If a:b =5:14 andb: c = 7 : 3, then find a:b:c

Answer

To solve the problem of finding the ratio \( a:b:c \) given the ratios \( a:b = 5:14 \) and \( b:c = 7:3 \), we can follow these steps:

### Step 1: Write down the given ratios
We are given:
- \( a:b = 5:14 \)
- \( b:c = 7:3 \)

### Step 2: Express the ratios in terms of a common variable
From the first ratio \( a:b = 5:14 \), we can express \( a \) and \( b \) in terms of a variable \( k \):
- Let \( a = 5k \)
- Let \( b = 14k \)

From the second ratio \( b:c = 7:3 \), we can express \( b \) and \( c \) in terms of another variable \( m \):
- Let \( b = 7m \)
- Let \( c = 3m \)

### Step 3: Equate the expressions for \( b \)
Since both expressions represent \( b \), we can set them equal to each other:
\[ 14k = 7m \]

### Step 4: Solve for one variable in terms of the other
To find a relationship between \( k \) and \( m \), we can rearrange the equation:
\[ m = 2k \]

### Step 5: Substitute back to find \( c \)
Now that we have \( m \) in terms of \( k \), we can find \( c \):
\[ c = 3m = 3(2k) = 6k \]

### Step 6: Write the ratios \( a:b:c \)
Now we can express \( a \), \( b \), and \( c \) in terms of \( k \):
- \( a = 5k \)
- \( b = 14k \)
- \( c = 6k \)

Thus, the ratio \( a:b:c \) can be written as:
\[ a:b:c = 5k : 14k : 6k \]

### Step 7: Simplify the ratio
Since \( k \) is a common factor, we can simplify the ratio:
\[ a:b:c = 5 : 14 : 6 \]

### Final Answer
The required ratio \( a:b:c \) is:
\[ \boxed{5:14:6} \]