Two numbers are in the ratio 1:4 and their LCM and HCF are 84 and 21. Find the smaller number.

Ratio of the two numbers = 1:4

LCM = 84

HCF = 21

​$$\text{Product of two numbers} = \text{LCM} \times \text{HCF}$$​​

Let the two numbers be x and 4x, where x is the smaller number.

​$$x \times 4x = \text{LCM} \times \text{HCF} \\4x^2 = 84 \times 21 \\x^2 = \frac{1764}{4} \\x = \sqrt{441} = 21$$​​

Thus, the smaller number is 21​.