If `a: b= 2: 3, b: c= 4: 5 and c: d= 6: 7`, find : `a: b: c:d`.

Answer

To find 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 ratios in terms of a common variable
From the given ratios, we can express \( a \), \( b \), \( c \), and \( d \) in terms of a common variable.

Let:
- \( a = 2x \)
- \( b = 3x \)

From the second ratio \( b : c = 4 : 5 \):
- Since \( b = 3x \), we can set up the equation:
\[ \frac{b}{c} = \frac{4}{5} \implies \frac{3x}{c} = \frac{4}{5} \]
Cross-multiplying gives:
\[ 5 \cdot 3x = 4c \implies c = \frac{15x}{4} \]

### Step 2: Find \( d \) in terms of \( x \)
From the third ratio \( c : d = 6 : 7 \):
- Since \( c = \frac{15x}{4} \), we can set up the equation:
\[ \frac{c}{d} = \frac{6}{7} \implies \frac{\frac{15x}{4}}{d} = \frac{6}{7} \]
Cross-multiplying gives:
\[ 7 \cdot \frac{15x}{4} = 6d \implies d = \frac{7 \cdot 15x}{24} = \frac{105x}{24} \]

### Step 3: Write all ratios together
Now we have:
- \( a = 2x \)
- \( b = 3x \)
- \( c = \frac{15x}{4} \)
- \( d = \frac{105x}{24} \)

### Step 4: Find a common denominator
To express all ratios in the same format, we can multiply each term by 24 (the least common multiple of the denominators):
- \( a = 2x \cdot 12 = 24x \)
- \( b = 3x \cdot 8 = 24x \)
- \( c = \frac{15x}{4} \cdot 6 = 90x \)
- \( d = \frac{105x}{24} \cdot 1 = 105x \)

### Step 5: Combine the ratios
Now we can express the ratios as:
\[ a : b : c : d = 24x : 36x : 90x : 105x \]
This simplifies to:
\[ a : b : c : d = 24 : 36 : 90 : 105 \]

### Step 6: Simplify the ratios
To simplify \( 24 : 36 : 90 : 105 \):
- Find the GCD (Greatest Common Divisor) of these numbers. The GCD is 3.
- Dividing each term by 3 gives:
\[ a : b : c : d = 8 : 12 : 30 : 35 \]

### Final Answer
Thus, the final ratio is:
\[ \boxed{8 : 12 : 30 : 35} \]