Determine the ratio requested based on the given information.

The monthly production volume of Factory B is 30% greater than that of Factory A, and the monthly production volume of Factory C is 25% greater than that of Factory B. What is the ratio of the production volume of Factory A to that of Factory C?

Correct Answer/Option:
The correct option is 8 : 13.

Step-by-Step Explanation:

Step 1: Define variables for the production volumes.
Let $P_A$, $P_B$, and $P_C$ represent the monthly production volumes of Factory A, Factory B, and Factory C, respectively.

Step 2: Express Factory B's production in terms of Factory A's production.
It is given that Factory B's production volume is 30% greater than that of Factory A.

$P_B=P_A+0.30P_A=1.30P_A=\frac{13}{10}{P}_A$

Step 3: Express Factory C's production in terms of Factory B's production.
It is also given that Factory C's production volume is 25% greater than that of Factory B.

$P_C=P_B+0.25P_B=1.25P_B=\frac{5}{4}{P}_B$

Step 4: Express Factory C's production directly in terms of Factory A's production.
Substitute the relation between $P_B$ and $P_A$ into the equation for $P_C$:

$P_C=\frac{5}{4}\times \left(\frac{13}{10}{P}_A\right)$

$P_C=\frac{5\times 13}{4\times 10}{P}_A=\frac{13}{8}{P}_A$

Step 5: Calculate the ratio of Factory A to Factory C.
Now, find the ratio $\frac{P_A}{P_C}$:

$\frac{P_A}{P_C}=\frac{P_A}{\frac{13}{8}{P}_A}=\frac{8}{13}$

Thus, the ratio of the production volume of Factory A to Factory C is 8 : 13.