Three numbers are in the ratio $ 1 : 2 : 5 $, and their LCM is $ 1600 $. The HCF of the numbers is:

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Examoogle · Accepted Answer

The correct answer is Option D: 160. Let the numbers be $x, 2x, 5x$. Their LCM is $10x$. Given $10x = 1600$, so $x = 160$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Let the numbers be $x, 2x, 5x$. Their LCM is $10x$. Given $10x = 1600$, so $x = 160$.

Formula Used: $$LCM = HCF 	imes 	ext{product of ratio terms} 	imes 	ext{coprime factors}$$

Given: Three numbers are in the ratio 1 : 2 : 5 and their LCM is 1600.

Steps: Define the numbers: Since the numbers are in the ratio $1 : 2 : 5$, let them be $1x, 2x,$ and $5x$, where $x$ is the HCF of the three numbers.. Calculate LCM of the variables: The LCM of $x, 2x,$ and $5x$ is the smallest number divisible by all three. Calculating the LCM of coefficients $(1, 2, 5)$ gives $10$. Thus, $LCM(x, 2x, 5x) = 10x$.. Solve for x (HCF): We are given that the LCM is 1600. Equating the calculated LCM to the given value: $10x = 1600$.

Answer: D is correct because the calculated HCF, represented by the variable x, is 160.

Exam Tip: This concept is fundamental for HCF/LCM problems in arithmetic; always remember that $Number = Ratio 	imes HCF$.

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