If `a:b = 5:9` and `b:c = 18:35`, then find a:c.

Answer

To find the ratio \( a:c \) given \( a:b = 5:9 \) and \( b:c = 18:35 \), we can follow these steps:

### Step 1: Express the ratios in terms of a common variable
We have:
\[ a:b = 5:9 \implies a = 5k \quad \text{and} \quad b = 9k \quad \text{(for some variable } k\text{)} \]
\[ b:c = 18:35 \implies b = 18m \quad \text{and} \quad c = 35m \quad \text{(for some variable } m\text{)} \]

### Step 2: Set the two expressions for \( b \) equal to each other
Since both expressions represent \( b \), we can set them equal:
\[ 9k = 18m \]

### Step 3: Solve for one variable in terms of the other
To find a relationship between \( k \) and \( m \), we can simplify the equation:
\[ k = 2m \]

### Step 4: Substitute \( k \) back into the expression for \( a \)
Now we can express \( a \) in terms of \( m \):
\[ a = 5k = 5(2m) = 10m \]

### Step 5: Substitute \( m \) back into the expression for \( c \)
We already have \( c \) in terms of \( m \):
\[ c = 35m \]

### Step 6: Write the ratio \( a:c \)
Now we can express the ratio \( a:c \):
\[ a:c = 10m : 35m \]

### Step 7: Simplify the ratio
We can simplify this ratio by dividing both terms by \( m \):
\[ a:c = 10 : 35 \]
Now, we can simplify this further by dividing both sides by 5:
\[ a:c = 2 : 7 \]

### Final Answer
Thus, the ratio \( a:c \) is:
\[ \boxed{2:7} \]