Yoko is hiking on a trail that is  miles from start to finish. The ratio of the distance she has walked from the starting point to the distance she has left to walk is . How many miles has Yoko walked?

Let's define the total distance of the trail as \( d \) miles. Let the distance Yoko has already walked be \( x \) miles. Therefore, the distance she has left to walk is \( d - x \).

According to the problem, the ratio of the distance Yoko has walked to the distance she has left to walk is given as:

\[
\frac{x}{d - x} = \frac{3}{5}
\]

To find how many miles Yoko has walked, we need to solve this equation.

### Step 1: Set up the proportion
We can cross-multiply to get rid of the fraction:

\[
x \cdot 5 = 3 \cdot (d - x)
\]

### Step 2: Expand the equation
Expand the right side:

\[
5x = 3d - 3x
\]

### Step 3: Combine like terms
Add \( 3x \) to both sides to get all the \( x \)'s on one side:

\[
5x + 3x = 3d
\]

\[
8x = 3d
\]

### Step 4: Solve for \( x \)
Now, divide both sides by 8 to solve for \( x \):

\[
x = \frac{3d}{8}
\]

Thus, Yoko has walked \( \frac{3}{8} \) of the total distance \( d \).

If you know the total distance \( d \), you can substitute it into the equation to find the specific distance Yoko has walked.

Would you like me to clarify anything or go into more detail?

Here are five related questions to expand your understanding:
1. What would the equation look like if the ratio were \( \frac{2}{3} \) instead of \( \frac{3}{5} \)?
2. How can you find the distance Yoko has left to walk in terms of \( d \)?
3. What happens if \( d = 8 \) miles? How far has Yoko walked?
4. How does changing the ratio affect the distance walked?
5. Can you write an equation for the distance walked if the total trail distance changes?

**Tip:** Solving ratios often involves setting up a proportion and then cross-multiplying to simplify the equation!

Learn how to calculate the distance Yoko has walked on a trail based on the given ratio problem. Understand the ratio of distance walked to distance left to walk and solve using a proportion method. Suitable for Grades 7-9.

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