**Given information:**

- Odds against A solving the problem are 4 to 3.
- Odds in favor of B solving the problem are 7 to 5.

**Understanding odds:**

Odds represent the likelihood of an event occurring. In this case, the odds are given as ratios. For example, odds against A solving the problem are 4 to 3 means that for every 4 times A fails to solve the problem, A succeeds 3 times.

**Calculating probabilities from odds:**

To calculate the probability from odds, we use the formula:

Probability = favorable outcomes / total outcomes

In this case, the favorable outcomes represent the number of times A or B can solve the problem, and the total outcomes represent all possible outcomes.

**Calculating the probability of A solving the problem:**

If the odds against A solving the problem are 4 to 3, it means that out of every 4+3=7 attempts, A succeeds 3 times. Therefore, the probability of A solving the problem can be calculated as:

Probability of A = 3 / 7

**Calculating the probability of B solving the problem:**

If the odds in favor of B solving the problem are 7 to 5, it means that out of every 7+5=12 attempts, B succeeds 7 times. Therefore, the probability of B solving the problem can be calculated as:

Probability of B = 7 / 12

**Calculating the probability of both A and B solving the problem:**

To calculate the probability that both A and B solve the problem, we multiply their individual probabilities:

Probability of both A and B = Probability of A * Probability of B
= (3 / 7) * (7 / 12)
= 21 / 84
= 1 / 4

**Simplifying the probability:**

To simplify the probability, we can divide both the numerator and denominator by their greatest common divisor, which is 1 in this case:

Probability of both A and B = 1 / 4

**Converting the probability to odds:**

To convert the probability to odds, we use the formula:

Odds = favorable outcomes : unfavorable outcomes

In this case, the favorable outcome is 1 (both A and B solving the problem) and the unfavorable outcome is 3 (A not solving the problem). Therefore, the odds in favor of both A and B solving the problem can be written as:

Odds in favor = 1 : 3

**Calculating the probability that the problem will be solved:**

The probability that the problem will be solved is the sum of the probabilities of A and B solving the problem, minus the probability of both A and B solving the problem (to avoid double counting):

Probability of problem solved = Probability of A + Probability of B - Probability of both A and B
= 3 / 7 + 7 / 12 - 1 / 4
= (36 + 49 - 21) / 84
= 64 / 84
= 16 / 21

Therefore, the probability that the problem will be solved if both A and B try is 16/21.

**Conclusion:**

The correct answer is option B) 16/21.

To find the probability that the problem will be solved if both A and B try, we need to consider the odds against A and the odds in favor of B.

Given:
Odds against A = 4 to 3
Odds in favor of B = 7 to 5

To find the probability of an event, we can use the formula:

Probability = favorable outcomes / total outcomes

Let's calculate the probabilities for A and B separately:

For A:
Odds against A = 4 to 3
This means that for every 4 unfavorable outcomes, there are 3 favorable outcomes. So, the probability of A solving the problem is given by:

Probability of A = favorable outcomes / total outcomes
Probability of A = 3 / (4 + 3)
Probability of A = 3 / 7

For B:
Odds in favor of B = 7 to 5
This means that for every 7 favorable outcomes, there are 5 unfavorable outcomes. So, the probability of B solving the problem is given by:

Probability of B = favorable outcomes / total outcomes
Probability of B = 7 / (7 + 5)
Probability of B = 7 / 12

Now, to find the probability that the problem will be solved if both A and B try, we need to find the probability that either A or B solves the problem. Since A and B are independent events, we can add their probabilities:

Probability of (A or B) = Probability of A + Probability of B - (Probability of A * Probability of B)

Probability of (A or B) = (3 / 7) + (7 / 12) - [(3 / 7) * (7 / 12)]
Probability of (A or B) = (36 + 49 - 21) / (7 * 12)
Probability of (A or B) = 64 / 84
Probability of (A or B) = 16 / 21

Therefore, the probability that the problem will be solved if both A and B try is 16/21, which corresponds to option B.