This question involves finding a specific number when ratios between numbers and their total sum are provided. We need to determine the value of the third number.

Understanding the Given Ratios

We are given two ratios:

• The ratio of the first number to the second number is \(3 : 4\).
• The ratio of the second number to the third number is \(8 : 5\).

We are also told that the sum of the three numbers is 190.

Combining the Ratios

To work with all three numbers together, we need a single combined ratio (First : Second : Third). The second number is common to both ratios. To combine them, we must make the value representing the second number the same in both ratios.

• First ratio: \( \text{First} : \text{Second} = 3 : 4 \)
• Second ratio: \( \text{Second} : \text{Third} = 8 : 5 \)

Notice that the second number is represented by 4 in the first ratio and 8 in the second. The least common multiple of 4 and 8 is 8. We can adjust the first ratio by multiplying both parts by 2:

Adjusted First Ratio: \( (3 \times 2) : (4 \times 2) = 6 : 8 \)

Now, the second number is represented by 8 in both ratios. We can combine them:

Combined Ratio: \( \text{First} : \text{Second} : \text{Third} = 6 : 8 : 5 \)

Representing the Numbers

We can represent the three numbers based on the combined ratio using a common multiplier, let's call it \(x\).

• First number = \(6x\)
• Second number = \(8x\)
• Third number = \(5x\)

Using the Sum to Find the Multiplier

The sum of the three numbers is given as 190. So, we can set up an equation:

\( \text{First number} + \text{Second number} + \text{Third number} = 190 \)

Substituting our expressions:

\( 6x + 8x + 5x = 190 \)

Combine the terms with \(x\):

\( (6 + 8 + 5)x = 190 \)

\( 19x = 190 \)

Now, solve for \(x\) by dividing both sides by 19:

\( x = \frac{190}{19} \)

\( x = 10 \)

Calculating the Third Number

We need to find the value of the third number, which we represented as \(5x\).

Third number = \( 5x \)

Substitute the value of \(x\) we found:

Third number = \( 5 \times 10 \)

Third number = \( 50 \)

Conclusion

The third number is 50. This corresponds to the first option.