`a:b = 1:2, b :c = 3:2 , and c:d = 1:3 ` Find `a:b:c:d` ?

Answer

To find the ratio \( a:b:c:d \) given the ratios \( a:b = 1:2 \), \( b:c = 3:2 \), and \( c:d = 1:3 \), we can follow these steps:

### Step 1: Write down the given ratios
We have:
- \( a:b = 1:2 \)
- \( b:c = 3:2 \)
- \( c:d = 1:3 \)

### Step 2: Express each variable in terms of a common variable
From the first ratio \( a:b = 1:2 \), we can express:
- \( a = 1x \)
- \( b = 2x \)

From the second ratio \( b:c = 3:2 \), we can express \( c \) in terms of \( b \):
- Since \( b = 2x \), we can set \( 2x = 3y \) (where \( y \) is a new variable for the second ratio).
- Therefore, \( y = \frac{2x}{3} \) and \( c = 2y = 2 \times \frac{2x}{3} = \frac{4x}{3} \).

From the third ratio \( c:d = 1:3 \), we can express \( d \) in terms of \( c \):
- Since \( c = \frac{4x}{3} \), we can set \( \frac{4x}{3} = 1z \) (where \( z \) is a new variable for the third ratio).
- Therefore, \( z = \frac{4x}{3} \) and \( d = 3z = 3 \times \frac{4x}{3} = 4x \).

### Step 3: Now we have all variables in terms of \( x \)
We have:
- \( a = 1x \)
- \( b = 2x \)
- \( c = \frac{4x}{3} \)
- \( d = 4x \)

### Step 4: Find a common denominator to combine the ratios
The common denominator for \( a, b, c, d \) can be found by taking the least common multiple (LCM) of the denominators. Here, the denominators are \( 1, 1, 3, 1 \), so the LCM is \( 3 \).

### Step 5: Multiply each term to eliminate fractions
Now we multiply each term by \( 3 \):
- \( a = 1x \times 3 = 3x \)
- \( b = 2x \times 3 = 6x \)
- \( c = \frac{4x}{3} \times 3 = 4x \)
- \( d = 4x \times 3 = 12x \)

### Step 6: Write the final ratio
Thus, the ratio \( a:b:c:d \) becomes:
- \( 3x : 6x : 4x : 12x \)

We can simplify this by dividing each term by \( x \):
- \( 3 : 6 : 4 : 12 \)

### Final Answer
The final ratio is:
\[ a:b:c:d = 3:6:4:12 \]