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**Hint**: Let us assume the common factor in the ratio be “x” and frame the mathematical expression for the total number parts and first find the value of “x” and then find the value of the given ratio.

**:**

__Complete step-by-step answer__Let us assume the common factor in the given ratio be “x”

Now, $ 2:3 $ is expressed as $ \dfrac{2}{3} = \dfrac{{2x}}{{3x}} $ …. (A)

Now given that we have to divide the number $ 1000 $ in a ratio of $ 2:3 $

Therefore, $ 2x + 3x = 1000 $

Simplify the above expression finding the sum of the terms on the left hand side of the equation –

$ 5x = 1000 $

Term multiplicative on one side if moved to the opposite side then it goes to the denominator.

$ x = \dfrac{{1000}}{5} $

Find the factors for the term in the numerator of the above equation –

$ x = \dfrac{{200 \times 5}}{5} $

Common factors from the numerator and the denominator cancel each other and therefore remove from the numerator and the denominator of the above expression.

$ x = 200 $

Place the above value in the equation A

$ 2:3 $ is expressed as $ \dfrac{2}{3} = \dfrac{{2x}}{{3x}} = \dfrac{{2(200)}}{{3(200)}} = \dfrac{{400}}{{600}} $

Hence, the ratio is expressed as $ 400:600 $

**So, the correct answer is “ $ 400:600 $ ”.**

**Note**: When there is ratio given always suppose any common variable and find its value. Always remember the sum of ratio along with the variable always gives the number of parts of the whole. Ratio can be well defined as the comparison between two numbers that are without any units. Whereas, when two ratios are set equal to each other are known as the proportion. Four numbers a, b, c, and d are called to be in the proportion. If $ a:b = c:d $ whereas, four numbers are called to be in continued proportion if the terms are expressed as $ a:b = b:c = c:d $

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