A number of students from SMS are participating in a community service project. The ratio of 8th graders to 6th graders is 5:3 and the ratio of 8th graders to 7th graders is 8:5. What is the smallest number of students that participated?

The smallest number of students that participated in community service project is 89Solution:Given that,The ratio of 8th graders to 6th graders is 5:3 and the ratio of 8th graders to 7th graders is 8:5We have to find what is the smallest number of students that participated?Now, given ratios can be converted to fractions. a : b can be converted to fraction as [tex]\frac{a}{b}[/tex][tex]\text { 8th graders to } 6 \text { th graders is } 5: 3 \rightarrow \frac{\text { 8th graders }}{6 t h \text { graders }}=\frac{5}{3}[/tex]On inversing we get,[tex]\rightarrow \frac{6 t h \text { graders }}{\text { 8th graders }}=\frac{3}{5}[/tex][tex]8 \text { th graders to } 7 \text { th graders is } 8: 5 \rightarrow \frac{\text { 8th graders }}{7 \text { th graders }}=\frac{8}{5}[/tex]On inversing we get,[tex]\rightarrow \frac{7 t h \text { graders }}{\text { 8th graders }}=\frac{5}{8}[/tex]Now, add both the fractions[tex]\frac{6 \text {th graders}}{\text { 8th graders }}+\frac{7 \text {th graders}}{\text { 8th graders }}=\frac{3}{5}+\frac{5}{8}[/tex][tex]\frac{6 t h \text { graders } + 7 \text { th graders }}{\text { 8th graders }}=\frac{24+25}{40}=\frac{49}{40}[/tex]Now, we want least number of students, then we have to take from above fraction,6th graders + 7th graders = 49  And 8th graders = 40,  Then, total students = 6th graders + 7th graders + 8th graders = 49 + 40 = 89Hence, 89 students participated.