\text{If}\mathrm{ a}:b=2:3,\text{then}(7a^3-2b^3):(b^3-a^3)=?

The correct option is 2 : 19.

Step-by-Step Explanation:

Step 1: Express the given ratio in terms of a constant variable

We are given the ratio of $a$ to $b$ as:

$a:b=2:3$

This means we can write:

$\frac{a}{b}=\frac{2}{3}$

Let $a=2k$ and $b=3k$, where $k$ is a non-zero constant.

Step 2: Calculate the values of $a^3$ and $b^3$

Now, substitute the expressions for $a$ and $b$ to find their cubes:

$a^3=(2k{)}^3=8k^3$

$b^3=(3k{)}^3=27k^3$

Step 3: Evaluate the components of the required ratio

We need to find the ratio $(7a^3-2b^3):(b^3-a^3)$.

First term:

$7a^3-2b^3=7\left(8k^3\right)-2\left(27k^3\right)$

$=56k^3-54k^3=2k^3$

Second term:

$b^3-a^3=27k^3-8k^3=19k^3$

Step 4: Determine the simplified ratio

Putting both terms together as a fraction:

$\frac{7a^3-2b^3}{b^3-a^3}=\frac{2k^3}{19k^3}$

Canceling out the common factor $k^3$:

$=\frac{2}{19}$

Therefore, the ratio is 2 : 19.