If two numbers are in the ratio 3:4:5 and their LCM is 1200, what is their HCF?

Steps to Solve

1. Define the Numbers Based on the Ratio In the ratio 3:4:5, let the two numbers be represented as $3x$, $4x$, and $5x$.

2. Calculate the LCM of the Numbers The LCM of the numbers $3x, 4x, 5x$ can be calculated using the relationship of LCM and the prime factorizations:

• Prime factorization of 3 is $3^1$.
• Prime factorization of 4 is $2^2$.
• Prime factorization of 5 is $5^1$.

Thus, $$ \text{LCM}(3x, 4x, 5x) = \text{LCM}(3, 4, 5) \cdot x = 60x $$

3. Set LCM to Given Value According to the problem, the LCM is given as 1200: $$ 60x = 1200 $$

4. Solve for x To find $x$, divide both sides of the equation by 60: $$ x = \frac{1200}{60} = 20 $$

5. Find the Actual Numbers Now substitute $x$ back into the expressions for the numbers:

• First number: $3x = 3 \times 20 = 60$
• Second number: $4x = 4 \times 20 = 80$
• Third number: $5x = 5 \times 20 = 100$

6. Determine the HCF of the Numbers The HCF of the numbers 60, 80, and 100 can be found by:

• Prime factorization of 60: $2^2 \cdot 3^1 \cdot 5^1$
• Prime factorization of 80: $2^4 \cdot 5^1$
• Prime factorization of 100: $2^2 \cdot 5^2$

The common factors are $2^2$ and $5^1$, thus: $$ \text{HCF} = 2^2 \cdot 5^1 = 4 \cdot 5 = 20 $$