When Tom moved to a new home, his distance to work decreased by `(1)/(2)` the original distance and the constant rate at which he travels to work increased by `(1)/(3)` the original rate. By what percent has the time it takes Tom to travel to work decreased?

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Define the Variables
Let:
- \( D \) = Original distance to work
- \( S \) = Original speed to work

### Step 2: Calculate Original Time
The original time taken to travel to work, denoted as \( T_1 \), can be calculated using the formula:
\[ T_1 = \frac{D}{S} \]

### Step 3: Calculate New Distance
After moving to the new home, the distance to work decreases by half of the original distance. Therefore, the new distance \( D' \) is:
\[ D' = D - \frac{D}{2} = \frac{D}{2} \]

### Step 4: Calculate New Speed
The speed increases by one third of the original speed. Therefore, the new speed \( S' \) is:
\[ S' = S + \frac{S}{3} = S + \frac{S}{3} = \frac{3S}{3} + \frac{S}{3} = \frac{4S}{3} \]

### Step 5: Calculate New Time
The new time taken to travel to work, denoted as \( T_2 \), can be calculated as:
\[ T_2 = \frac{D'}{S'} = \frac{\frac{D}{2}}{\frac{4S}{3}} = \frac{D}{2} \times \frac{3}{4S} = \frac{3D}{8S} \]

### Step 6: Relate New Time to Original Time
From Step 2, we know that \( T_1 = \frac{D}{S} \). Thus, we can express \( T_2 \) in terms of \( T_1 \):
\[ T_2 = \frac{3}{8} T_1 \]

### Step 7: Calculate Percentage Decrease in Time
The percentage decrease in time can be calculated using the formula:
\[ \text{Percentage Decrease} = \frac{T_1 - T_2}{T_1} \times 100\% \]
Substituting \( T_2 \):
\[ \text{Percentage Decrease} = \frac{T_1 - \frac{3}{8} T_1}{T_1} \times 100\% = \frac{T_1 - \frac{3T_1}{8}}{T_1} \times 100\% \]
This simplifies to:
\[ = \frac{\frac{8T_1}{8} - \frac{3T_1}{8}}{T_1} \times 100\% = \frac{\frac{5T_1}{8}}{T_1} \times 100\% = \frac{5}{8} \times 100\% = 62.5\% \]

### Final Answer
The percentage decrease in the time it takes Tom to travel to work is:
\[ \boxed{62.5\%} \]