Belle had some roses and carnations. The number of roses was , of the total number of roses and carnations. After 247 carnations were sold, the number of roses became 5 of the total number of roses and carnations. How many more carnations than roses did Belle have at first?

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 

Belle had some roses and carnations. The number of roses was one-third of the total number of roses and carnations. After 247 carnations were sold, the number of roses became five-sixths of the total number of roses and carnations. How many more carnations than roses did Belle have at first?

Learn how to solve a flower ratio problem where Belle has a combination of roses and carnations. Using algebraic linear equations, we find the original number of flowers and the relationship between the roses and carnations. Suitable for students in Grades 8-10.

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