4. mai, clare, and tyler are hiking from a parking lot to the summit of a mountain. they pass a sign that…

4. mai, clare, and tyler are hiking from a parking lot to the summit of a mountain. they pass a sign that gives distances.\nparking lot: $\frac{3}{4}$ mile\nsummit: $1\frac{1}{2}$ miles\nmai says: \we are one third of the way there.\\nclare says: \we have to go twice as far as we have already gone.\\ntyler says: \the total hike is three times as long as what we have already gone.\\ndo you agree with any of them? explain your reasoning.\ntotal distances =

4. mai, clare, and tyler are hiking from a parking lot to the summit of a mountain. they pass a sign that gives distances.\nparking lot: $\frac{3}{4}$ mile\nsummit: $1\frac{1}{2}$ miles\nmai says: \we are one third of the way there.\\nclare says: \we have to go twice as far as we have already gone.\\ntyler says: \the total hike is three times as long as what we have already gone.\\ndo you agree with any of them? explain your reasoning.\ntotal distances =

Answer

Explanation:

Step1: Calculate total hike distance

Add distance from parking lot to sign and sign to summit. First convert $1\frac{1}{2}$ to improper fraction: $1\frac{1}{2}=\frac{3}{2}$. $$\text{Total distance} = \frac{3}{4} + \frac{3}{2} = \frac{3}{4} + \frac{6}{4} = \frac{9}{4} = 2\frac{1}{4} \text{ miles}$$

Step2: Find fraction of hike completed

Divide distance from parking lot to sign by total distance. $$\text{Fraction completed} = \frac{\frac{3}{4}}{\frac{9}{4}} = \frac{3}{4} \times \frac{4}{9} = \frac{1}{3}$$

Step3: Verify Mai's statement

Mai claims they are $\frac{1}{3}$ of the way there. The calculated fraction completed is $\frac{1}{3}$, so this matches.

Step4: Verify Clare's statement

Clare claims remaining distance is twice the distance traveled. Remaining distance is $\frac{3}{2}$ miles, distance traveled is $\frac{3}{4}$ miles. $$2 \times \frac{3}{4} = \frac{6}{4} = \frac{3}{2} \text{ miles}$$ This equals the remaining distance, so Clare's statement is correct.

Step5: Verify Tyler's statement

Tyler claims total distance is 3 times the distance traveled. $$3 \times \frac{3}{4} = \frac{9}{4} = 2\frac{1}{4} \text{ miles}$$ This equals the total distance, so Tyler's statement is correct.

Answer:

I agree with Mai, Clare, and Tyler.

  • The total hike distance is $2\frac{1}{4}$ miles, and the group has completed $\frac{1}{3}$ of the hike, which matches Mai's statement.
  • The remaining distance ($1\frac{1}{2}$ miles) is twice the distance already traveled ($\frac{3}{4}$ miles), which matches Clare's statement.
  • The total hike distance ($2\frac{1}{4}$ miles) is three times the distance already traveled ($\frac{3}{4}$ miles), which matches Tyler's statement.