Nothing stays the same

A florist had roses and tulips. The number of roses to the number of tulips was 3 to 2. Two ninths of the roses and one quarter of the tulips were then sold. After that, there were 55 more roses than tulips left. How many roses did the florist have at first? Try it first. Pause if you like. Write what we know. Roses to tulips is 3 to 2. Now draw it. Roses, 3 blue units. Tulips, 2. Two ninths of the roses means cutting the roses into 9 equal parts. But the rose bar only has 3 equal units. So share the 9 parts equally among the 3 units. 9 divided by 3 is 3. Each rose unit is cut into 3 parts. One quarter of the tulips means cutting the tulips into 4 equal parts. The tulip bar has 2 equal units. Share the 4 parts equally among the 2 units. 4 divided by 2 is 2. Each tulip unit is cut into 2 parts. But every unit has to stay the same size, the roses and the tulips together. So we need one number of parts that both 3 and 2 reach. Count up in threes. 3, 6. Count up in twos. 2, 4, 6. 6 is in both lists. It is the lowest common multiple of 3 and 2. So cut every unit into 6. The roses were 3 units, and each one is now 6 small units. 3 times 6 is 18 small units. The tulips were 2 units. 2 times 6 is 12 small units. Write the new ratio. Two ninths of the roses were sold, and the roses are 18 units. So write it as 2 ninths times 18. 9 and 18 both divide by 9. Cross them out. Write 1 under the 9, and 2 over the 18. 2 times 2 is 4. So 4 units of roses were sold. One quarter of the tulips were sold, and the tulips are 12 units. So write it as 1 quarter times 12. 4 and 12 both divide by 4. Cross them out. Write 1 under the 4, and 3 over the 12. 1 times 3 is 3. So 3 units of tulips were sold. Now count what is left in each bar. The roses were 18 units, and 4 units were sold. 18 minus 4 is 14. The tulips were 12 units, and 3 units were sold. 12 minus 3 is 9. So 14 units of roses are left, and 9 units of tulips. 14 units minus 9 units. That's 5 units. That's the difference after the selling. And the question says there are 55 more roses than tulips. So those 5 units are 55. So 1 unit is 55 divided by 5. That's 11. The roses are 18 units. 18 times 11 is one hundred and ninety eight. That's how many roses the florist had at first. In this one, nothing stayed the same. So we didn't look for something that did. We cut the units small enough for both fractions to be whole numbers of the small units, and counted from there.