Math Problem Statement

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?

Solution

Let's define the total distance of the trail as dd miles. Let the distance Yoko has already walked be xx miles. Therefore, the distance she has left to walk is dxd - x.

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

xdx=35\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:

x5=3(dx)x \cdot 5 = 3 \cdot (d - x)

Step 2: Expand the equation

Expand the right side:

5x=3d3x5x = 3d - 3x

Step 3: Combine like terms

Add 3x3x to both sides to get all the xx's on one side:

5x+3x=3d5x + 3x = 3d

8x=3d8x = 3d

Step 4: Solve for xx

Now, divide both sides by 8 to solve for xx:

x=3d8x = \frac{3d}{8}

Thus, Yoko has walked 38\frac{3}{8} of the total distance dd.

If you know the total distance dd, 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 23\frac{2}{3} instead of 35\frac{3}{5}?
  2. How can you find the distance Yoko has left to walk in terms of dd?
  3. What happens if d=8d = 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!