Question

Two numbers are in the ratio 3 : 5 and their LCM is 180. Find the HCF of these two numbers.

Hint:

Alternatively, use the formula:
\[ \text{Product of numbers} = \text{HCF} \times \text{LCM} \]
\[ (3x) \times (5x) = x \times 180 \implies 15x^2 = 180x \implies 15x = 180 \implies x = 12 \]

Answer 1

Step 1: Represent the numbers using their ratio.
Since the numbers are in the ratio $3:5$ and $3,5$ share no common factor, we can write the numbers as $3x$ and $5x$, where $x$ itself is their HCF.
Step 2: Use the product identity instead of the direct LCM formula.
We know Product of two numbers $=$ HCF $\times$ LCM, so:
\[ (3x)(5x) = x \times 180 \]
Step 3: Solve the resulting equation.
\[ 15x^2 = 180x \implies 15x = 180 \implies x = 12 \]
Since $x$ is the HCF:
\[ \boxed{\text{HCF} = 12} \]