inverse proportion help!

\(a = p b\\ a = \dfrac q c\\ c = \dfrac{q}{pb}\)

\(2=5p \Rightarrow p=\dfrac 2 5\\ 2 = \dfrac q 9 \Rightarrow q = 18\\~\\ c = \dfrac{18}{\frac 2 5 \cdot 3} = 15 \)

Suppose a is directly proportional to b, but inversely proportional to c. If a=2 when b=5 and c=9, then what is c when b=3?

\(a:b=2:5\\ a:c=9:2\)                    Nonsense. How blamable 
 

\(b=\frac{5a}{2}\\ c=\frac{2a}{9}\)

b=3

\(a=\frac{2b}{5 }\\ a=\frac{2\cdot 3}{5}\\ a=\frac{6}{5}\\ c=\frac{2a}{9}=\frac{2\cdot6}{9\cdot 5}\\ \color{blue}c=\frac{4}{15}\ \color{black}Not\ correct.\)      

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Suppose a is directly proportional to b, but inversely proportional to c. If a=2 when b=5 and c=9, then what is c when b=3?

\(\color{BrickRed}a:b=2:5\\ \color{BrickRed}a:\frac{1}{c}=2:\frac{1}{9}\\ a=\frac{2b}{5}\\ \frac{2}{c}=\frac{a}{9}\\ c=\frac{18}{a}\)

b=3

\(a=\frac{2b}{5}\\ a=\frac{6}{5}\\ c=\frac{18}{a}\\ c=\frac{18\cdot 5}{6}\\ \color{blue}c=15\)

Thanks Melody! I have "looked up".

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