If A:B=2:3 , B:C = 4:5, C:D = 6:7. then A : B : C : D is-

Options

A. `18 : 24 : 30 : 35`
B. `16 : 24: 30 : 35`
C. `16 : 22 : 30 : 35`
D. None of these

Answer

To solve the problem of finding the ratio A : B : C : D given the ratios A : B = 2 : 3, B : C = 4 : 5, and C : D = 6 : 7, we can follow these steps:

### Step 1: Express the Ratios in Terms of a Common Variable
We start by expressing each variable in terms of a common variable.

Let:
- A = 2x (from A : B = 2 : 3)
- B = 3x (from A : B = 2 : 3)

Next, we use the ratio B : C = 4 : 5:
- B = 4y
- C = 5y

Since we already have B = 3x, we can set 3x = 4y.

### Step 2: Find the Relationship Between x and y
From the equation 3x = 4y, we can express y in terms of x:
- y = (3/4)x

### Step 3: Substitute y Back to Find C
Now, we can substitute y back into the expression for C:
- C = 5y = 5 * (3/4)x = (15/4)x

### Step 4: Express C in Terms of a New Variable
Next, we need to work with the ratio C : D = 6 : 7:
- C = 6z
- D = 7z

Since we have C = (15/4)x, we can set (15/4)x = 6z.

### Step 5: Find the Relationship Between x and z
From the equation (15/4)x = 6z, we can express z in terms of x:
- z = (15/24)x = (5/8)x

### Step 6: Substitute z Back to Find D
Now, we can substitute z back into the expression for D:
- D = 7z = 7 * (5/8)x = (35/8)x

### Step 7: Write All Ratios in Terms of x
Now we have:
- A = 2x
- B = 3x
- C = (15/4)x
- D = (35/8)x

### Step 8: Find a Common Denominator
To express A : B : C : D in the simplest form, we need a common denominator. The denominators are 1, 1, 4, and 8. The least common multiple (LCM) of these is 8.

Now we convert each term:
- A = 2x = (16/8)x
- B = 3x = (24/8)x
- C = (15/4)x = (30/8)x
- D = (35/8)x = (35/8)x

### Step 9: Write the Final Ratio
Now we can write the ratio:
A : B : C : D = 16 : 24 : 30 : 35

### Step 10: Simplify the Ratio
To simplify, we can divide each term by the greatest common divisor (GCD). The GCD of 16, 24, 30, and 35 is 1, so the ratio remains:
A : B : C : D = 16 : 24 : 30 : 35

### Final Answer
Thus, the final answer is:
**A : B : C : D = 16 : 24 : 30 : 35**