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

Answer

To find the value of A : B : C : D given the ratios A : B = 1 : 3, B : C = 2 : 5, and C : D = 2 : 3, we can follow these steps:

### Step 1: Express the ratios in terms of a common variable
1. From the ratio A : B = 1 : 3, we can express B in terms of A:
- Let A = 1x, then B = 3x.

2. From the ratio B : C = 2 : 5, we can express C in terms of B:
- Let B = 2y, then C = 5y.

3. From the ratio C : D = 2 : 3, we can express D in terms of C:
- Let C = 2z, then D = 3z.

### Step 2: Find a common variable for B
Now we need to find a common value for B from the two expressions we have:
- From A : B, we have B = 3x.
- From B : C, we have B = 2y.

Setting these equal gives:
\[ 3x = 2y \]
From this, we can express y in terms of x:
\[ y = \frac{3}{2}x \]

### Step 3: Find a common variable for C
Next, we can express C in terms of x:
- From B : C, we have C = 5y = 5(\frac{3}{2}x) = \frac{15}{2}x.
- From C : D, we have C = 2z.

Setting these equal gives:
\[ \frac{15}{2}x = 2z \]
From this, we can express z in terms of x:
\[ z = \frac{15}{4}x \]

### Step 4: Find D in terms of A
Now we can express D in terms of x:
- From C : D, we have D = 3z = 3(\frac{15}{4}x) = \frac{45}{4}x.

### Step 5: Write the final ratios
Now we have:
- A = 1x
- B = 3x
- C = \(\frac{15}{2}x\)
- D = \(\frac{45}{4}x\)

To express A : B : C : D in the simplest form, we can find a common denominator. The least common multiple of the denominators (1, 1, 2, 4) is 4.

Multiplying each term by 4 gives:
- A = \(4 \cdot 1x = 4x\)
- B = \(4 \cdot 3x = 12x\)
- C = \(4 \cdot \frac{15}{2}x = 30x\)
- D = \(4 \cdot \frac{45}{4}x = 45x\)

Thus, the ratio A : B : C : D is:
\[ A : B : C : D = 4 : 12 : 30 : 45 \]

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