Given that \( a: b=2: 3 \) and that \( b: c=1: 7 \), find the ratio \( a: b: c \) Give your ratio in its simplest form with integer parts.

To find the ratio \( a: b: c \) given the ratios \( a: b = 2: 3 \) and \( b: c = 1: 7 \), we can start by expressing \( a \), \( b \), and \( c \) in terms of a common variable. From the ratio \( a: b = 2: 3 \), we can express \( a \) and \( b \) as: \[ a = 2k \quad \text{and} \quad b = 3k \] for some positive integer \( k \). Next, from the ratio \( b: c = 1: 7 \), we can express \( b \) and \( c \) as: \[ b = 1m \quad \text{and} \quad c = 7m \] for some positive integer \( m \). Since both expressions represent \( b \), we can set them equal to each other: \[ 3k = 1m \] This implies: \[ m = 3k \] Now we can substitute \( m \) back into the expression for \( c \): \[ c = 7m = 7(3k) = 21k \] Now we have all three variables expressed in terms of \( k \): \[ a = 2k, \quad b = 3k, \quad c = 21k \] Now we can write the ratio \( a: b: c \): \[ a: b: c = 2k: 3k: 21k \] Since \( k \) is a common factor, we can simplify this to: \[ a: b: c = 2: 3: 21 \] Thus, the ratio \( a: b: c \) in its simplest form is: \[ \boxed{2: 3: 21} \]