Let's break down the problem step by step.

Step 1: Understanding the Ratio for Anna and Bill

Anna and Bill share some money in the ratio $2:5$.

• Let the total amount of money Anna and Bill share be $x$.
• Anna's share, $A$, is $\frac{2}{2+5} \times x = \frac{2}{7} \times x$.
• Bill's share, $B$, is $\frac{5}{7} \times x$.

Step 2: Understanding the Ratio for Carl and Donna

Carl and Donna share twice as much money as Anna and Bill in the ratio $3:1$.

• The total amount Carl and Donna share is $2x$ (since it's twice as much as Anna and Bill's total).
• Carl's share, $C$, is $\frac{3}{3+1} \times 2x = \frac{3}{4} \times 2x = \frac{3}{2} \times x$.
• Donna's share, $D$, is $\frac{1}{4} \times 2x = \frac{1}{2} \times x$.

Step 3: Finding the Ratio $A:B:C:D$

Now, we need to find the ratio $A : B : C : D$:

$A = \frac{2}{7}x, \quad B = \frac{5}{7}x, \quad C = \frac{3}{2}x, \quad D = \frac{1}{2}x$

The ratio $A : B : C : D$ is:

$\frac{2}{7} : \frac{5}{7} : \frac{3}{2} : \frac{1}{2}$

Step 4: Simplifying the Ratio

To simplify, multiply all parts by the least common denominator (LCD), which in this case is 14 (to clear the fractions):

$A : B : C : D = \frac{2}{7} \times 14 : \frac{5}{7} \times 14 : \frac{3}{2} \times 14 : \frac{1}{2} \times 14$

This simplifies to:

$4 : 10 : 21 : 7$

So, the simplified ratio is $A : B : C : D = 4 : 10 : 21 : 7$.

Would you like any further details or have any questions?

Here are 5 related questions to further your understanding:

1. How do you determine the least common denominator when simplifying ratios involving fractions?
2. What would the ratio be if Carl and Donna shared three times as much as Anna and Bill?
3. How would the solution change if the ratio between Carl and Donna were 2:3 instead of 3:1?
4. If Anna and Bill shared their money in a 3:2 ratio instead, what would the final ratio $A : B : C : D$ be?
5. How can you verify that the ratios calculated are correct?

Tip: Always double-check the simplification of ratios by ensuring that each term is in its simplest form before concluding.