Given information:
- Quadrilateral ABCD is inscribed in a circle.
- AB:CD is 2:1.
- BC:AD is 5:4.
- Diagonal AC and BD meet at point E.

Objective:
To find the ratio of AE:CE.

Approach:
1. Let's assume the length of AB as 2x and the length of CD as x.
2. As ABCD is an inscribed quadrilateral, opposite angles are supplementary. So, ∠ABC + ∠ADC = 180° and ∠ABD + ∠ACD = 180°.
3. Let's assume the length of BC as 5y and the length of AD as 4y.
4. Using the length ratios, we can find the length of AC and BD.
- AB:CD = AC:BD
- 2x:x = AC:BD
- AC = 2x and BD = x
5. Let's assume the length of AE as a and the length of CE as b.
6. Using the length ratios, we can find the length of AE and CE.
- BC:AD = BE:ED
- 5y:4y = BE:ED
- BE = 5y and ED = 4y
7. Applying the intersecting chords theorem, we have:
- AE * CE = BE * ED
- a * b = 5y * 4y = 20y^2
8. As AE:CE is the same as a:b, we can conclude that AE:CE = a:b = 20y^2.
9. To find the ratio of AE:CE, we need to express it in terms of x.
10. Using similarity of triangles ABE and CDE, we have:
- AB/CD = AE/CE
- 2x/x = a/b
- a/b = 2
11. From step 8, we know that a/b = 20y^2.
12. Equating the values of a/b from steps 10 and 11, we have:
- 2 = 20y^2
- y^2 = 1/10
- y = 1/√10
13. Substituting the value of y in the ratio AE:CE = 20y^2, we get:
- AE:CE = 20 * (1/√10)^2 = 20/10 = 2:1.

Conclusion:
The ratio of AE:CE is 2:1.