Three number are in the ratio of 3 : 4 : 5 and their L.C.M. is 2400. Their H.C.F. is:

Step 1: Understand the problem The problem is asking for the Highest Common Factor (HCF) of three numbers that are in the ratio 3:4:5 and have a Least Common Multiple (LCM) of 2400.

Step 2: Use the relationship between HCF, LCM, and the numbers The relationship between the HCF, LCM, and the numbers is given by the formula: Product of numbers = HCF * LCM

Step 3: Find the actual numbers The numbers are in the ratio 3:4:5, so we can represent the numbers as 3x, 4x, and 5x. The LCM of these numbers is given as 2400.

Step 4: Use the LCM to find the value of x The LCM of numbers in the ratio 3:4:5 is calculated by multiplying the ratio numbers together, so 345 = 60. Since the LCM is given as 2400, we can set up the equation 60x = 2400 to find the value of x. Solving for x gives x = 2400 / 60 = 40.

Step 5: Substitute x into the numbers Substituting x = 40 into the numbers 3x, 4x, and 5x gives the actual numbers as 120, 160, and 200.

Step 6: Find the HCF The HCF of the numbers 120, 160, and 200 is 40. This is because 40 is the largest number that can divide all three numbers without leaving a remainder.

So, the HCF of the three numbers is 40.