`a:b=3:4 and b:c = 2:5 ` Find `a:b:c` ?

Answer

To find the ratio \( a:b:c \) given that \( a:b = 3:4 \) and \( b:c = 2:5 \), we can follow these steps:

### Step 1: Write down the given ratios
The given ratios are:
- \( a:b = 3:4 \)
- \( b:c = 2:5 \)

### Step 2: Express the ratios in terms of a common variable
From the first ratio \( a:b = 3:4 \), we can express \( a \) and \( b \) as:
- \( a = 3k \)
- \( b = 4k \)
(where \( k \) is a common multiplier)

From the second ratio \( b:c = 2:5 \), we can express \( b \) and \( c \) as:
- \( b = 2m \)
- \( c = 5m \)
(where \( m \) is another common multiplier)

### Step 3: Equate the value of \( b \)
Since \( b \) is common in both expressions, we can set the two expressions for \( b \) equal to each other:
\[ 4k = 2m \]

### Step 4: Solve for one variable in terms of the other
From \( 4k = 2m \), we can simplify this to:
\[ 2k = m \quad \text{or} \quad m = 2k \]

### Step 5: Substitute back to find \( c \)
Now substitute \( m = 2k \) into the expression for \( c \):
\[ c = 5m = 5(2k) = 10k \]

### Step 6: Write the ratios in terms of \( k \)
Now we have:
- \( a = 3k \)
- \( b = 4k \)
- \( c = 10k \)

### Step 7: Combine the ratios
Thus, the combined ratio \( a:b:c \) can be expressed as:
\[ a:b:c = 3k : 4k : 10k \]
We can simplify this by dividing each term by \( k \):
\[ a:b:c = 3 : 4 : 10 \]

### Final Answer
The ratio \( a:b:c \) is \( 3:4:10 \).

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