PREVIOUSLY ASKED IN:
PSC Miscellaneous Prelims 2018
Answer
45
Explanation
Given the ratio of the two numbers is 3 : 4, let the common multiplier (which is also the HCF) be 'x'. Thus, the two numbers are 3x and 4x. The Least Common Multiple (LCM) of 3x and 4x is = 3 × 4 × x = 12x. According to the problem, the LCM is 180. So, we set up the equation: 12x = 180. Solving for x, we get x = 180 / 12 = 15. The first number is 3x = 3 × 15 = 45. (The second number would be 4x = 60).
Key Points
- > When the ratio is a:b, the numbers can be assumed as ax and bx (x is the HCF).
- > The LCM of ax and bx is (a × b × x) if a and b are co-prime.
- > Here, ratio is 3:4. So, LCM = 3 × 4 × x = 12x.
- > Given LCM = 180. Therefore, 12x = 180 => x = 15.
- > First number = 3 × 15 = 45.
- > Second number = 4 × 15 = 60.
- > Verification: LCM of 45 and 60 is indeed 180.
Additional Information
Key Formulas for LCM and HCF
| Rule / Formula | Explanation |
|---|---|
| Product of two numbers | Equals the product of their LCM and HCF |
| LCM | (Product of ratio terms) × HCF |
| HCF | LCM ÷ (Product of ratio terms) |
| LCM of Fractions | LCM of Numerators ÷ HCF of Denominators |
Memory Tips
- Quick Mental Math: Multiply the ratio numbers (3 × 4 = 12). Divide the given LCM by this product (180 / 12 = 15). This quotient 15 is your baseline multiplier. Simply multiply the ratio terms by 15 to get the original numbers (15×3=45 and 15×4=60).