Three numbers are in the ratio of 3:4:5 and their LCM is 2400. What is their HCF?

Steps to Solve

1. Define the Numbers Based on the Ratio

Let the three numbers be $3x$, $4x$, and $5x$ where $x$ is a common factor.

2. Use the Relationship Between LCM and HCF

The relationship states that:

$$ \text{LCM}(a, b, c) \cdot \text{HCF}(a, b, c) = a \cdot b \cdot c $$

Here, we know:

• LCM = 2400
• The three numbers are $3x$, $4x$, and $5x$.

3. Calculate the Product of the Numbers

The product of the three numbers can be calculated as follows:

$$ (3x) \cdot (4x) \cdot (5x) = 60x^3 $$

4. Set up the Equation

Using the relationship, set up the equation:

$$ 2400 \cdot \text{HCF} = 60x^3 $$

5. Solve for HCF

Let HCF be denoted as $h$:

$$ h = \frac{60x^3}{2400} $$ $$ h = \frac{x^3}{40} $$

6. Find x Using the LCM

Next, we need to find $x$. To find $x$, we start by determining the LCM of the three numbers:

The LCM of $3x$, $4x$, and $5x$ is given by:

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

Now, we can set the LCM equal to 2400:

$$ 60x = 2400 $$

7. Calculate x

Solve for $x$:

$$ x = \frac{2400}{60} = 40 $$

8. Substitute x Back to Find HCF

Using the value of $x$, find HCF:

$$ h = \frac{x^3}{40} = \frac{40^3}{40} = 40^2 = 1600 $$

Now we have the highest common factor.

9. Final Result

Thus, the highest common factor is $h = 1600$.