Each book on a certain shelf is labeled by a single category. For every 2 history books, there are 7 fantasy books and for every 3 fantasy books, there are 5 reference books. If the proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained, which of the following could be the number of history books if there are fewer than 60 fantasy books on the shelf after the changes?

6
21
24
35
36

H/F = 2/7.
F/R = 3/5.

To combine the ratios, the COMMON ELEMENT -- F -- must be represented by the SAME VALUE in each ratio:
H/F = (2*3)/(7*3) = 6/21.
F/R = (3*7)/(5*7) = 21/35.
Combining the ratios, we get:
H/F/R = 6/21/35.

After H/R is doubled from 6/35 to 12/35, the new ratio is as follows:
H/F/R = 12/21/35.

Since the actual value of F is less than 60, only two cases are possible.
H/F/R = 12/21/35.
H/F/R = 24/42/70.

Thus, the value of H could be 12 or 24.

The correct answer is C.

H:F = 2:7

F:R = 3:5

According to the question proportion of the history to fantasy book is doubled

so H:F = 4:14 right?

Careful!
To yield 4/14, you multiplied H/F = 2/7 by 2/2.
Since 2/2 = 1, multplying a ratio by 2/2 does not change its value..
The result is that the new ratio (4/14) is the same as the original ratio (2/7):
4/14 = 2/7.

To DOUBLE A RATIO means to MULTIPLY IT BY 2/1.
Thus, when we double the value of H/F, we get:
(2/7)(2/1) = 4/7.

Implication:
Instead of 2 history books for every 7 fantasy books, now there are 4 history books for every 7 fantasy books, DOUBLING the concentration of history books per fantasy books.