Question

kylie biked at a constant speed. she biked 12 miles in 4 hours. what does the constant of proportionality of \\(\frac{1}{3}\\) mean? \
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number of miles	number of hours	\
---	---	\
12	\\(\times \frac{1}{3}\\)	4	\
6	\\(\times \frac{1}{3}\\)	2	\

\
the constant of proportionality is \\(\frac{1}{3}\\).\
the constant of proportionality of \\(\frac{1}{3}\\) means that kylie dropdown options: biked \\(\frac{1}{3}\\) mile every hour, biked 1 mile every \\(\frac{1}{3}\\) hour.

Explanation:

Step1: Understand Proportionality

The relationship between miles (m) and hours (h) is proportional, so \( m = k \times h \), where \( k \) is the constant of proportionality. Here, \( k=\frac{1}{3} \) when relating miles to hours? Wait, no—wait, the table shows miles \( \times \frac{1}{3} = \) hours. So hours \( = \frac{1}{3} \times \) miles. So if we solve for miles per hour (speed), speed \( = \frac{\text{miles}}{\text{hours}} \). From the first row: \( \frac{12}{4} = 3 \) mph. But the constant here is \( \frac{1}{3} \) (miles \( \times \frac{1}{3} = \) hours). So if miles is \( m \), hours \( h = \frac{1}{3}m \), which means for each mile, time is \( \frac{1}{3} \) hour. Let's check: 12 miles \( \times \frac{1}{3} = 4 \) hours (correct). 6 miles \( \times \frac{1}{3} = 2 \) hours (correct). So the constant \( \frac{1}{3} \) means that to find hours, multiply miles by \( \frac{1}{3} \), i.e., for each mile, it takes \( \frac{1}{3} \) hour. So "biked 1 mile every \( \frac{1}{3} \) hour" is correct? Wait, no—wait, speed is miles per hour. Wait, maybe I mixed up. Let's re-express: If \( h = \frac{1}{3}m \), then \( m = 3h \), so speed is 3 miles per hour. But the constant of proportionality here (between hours and miles) is \( \frac{1}{3} \). So when miles increase, hours increase by \( \frac{1}{3} \) per mile. So for 1 mile, hours are \( \frac{1}{3} \), meaning she bikes 1 mile every \( \frac{1}{3} \) hour. Let's verify: In 1 hour, how many miles? If 1 mile takes \( \frac{1}{3} \) hour, then in 1 hour, she bikes \( \frac{1}{\frac{1}{3}} = 3 \) miles, which matches \( 12 \) miles in \( 4 \) hours (\( 12/4 = 3 \) mph). So the constant \( \frac{1}{3} \) (hours per mile) means 1 mile every \( \frac{1}{3} \) hour. The other option: "biked \( \frac{1}{3} \) mile every hour" would mean speed \( \frac{1}{3} \) mph, which is wrong (since 12 miles in 4 hours is 3 mph). So the correct meaning is "biked 1 mile every \( \frac{1}{3} \) hour".

Answer:

biked 1 mile every \( \frac{1}{3} \) hour