Question

If \[yz:zx:xy=1:2:3\], then \[\dfrac{x}{yz}:\dfrac{y}{zx}\]is equal to
(a) 3 : 2
(b) 1 : 4
(c) 2 : 1
(d) 4 : 1

Answer 1

Hint: In this question, we can get the ratio of x and y, y and z from the given ratios. Then, from the corresponding ratios obtained we can get the ratio of x, y, z . Now, we need to substitute those ratio terms obtained to get the result.

Complete step-by-step answer:
RATIO:
Ratio is the relation between one quantity and another quantity, given that both quantity must be of the same kind and same unit, denoted by \[x:y\], read as x is to y
If the ratio between first and second quantity is \[a:b\] and the ratio between second and third quantity is \[b:c\], then ratio among first, second and third quantity is given by
\[ac:bc:ab\]
Now, from the given ratio in the question we have
\[yz:zx:xy=1:2:3\]
Now, using the above formula let us first consider the first two quantities
\[\Rightarrow \dfrac{yz}{zx}=\dfrac{1}{2}\]
Now, on simplifying this further we get,
\[\Rightarrow y:x=1:2\]
Let us now consider that next two quantities then we get,
\[\Rightarrow \dfrac{zx}{xy}=\dfrac{2}{3}\]
Now, on cancelling the common term sand simplifying further we get,
\[\Rightarrow z:y=2:3\]
Now, on writing x and z in terms of y we get,
\[\Rightarrow x=2y,z=\dfrac{2y}{3}\]
Now, let us take the ratio between x, y and z
\[\Rightarrow x:y:z=2y:y:\dfrac{2y}{3}\]
Now, on further simplification we get,
\[\Rightarrow x:y:z=6:3:2\]
Now, let us find the value of
\[\Rightarrow \dfrac{x}{yz}:\dfrac{y}{zx}\]
Now, on further substituting the values we get,
\[\Rightarrow \dfrac{6}{3\times 2}:\dfrac{3}{6\times 2}\]
Now, on cancelling out the common terms we get,
\[\Rightarrow 1:\dfrac{1}{4}\]
Now, this can be further written as
\[\Rightarrow 4:1\]
Hence, the correct option is (d).

Note:
Instead of finding the ratio between x, y and z we can also get the result just by substituting the value of x and z in terms of y and simplify further accordingly. Both the methods will give the same result but the number of steps gets reduced.
It is important to note that in the given ratio terms we have common terms which gives us the individual ratios and helps in getting the result.
 It is also to be noted that when there is a fraction term in the ratio then that denominator gets carried to the numerator of the remaining terms in the ratio which does not affect the final ratio.