If a : b = 3 : 5 and b : c = 4 : 7 , find a : b : c

Answer

To solve the problem of finding the ratio \( a : b : c \) given \( a : b = 3 : 5 \) and \( b : c = 4 : 7 \), we can follow these steps:

### Step 1: Write the given ratios
We start with the ratios provided in the question:
- \( a : b = 3 : 5 \)
- \( b : c = 4 : 7 \)

### Step 2: Express the ratios in fractional form
From the first ratio:
\[ \frac{a}{b} = \frac{3}{5} \]
From the second ratio:
\[ \frac{b}{c} = \frac{4}{7} \]

### Step 3: Make the value of \( b \) equal in both ratios
To combine these ratios, we need to express \( b \) in a way that it is the same in both ratios.

From the first ratio, we can express \( b \) in terms of \( a \):
\[ b = \frac{5}{3}a \]

From the second ratio, we can express \( c \) in terms of \( b \):
\[ c = \frac{7}{4}b \]

### Step 4: Substitute \( b \) into the expression for \( c \)
Now, substitute \( b \) from the first ratio into the equation for \( c \):
\[ c = \frac{7}{4} \left(\frac{5}{3}a\right) = \frac{35}{12}a \]

### Step 5: Write the ratios in terms of \( a \)
Now we can express \( a \), \( b \), and \( c \) in terms of \( a \):
- \( a = a \)
- \( b = \frac{5}{3}a \)
- \( c = \frac{35}{12}a \)

### Step 6: Write the combined ratio \( a : b : c \)
Now we can write the ratio \( a : b : c \) as:
\[ a : b : c = a : \frac{5}{3}a : \frac{35}{12}a \]

### Step 7: Eliminate \( a \) and simplify
We can eliminate \( a \) from the ratio:
\[ 1 : \frac{5}{3} : \frac{35}{12} \]

To simplify this, we can convert all parts to have a common denominator. The least common multiple of 3 and 12 is 12. Thus, we can write:
\[ 1 = \frac{12}{12}, \quad \frac{5}{3} = \frac{20}{12}, \quad \frac{35}{12} = \frac{35}{12} \]

So the combined ratio becomes:
\[ \frac{12}{12} : \frac{20}{12} : \frac{35}{12} = 12 : 20 : 35 \]

### Final Answer
Thus, the ratio \( a : b : c \) is:
\[ \boxed{12 : 20 : 35} \]