Two horses A and B run at a speed of 3:2 ratio in the first lap, during the second lap the ratio differs by 4:7 : during the third lap ther ratio differs by 8:9. What is the difference in ratio of speed altogether between the two horses ?

Options

A. 4
B. 2
C. 3
D. 1

Explanation

To solve the problem step by step, we will analyze the speed ratios of the two horses A and B across the three laps and calculate the overall difference in their speeds.

### Step 1: Identify the speed ratios for each lap
- **Lap 1:** The speed ratio of horses A and B is 3:2.
- **Lap 2:** The speed ratio of horses A and B is 4:7.
- **Lap 3:** The speed ratio of horses A and B is 8:9.

### Step 2: Convert the ratios into a common format
To work with these ratios, we can express them in terms of a common variable (let's use 'X').

- For **Lap 1:**
- Speed of A = 3X
- Speed of B = 2X

- For **Lap 2:**
- Speed of A = 4Y (where Y is another variable)
- Speed of B = 7Y

- For **Lap 3:**
- Speed of A = 8Z (where Z is another variable)
- Speed of B = 9Z

### Step 3: Find a common variable for all laps
To compare the speeds across all laps, we need to express them in terms of a single variable. We can set the variables such that they are proportional to each other.

Let’s take the least common multiple (LCM) of the denominators of the ratios to find a common variable:
- The ratios are 3:2, 4:7, and 8:9.

To find a common variable, we can express the speeds as follows:
- For Lap 1: 3X and 2X
- For Lap 2: 4(3) and 7(3) → 12 and 21 (using 3 as a common multiple)
- For Lap 3: 8(2) and 9(2) → 16 and 18 (using 2 as a common multiple)

### Step 4: Calculate the total speeds
Now we can sum the speeds of A and B over the three laps:
- Total speed of A = (3X + 12 + 16) = 3X + 28
- Total speed of B = (2X + 21 + 18) = 2X + 39

### Step 5: Calculate the difference in speeds
Now we can find the difference in their speeds:
- Difference = Total speed of B - Total speed of A
- Difference = (2X + 39) - (3X + 28)
- Difference = 2X + 39 - 3X - 28
- Difference = -X + 11

### Step 6: Interpret the result
The difference in speeds can be expressed as:
- Difference = 11 - X

If we assume X = 1 (for simplicity), the difference in speeds becomes:
- Difference = 11 - 1 = 10

### Final Answer
Thus, the difference in the ratio of speed altogether between the two horses A and B is **10**.

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