A,B and C together start a business. Three times the investment of A equals four times the Investment of B and the Capital of B is twice that of C. The ratio of share of each in the profit.

Let the capitals (investments) of A, B and C be denoted by $$A$$, $$B$$ and $$C$$ respectively. Since all three partners start the business together, the period of investment for each is the same; therefore their shares in the profit will be in the same ratio as their capitals.

Step 1: Translate the given conditions into equations.
Three times A’s investment equals four times B’s investment: $$3A = 4B \,\,\,\Rightarrow\,\,\, \frac{A}{B} = \frac{4}{3}$$ $$-(1)$$
B’s capital is twice C’s capital: $$B = 2C \,\,\,\Rightarrow\,\,\, \frac{B}{C} = \frac{2}{1}$$ $$-(2)$$

Step 2: Express every capital in terms of a single variable (choose $$B$$).
From $$(1)$$: $$A = \frac{4}{3}B$$
From $$(2)$$: $$C = \frac{B}{2}$$

Step 3: Write the ratio $$A : B : C$$ using these expressions.
$$A : B : C = \frac{4}{3}B : B : \frac{B}{2}$$

Step 4: Remove the common factor $$B$$.
$$A : B : C = \frac{4}{3} : 1 : \frac{1}{2}$$

Step 5: Clear the fractions to obtain whole numbers. The LCM of the denominators 3 and 2 is 6. Multiply each term by 6:
$$\left(\frac{4}{3}\times6\right) : \left(1\times6\right) : \left(\frac{1}{2}\times6\right) = 8 : 6 : 3$$

Therefore, the partners will share the profit in the ratio $$8 : 6 : 3$$.

Option D which is: 8 : 6 : 3