac{1}{24} \), and the fraction \( \frac{\frac{2}{3}}{16} \) is equal to \( \frac{2}{3\times16}=\frac{2}{48} \), and the option that is \( \frac{\frac{2}{3}}{16} \) (the first option) or \( \frac{2}{3\times16} \), but in the options, the first option is \( \frac{\frac{2}{3}}{16} \) and the second option is \( \frac{2}{16} \) (which is wrong), the third is distance, the fourth is time over distance. Wait, maybe I misread the options. Let's look again:

Wait, the options are:

1. \( \frac{\frac{2}{3}}{16} \) (which is \( \frac{2}{3}\) divided by 16)

2. \( \frac{2}{16} \) (this is 2 divided by 16, but the distance is \( \frac{2}{3} \), not 2)

3. \( \frac{2}{3} \) (distance, not speed)

4. \( \frac{16}{\frac{2}{3}} \) (time divided by distance)

So the correct one is the first option \( \frac{\frac{2}{3}}{16} \) (or written as \( \frac{2}{3\times16}=\frac{2}{48} \)), but also, \( \frac{2}{3}\div16=\frac{2}{3}\times\frac{1}{16}=\frac{2}{48}=\frac{1}{24} \), and the fraction \( \frac{2}{3\times16} \) can be written as \( \frac{2}{3}\times\frac{1}{16}=\frac{2}{48} \), and the option that represents speed (distance over time) is \( \frac{\text{Distance}}{\text{Time}}=\frac{\frac{2}{3}}{16} \), which is the same as \( \frac{2}{3\times16}=\frac{2}{48} \), and among the options, the first option is \( \frac{\frac{2}{3}}{16} \) (written as \( \frac{\frac{2}{3}}{16} \)) and the second option is \( \frac{2}{16} \) (which is incorrect as the distance is \( \frac{2}{3} \), not 2). Wait, maybe there is a typo in my initial analysis. Let's re - express the speed formula clearly:

Speed \( s=\frac{d}{t} \), where \( d = \frac{2}{3} \) mile and \( t = 16 \) minutes. So \( s=\frac{\frac{2}{3}}{16}=\frac{2}{3\times16}=\frac{2}{48}=\frac{1}{24} \) miles per minute.

Now let's look at the options:

- Option 1: \( \frac{\frac{2}{3}}{16} \) (this is equal to \( \frac{2}{3\times16}=\frac{2}{48}=\frac{1}{24} \))

- Option 2: \( \frac{2}{16}=\frac{1}{8} \) (incorrect, as it uses \( d = 2 \) instead of \( d=\frac{2}{3} \))

- Option 3: \( \frac{2}{3} \) (this is the distance, not speed)

- Option 4: \( \frac{16}{\frac{2}{3}}=16\times\frac{3}{2} = 24 \) (this is \( \frac{t}{d} \), not speed)

So the correct fraction representing the speed is \( \frac{\frac{2}{3}}{16} \) (the first option) or in the form of \( \frac{2}{3\times16} \), and the option that matches \( \frac{d}{t} \) with \( d = \frac{2}{3} \) and \( t = 16 \) is the first option \( \frac{\frac{2}{3}}{16} \) (which is the same as \( \frac{2}{3}\div16 \)). But also, \( \frac{2}{3}\div16=\frac{2}{3}\times\frac{1}{16}=\frac{2}{48}=\frac{1}{24} \), and the fraction \( \frac{2}{3\times16} \) is equal to \( \frac{2}{48} \), and the option that is \( \frac{\frac{2}{3}}{16} \) (the first option) is the correct one. Wait, but the second option is \( \frac{2}{16} \), maybe the problem has a typo and the distance is 2 miles? No, the problem says \( \frac{2}{3} \) mile. So according to the formula \( s=\frac{d}{t} \), with \( d=\frac{2}{3} \) and \( t = 16 \), the speed is \( \frac{\frac{2}{3}}{16} \), which is the first option. But let's check the second option: \( \frac{2}{16} \) is \( \frac{1}{8} \), which is not equal to \( \frac{1}{24} \). The first option \( \frac{\frac{2}{3}}{16}=\frac{2}{3\times16}=\frac{2}{48}=\frac{1}{24} \), which is correct.

Answer:

The fraction representing Paige's speed in miles per minute is \( \boldsymbol{\frac{\frac{2}{3}}{16}} \) (the first option among the given options: \( \frac{\frac{2}{3}}{16} \)). If we consider the simplified form of the fraction \( \frac{\frac{2}{3}}{16}=\frac{2}{3\times16}=\frac{2}{48}=\frac{1}{24} \), but the question asks for the fraction that represents the speed, so the answer is \( \frac{\frac{2}{3}}{16} \) (or the first option in the list of options provided).