Let the total number of flowers Belle had at first (both roses and carnations) be denoted by $T$, and let the number of roses be $R$. We are given the following information:

1. The number of roses $R$ was one-third of the total number of roses and carnations, so: $R = \frac{1}{3} T$

2. After 247 carnations were sold, the number of roses became five-sixths of the total number of flowers. Let the new total number of flowers after the sale of 247 carnations be $T - 247$. At this point, the number of roses is still $R$, and we are told that: $R = \frac{5}{6} (T - 247)$

We now have two equations:

1. $R = \frac{1}{3} T$
2. $R = \frac{5}{6} (T - 247)$

Step 1: Substitute $R = \frac{1}{3} T$ into the second equation

Substituting $R = \frac{1}{3} T$ into the second equation: $\frac{1}{3} T = \frac{5}{6} (T - 247)$

Step 2: Solve for $T$

To solve for $T$, first eliminate the fractions by multiplying both sides by 6: $6 \times \frac{1}{3} T = 6 \times \frac{5}{6} (T - 247)$ $2T = 5(T - 247)$ Now, expand the right-hand side: $2T = 5T - 1235$ Subtract $2T$ from both sides: $0 = 3T - 1235$ Add 1235 to both sides: $3T = 1235$ Now, divide by 3: $T = \frac{1235}{3} = 411.67$

It seems that this is not an integer, which indicates that there might be an issue with interpreting the problem's information.

Let's revisit the problem statement:

• It was stated that the roses became 5 of the