Two numbers are in the ratio $4 : 5$ and their L.C.M. is $180$. The smaller number is

### Concept & L.C.M. of Ratios If two numbers are in the ratio $a:b$ (where $a$ and $b$ are co-prime), let the numbers be $ax$ and $bx$. Their Least Common Multiple (L.C.M.) is given by the formula $a \times b \times x$. ### Step-by-Step Solution Given: * Ratio of two numbers = $4 : 5$ * L.C.M. = $180$ 1. Let the two numbers be $4x$ and $5x$. Since $4$ and $5$ share no common factors other than $1$ (they are co-prime), their L.C.M. is simply their product multiplied by $x$. 2. Set up the L.C.M. equation: $\text{L.C.M.} = 4 \times 5 \times x = 20x$ 3. Equate to the given L.C.M.: $20x = 180$ 4. Solve for $x$: $x = \frac{180}{20} = 9$ 5. Calculate the smaller number ($4x$): Smaller number = $4 \times 9 = 36$ ### Exam Strategy & Shortcut Divide the given L.C.M. by the product of the ratio digits: $180 \div (4 \times 5) = 180 \div 20 = 9$. The smaller number is $4 \times 9 = 36$. This can be done entirely mentally in seconds. ### Common Pitfall Confusing L.C.M. with the product. If you treat L.C.M. as the product, you would erroneously do $20x^2 = 180$, leading to incorrect fractional square roots. ### Final Answer Therefore, the correct answer is **36**.