Amy and Bella bought some cookies in the ratio of $$3:4$$. Bella gave $$9$$ cookies to Amy. Then the ratio of the number of cookies Amy had to that of Bella was $$1:1$$. How many cookies did they have at first? 
Overseas Competition->Knowledge Point->Word Problem Modules->Solving Problems Involving Proportions->Ratio Word Problems with Invariant Sums
Total number of cookies both of them had did not change.  Make total the same number of units by finding LCM.  Before $\to$ $A:B:Total$ $\to$ $3:4:7$ $\to$ $6:8:14$  After $\to$ $A:B:Total$ $\to$ $1:1:2$ $\to$ $7:7:14$  $1u=9$  $14u=14\times9=126$ cookies at first (and also in the end) in total. 