Question

The sub-triplicate ratio of $3:81$ is ?
A) $81:3$
B) $\sqrt 3 :3$
C) $1:3$
D) $\sqrt 3 :9$

Answer 1

Hint: The given ratio has two numbers multiples of three. So we can cancel three from both. The ratio remains the same. Then we can find the sub-triplicate ratio by finding the cube root of both the numbers.

Formula used:
For the ratio, $a:b$, the sub triplicate ratio is given by $\sqrt[3]{a}:\sqrt[3]{b}$.

Complete step-by-step answer:
The given ratio is $3:81$.
Dividing with the common factor three, we have,
$3:81 = 1:27$ since $3 = 1 \times 3,81 = 27 \times 3$
So now we have the ratio $1:27$.
For the ratio, $a:b$, the sub triplicate ratio is $\sqrt[3]{a}:\sqrt[3]{b}$.
That is, the sub-triplicate ratio is the ratio between the cube roots of the two numbers.
So the sub triplicate ratio of $1:27$ is $\sqrt[3]{1}:\sqrt[3]{{27}}$.
We know that the cube root of one is one itself and the cube root of $27$ is three.
This gives the sub triplicate ratio is $1:3$.
Therefore the answer is option C.

Additional information:
Like the sub-triplicate ratio we have a triplicate ratio as well.
For the ratio, $a:b$, the triplicate ratio is ${a^3}:{b^3}$.
Also we have duplicate and sub-duplicate ratios.
For the ratio, $a:b$, the duplicate ratio is ${a^2}:{b^2}$.
And the sub duplicate ratio is $\sqrt a :\sqrt b $.

Note: When we apply the definition directly on the given numbers, we get $\sqrt[3]{3}:\sqrt[3]{{81}}$. But $3$ and $81$ are not perfect cubes. So it is difficult to find the answer. So we convert the ratio to its simplest form by cancelling the common term and then find its sub-triplicate ratio. Thus we get the answer easily.