We multiply the first ratio by 2 on both sides, and the second ratio by 3 to get the same number for 8 th graders, in order that we can put the two ratios together: $$ \begin{aligned} \&7: 6=7(2): 6(2)=14: 12 \textbackslash\textbackslash{} \&4: 7=4(3): 7(3)=12: 21\end{aligned} $$ Therefore, the ratio of 8th graders to 7th graders to 6th graders is $14: 12: 21$. Since the ratio is in lowest terms, the smallest number of students participating in the project is $$ 14+12+21=\text { (B) } 47 $$  Notice if you get $$28:24:42$$, you have to simplfy to $14: 12: 21$ for the smallest number of students. 
A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of $8^{\text {th }}$-graders to $6^{\text {th }}$-graders is $7: 6$, and the the ratio of $8^{\text {th }}$-graders to $7^{\text {th }}$ graders is $4: 7$. What is the smallest number of students that could be participating in the project? (2013 AMC 8, Question 16) 

[a] $$21$$ 
[b] $$47$$ 
[c] $$56$$ 
[d] $$94$$ 
[e] $$100$$ 