Step 1: Understanding the Question:
The question asks us to find the Highest Common Factor (HCF) of three numbers whose ratios are given as \(3 : 4 : 5\) and whose Least Common Multiple (LCM) is known to be 2400.
This involves understanding the relationship between numbers, their ratios, their HCF, and their LCM.
Step 2: Key Formula or Approach:
Let the three numbers be \(3x\), \(4x\), and \(5x\), where \(x\) is the Highest Common Factor (HCF) of the numbers.
The LCM of numbers expressed as \(ax, bx, cx\) is \(x \times \text{LCM}(a, b, c)\), provided \(a, b, c\) are relative integers or we calculate the combined LCM.
Formula: \[\text{LCM}(3x, 4x, 5x) = x \cdot \text{LCM}(3, 4, 5)\]
Step 3: Detailed Explanation:
Identification of Numbers: We represent the three numbers based on the given ratio. Let the numbers be \(n_1 = 3x\), \(n_2 = 4x\), and \(n_3 = 5x\). Here, \(x\) represents the common multiplier, which in this case is the HCF because the remaining factors (3, 4, and 5) are co-prime (they share no common factors other than 1).
Calculating LCM of the Ratio Parts: First, we need to find the LCM of the numerical parts of the ratio, which are 3, 4, and 5. Since 3, 4, and 5 have no common factors (they are pairwise co-prime except 4 which is \(2^2\)), their LCM is simply their product: \[3 \times 4 \times 5 = 60\]
Relating to the Given LCM: According to the property of LCM, the LCM of \(3x, 4x,\) and \(5x\) will be \(60x\). The problem states that the total LCM is 2400.
Solving for x: We set up the equation: \[60x = 2400\]
Divide both sides by 60: \[x = \frac{2400}{60} = 40\]
Verification: If the HCF is 40, the numbers are \(3 \times 40 = 120\), \(4 \times 40 = 160\), and \(5 \times 40 = 200\).
The LCM of 120, 160, and 200:
\(120 = 2^3 \times 3 \times 5\)
\(160 = 2^5 \times 5\)
\(200 = 2^3 \times 5^2\)
\(\text{LCM} = 2^5 \times 3 \times 5^2 = 32 \times 3 \times 25 = 2400\). This matches the given value.
Step 4: Final Answer:
The Highest Common Factor (HCF) of the three numbers is 40.