What must be added to each of the numbers 7, 11 and 19, so that the resulting numbers may be in continued proportion?

To solve this problem, we need to find a common ratio for the numbers to be in continued proportion. This means that the ratio of the second number to the first number should be the same as the ratio of the third number to the second number.

Let's denote the number to be added as x. So, the numbers become (7+x), (11+x), and (19+x).

The condition for the numbers to be in continued proportion is:

(11+x) / (7+x) = (19+x) / (11+x)

Cross-multiplying gives us:

(11+x)² = (7+x) * (19+x)

Expanding and simplifying gives us:

121 + 22x + x² = 133 + 26x + x²

Subtracting x² and 121 from both sides gives us:

22x = 12 + 26x

Subtracting 22x from both sides gives us:

0 = 12 + 4x

Subtracting 12 from both sides gives us:

-12 = 4x

Finally, dividing both sides by 4 gives us:

x = -3

So, -3 must be added to each of the numbers 7, 11, and 19 so that the resulting numbers may be in continued proportion.