1. **Problem statement:** Three numbers are in the ratio 3:4:5 and their LCM is 1200. We need to find their HCF. 2. **Understanding the problem:** If three numbers are in ratio 3:4:5, we can represent them as $3x$, $4x$, and $5x$ where $x$ is the common factor. 3. **Formula for LCM of numbers in ratio:** The LCM of $3x$, $4x$, and $5x$ is given by $$\text{LCM} = x \times \text{LCM}(3,4,5)$$ 4. **Calculate LCM of 3, 4, and 5:** - Prime factors: - 3 is prime - 4 = $2^2$ - 5 is prime - LCM is product of highest powers of primes: $$\text{LCM}(3,4,5) = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60$$ 5. **Use given LCM to find $x$:** $$1200 = x \times 60$$ $$x = \frac{1200}{60} = 20$$ 6. **Find the three numbers:** $$3x = 3 \times 20 = 60$$ $$4x = 4 \times 20 = 80$$ $$5x = 5 \times 20 = 100$$ 7. **Find HCF of 60, 80, and 100:** - Prime factors: - 60 = $2^2 \times 3 \times 5$ - 80 = $2^4 \times 5$ - 100 = $2^2 \times 5^2$ - Common prime factors with lowest powers: - $2^2$ (since minimum power of 2 is 2) - $5^1$ (minimum power of 5 is 1) - So, $$\text{HCF} = 2^2 \times 5 = 4 \times 5 = 20$$ **Final answer:** The HCF of the three numbers is $20$.