The ratio of `A:B=1:3,B:C=2:5,C:D=2:3`. Find the value of `A:B:C:D`

Answer

To solve the problem, we need to find the combined ratio of A, B, C, and D based on the given ratios.

### Step 1: Write down the given ratios
We have the following ratios:
1. A : B = 1 : 3
2. B : C = 2 : 5
3. C : D = 2 : 3

### Step 2: Express B in terms of A
From the first ratio, we can express B in terms of A:
- Let A = 1x (where x is a common multiplier)
- Then, B = 3x

### Step 3: Express C in terms of B
From the second ratio, we can express C in terms of B:
- We have B = 2y (where y is another common multiplier)
- Therefore, to make B consistent, we can set 3x = 2y.
- Solving for y gives us y = (3/2)x.
- Now, substituting y back, we find C:
- C = 5y = 5 * (3/2)x = (15/2)x

### Step 4: Express D in terms of C
From the third ratio, we can express D in terms of C:
- We have C = 2z (where z is another common multiplier)
- Therefore, to make C consistent, we can set (15/2)x = 2z.
- Solving for z gives us z = (15/4)x.
- Now, substituting z back, we find D:
- D = 3z = 3 * (15/4)x = (45/4)x

### Step 5: Combine all ratios
Now we have:
- A = 1x
- B = 3x
- C = (15/2)x
- D = (45/4)x

To express A : B : C : D in a simple ratio, we need a common denominator. The common denominator for 1, 3, (15/2), and (45/4) is 4.

### Step 6: Convert to a common denominator
- A = 1x = (4/4)x
- B = 3x = (12/4)x
- C = (15/2)x = (30/4)x
- D = (45/4)x = (45/4)x

Now we can express the ratios:
A : B : C : D = (4/4)x : (12/4)x : (30/4)x : (45/4)x = 4 : 12 : 30 : 45

### Step 7: Simplify the ratio
To simplify the ratio, we can divide each term by the greatest common divisor (GCD), which is 3:
- 4 ÷ 3 = 4/3
- 12 ÷ 3 = 4
- 30 ÷ 3 = 10
- 45 ÷ 3 = 15

Thus, the simplified ratio is:
A : B : C : D = 4 : 12 : 30 : 45

### Final Answer
The final value of A : B : C : D is 4 : 12 : 30 : 45.

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