Question

If $$a\colon b=3\colon 4$$ and $$b\colon c=5\colon 9$$, then $$a\colon b\colon c$$ is
A. $$15\colon 20\colon 36$$
B. $$20\colon 36\colon 15$$
C. $$3\colon 20\colon 9$$
D. None of these

Answer 1

Hint: In this question it is given that $$a\colon b=3\colon 4$$ and $$b\colon c=5\colon 9$$, we have to find $$a\colon b\colon c$$.
So to find the solution we have to take the corresponding values of b and after that we have to take LCM of the values of b, because b is the common part in both of the ratios. Now by multiplying we have to convert the corresponding values of ‘b’ into LCM values, from where we can directly write $$a\colon b\colon c$$.

Complete step-by-step answer:
given,
$$a\colon b=3\colon 4$$ and $$b\colon c=5\colon 9$$...........(1)
The corresponding values of b are 4 and 5, so we have to find the LCM of 4 and 5.
Since there is no common factor of 4 and 5, therefore the LCM of 4 and 5, i.e, LCM (4,5) = $4\times 5$ = 20
Now we have to convert the above ratios in such a way that the corresponding values of ‘b’ becomes 20.
So, $$a\colon b=3\colon 4$$
$$\Rightarrow \dfrac{a}{b} =\dfrac{3}{4}$$
Now to convert the denominator 20, we have to multiplying numerator and denominator by 5, so by multiplying we get,
$$\dfrac{a}{b} =\dfrac{3\times 5}{4\times 5}$$
$$\Rightarrow \dfrac{a}{b} =\dfrac{15}{20}$$
$$\therefore a\colon b=15\colon 20$$.........(2)

Another ratio,
$$b\colon c=5\colon 9$$
$$\Rightarrow \dfrac{b}{c} =\dfrac{5}{9}$$
Now here also, in order to get 20 in numerator we have to multiply 4 with numerator as well as denominator, so by multiplying we get,
$$\dfrac{b}{c} =\dfrac{5\times 4}{9\times 4}$$
$$\Rightarrow \dfrac{b}{c} =\dfrac{20}{36}$$
$$\therefore b\colon c=20\colon 36$$.........(3)
Now from equation (2) and (3) we can write,
$$a\colon b\colon c=15\colon 20\colon 36$$
Which is our required solution.
Hence the correct option is option A.

Note: While solving you need to know that after multiplying with the suitable term when we get $$a\colon b\colon c$$ then if we get any common factor in the corresponding terms then we have to simplify it by cancelling or dividing.
e.g: if $$a\colon b\colon c=2\colon 6\colon 10$$, where common factor is 2, then the ratio will be $$a\colon b\colon c=\dfrac{2}{2} \colon \dfrac{6}{2} \colon \dfrac{10}{2} =1\colon 3\colon 5$$.