Question

Find a : b : c if 6a = 9b = 10c.

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{6a=9b=10c}$

$\underline{\textbf{To find:}}$

$\mathsf{a:b:c}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{6a=9b=10c=k(say)}$

$\implies\mathsf{6a=k,\;\;9b=k,\;\;10c=k}$

$\implies\mathsf{a=\dfrac{k}{6},\;\;b=\dfrac{k}{9},\;\;c=\dfrac{k}{10}}$

$\mathsf{Now,}$

$\mathsf{a:b:c}$

$\mathsf{=\dfrac{k}{6}:\dfrac{k}{9}:\dfrac{k}{10}}$

$\textsf{Divide through out by k}$

$\mathsf{=\dfrac{1}{6}:\dfrac{1}{9}:\dfrac{1}{10}}$

$\textbf{But LCM(6,9,10)=90}$

$\textsf{Multiply through out by 90}$

$\mathsf{=\dfrac{90}{6}:\dfrac{90}{9}:\dfrac{90}{10}}$

$\mathsf{=15:10:9}$

$\implies\boxed{\mathsf{a:b:c=15:10:9}}$

Answer 2

If 6a = 9b = 10c then a : b : c = 15 : 10 : 9

Given :

6a = 9b = 10c

To find :

a : b : c

Solution :

Step 1 of 2 :

Express a and b in terms of c

6a = 9b = 10c

$\displaystyle \sf{ \implies a = \frac{10c}{6} \: \: and \: \: b = \frac{10c}{9} }$

Step 2 of 2 :

Find a : b : c

a : b : c

$\displaystyle \sf{ = \frac{10c}{6} : \frac{10c}{9} : c }$

$\displaystyle \sf{ = \frac{10}{6} : \frac{10}{9} : 1}$

$\displaystyle \sf{ = \frac{10 \times 18}{6} : \frac{10 \times 18}{9} :( 1 \times 18)}$

$\displaystyle \sf{ = 30: 20 :18}$

$\displaystyle \sf{ = 15: 10 :9}$

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