To find the ratio \( a:b:c \) given the ratios \( a:b = 5:2 \) and \( b:c = 3:4 \), we can start by expressing each variable in terms of \( b \).
From the first ratio \( a:b = 5:2 \), we can express \( a \) as: \[ a = \frac{5}{2}b \]
From the second ratio \( b:c = 3:4 \), we can express \( c \) as: \[ c = \frac{4}{3}b \]
Now, we need to express the ratios \( a:b:c \) in terms of \( b \):
- From \( b \) we know that \( b = b \).
- Replace \( a \) using \( a = \frac{5}{2}b \).
- Replace \( c \) using \( c = \frac{4}{3}b \).
Now, expressing everything in terms of a common base (let's use \( b \)),
- \( a = \frac{5}{2}b \)
- \( b = b \)
- \( c = \frac{4}{3}b \)
In order to combine these into a single ratio, we can find a common denominator for the coefficients of \( a \) and \( c \):
- The denominators are 2 and 3, hence the least common multiple (LCM) is 6.
Now we can convert each component:
- For \( a = \frac{5}{2}b \), convert to a base of 6: \[ a = \frac{5}{2}b = \frac{15}{6}b \]
- For \( b = b \): \[ b = \frac{6}{6}b \]
- For \( c = \frac{4}{3}b \), convert to a base of 6: \[ c = \frac{4}{3}b = \frac{8}{6}b \]
Now we can write the full ratio \( a:b:c \): \[ a:b:c = \frac{15}{6}b : \frac{6}{6}b : \frac{8}{6}b \] We can drop \( b \) (since it's common) and we get: \[ a:b:c = 15:6:8 \]
Thus, the final result is: \[ \boxed{15:6:8} \]