Scott starts jogging from point X to point Y. A half-hour later his friend Garrett who jogs 1 mile per hour slower than twice Scott's rate starts from the same point and follow the same path. If Garrett overtakes Scott in 2 hours, how many miles will Garrett have covered?

Options

A. `2 1/5`
B. `3 1/3`
C. `4`
D. 6

Answer

To solve the problem step by step, let's define the variables and break down the information given in the question.

### Step 1: Define Scott's Speed
Let Scott's jogging speed be \( a \) miles per hour.

### Step 2: Define Garrett's Speed
Garrett jogs 1 mile per hour slower than twice Scott's speed. Therefore, Garrett's speed can be expressed as:
\[ \text{Garrett's speed} = 2a - 1 \text{ miles per hour} \]

### Step 3: Calculate the Distance Scott Covers Before Garrett Starts
Scott starts jogging 0.5 hours (30 minutes) before Garrett. In that time, the distance Scott covers is:
\[ d_1 = a \times 0.5 = \frac{a}{2} \text{ miles} \]

### Step 4: Time Taken by Garrett to Overtake Scott
Garrett overtakes Scott after 2 hours of jogging. Therefore, the total time Scott has been jogging when Garrett overtakes him is:
\[ \text{Total time for Scott} = 2 + 0.5 = 2.5 \text{ hours} \]

### Step 5: Calculate the Total Distance Covered by Scott
The total distance Scott covers in 2.5 hours is:
\[ d_1 + d_2 = a \times 2.5 = 2.5a \text{ miles} \]
where \( d_2 \) is the distance Scott covers in the 2 hours after Garrett starts.

### Step 6: Calculate the Distance Garrett Covers
In 2 hours, Garrett covers the distance:
\[ d_2 = \text{Garrett's speed} \times \text{time} = (2a - 1) \times 2 = 4a - 2 \text{ miles} \]

### Step 7: Set Up the Equation for Distances
Since both Scott and Garrett cover the same distance when Garrett overtakes Scott, we can set up the equation:
\[ d_1 + d_2 = d_1 + (4a - 2) \]
Substituting \( d_1 \):
\[ \frac{a}{2} + (4a - 2) = 2.5a \]

### Step 8: Simplify the Equation
Now, we simplify the equation:
\[ \frac{a}{2} + 4a - 2 = 2.5a \]
Multiplying through by 2 to eliminate the fraction:
\[ a + 8a - 4 = 5a \]
This simplifies to:
\[ 9a - 5a = 4 \]
\[ 4a = 4 \]
\[ a = 1 \text{ mile per hour} \]

### Step 9: Calculate Garrett's Speed
Now that we have Scott's speed, we can find Garrett's speed:
\[ \text{Garrett's speed} = 2(1) - 1 = 1 \text{ mile per hour} \]

### Step 10: Calculate the Distance Garrett Covered
Garrett jogs for 2 hours, so the distance he covers is:
\[ \text{Distance covered by Garrett} = \text{Garrett's speed} \times \text{time} = 1 \times 2 = 2 \text{ miles} \]

### Final Answer
Garrett will have covered **2 miles**.

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