In a given race , the odds in favour of four horses A,B C , and D are 1 : 3 , 1 : 4 , 1 : 5 , and 1 : 6 respectively .Assuming that a dead heat is impossible , find the chance that one of them wins the race .

Answer

To solve the problem, we need to find the probability that one of the four horses (A, B, C, or D) wins the race, given their odds in favor.

### Step-by-Step Solution:

1. **Understanding Odds**:
- The odds in favor of a horse represent the ratio of the probability of the horse winning to the probability of it losing.
- For horse A, the odds are 1:3, which means if we consider the total outcomes as 4 (1 win + 3 losses), the probability of A winning is \( \frac{1}{4} \).
- For horse B, the odds are 1:4, meaning the probability of B winning is \( \frac{1}{5} \).
- For horse C, the odds are 1:5, meaning the probability of C winning is \( \frac{1}{6} \).
- For horse D, the odds are 1:6, meaning the probability of D winning is \( \frac{1}{7} \).

2. **Calculating Individual Probabilities**:
- For horse A:
\[ P(A) = \frac{1}{1 + 3} = \frac{1}{4} \]
- For horse B:
\[ P(B) = \frac{1}{1 + 4} = \frac{1}{5} \]
- For horse C:
\[ P(C) = \frac{1}{1 + 5} = \frac{1}{6} \]
- For horse D:
\[ P(D) = \frac{1}{1 + 6} = \frac{1}{7} \]

3. **Finding a Common Denominator**:
- The common denominator for 4, 5, 6, and 7 is 420.

4. **Converting Probabilities**:
- Convert each probability to have the common denominator:
\[ P(A) = \frac{1}{4} = \frac{105}{420} \]
\[ P(B) = \frac{1}{5} = \frac{84}{420} \]
\[ P(C) = \frac{1}{6} = \frac{70}{420} \]
\[ P(D) = \frac{1}{7} = \frac{60}{420} \]

5. **Summing the Probabilities**:
- Now, add the probabilities:
\[ P(A) + P(B) + P(C) + P(D) = \frac{105}{420} + \frac{84}{420} + \frac{70}{420} + \frac{60}{420} \]
\[ = \frac{105 + 84 + 70 + 60}{420} = \frac{319}{420} \]

6. **Final Probability**:
- The probability that one of the horses wins the race is:
\[ P(\text{one of A, B, C, D wins}) = \frac{319}{420} \]

### Final Answer:
The chance that one of the horses A, B, C, or D wins the race is \( \frac{319}{420} \).