X covers 1 km in 8 min 40 s while Y covers the same distance in 10 min. By what distance does X defeat Y?

Options

A. `13 1/3 m`
B. `133 2/3 m`
C. `133 2/5`m
D. `133 1/3 m`

Answer

To solve the problem of how much distance X defeats Y, we will follow these steps:

### Step 1: Convert the time taken by X into seconds
X covers 1 km in 8 minutes 40 seconds.
- First, convert 8 minutes into seconds:
\[ 8 \text{ minutes} = 8 \times 60 = 480 \text{ seconds} \]
- Now, add the remaining 40 seconds:
\[ 480 \text{ seconds} + 40 \text{ seconds} = 520 \text{ seconds} \]
Thus, X takes 520 seconds to cover 1000 meters.

### Step 2: Convert the time taken by Y into seconds
Y covers the same distance in 10 minutes.
- Convert 10 minutes into seconds:
\[ 10 \text{ minutes} = 10 \times 60 = 600 \text{ seconds} \]
So, Y takes 600 seconds to cover 1000 meters.

### Step 3: Calculate the speeds of X and Y
- Speed of X:
\[ \text{Speed of X} = \frac{\text{Distance}}{\text{Time}} = \frac{1000 \text{ meters}}{520 \text{ seconds}} = \frac{1000}{520} = \frac{25}{13} \text{ meters/second} \]

- Speed of Y:
\[ \text{Speed of Y} = \frac{\text{Distance}}{\text{Time}} = \frac{1000 \text{ meters}}{600 \text{ seconds}} = \frac{1000}{600} = \frac{5}{3} \text{ meters/second} \]

### Step 4: Calculate the distance covered by Y in the time taken by X
Now, we need to find out how far Y travels in the time it takes X to finish the race (520 seconds).
\[ \text{Distance covered by Y} = \text{Speed of Y} \times \text{Time taken by X} = \frac{5}{3} \times 520 \]
Calculating this:
\[ \frac{5}{3} \times 520 = \frac{5 \times 520}{3} = \frac{2600}{3} \text{ meters} \]

### Step 5: Calculate the distance by which X defeats Y
Now, we find the distance by which X defeats Y:
\[ \text{Distance defeated} = \text{Distance covered by X} - \text{Distance covered by Y} \]
\[ = 1000 - \frac{2600}{3} \]
To perform this subtraction, convert 1000 into a fraction with a denominator of 3:
\[ 1000 = \frac{3000}{3} \]
Now, subtract:
\[ \frac{3000}{3} - \frac{2600}{3} = \frac{3000 - 2600}{3} = \frac{400}{3} \text{ meters} \]

### Step 6: Convert the fraction into a mixed number
To convert \(\frac{400}{3}\) into a mixed number:
\[ 400 \div 3 = 133 \text{ remainder } 1 \]
Thus, \(\frac{400}{3} = 133 \frac{1}{3}\) meters.

### Final Answer
X defeats Y by \(133 \frac{1}{3}\) meters.

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