If the sum of the two numbers is 27 and their HCF and LCM is 3 and 60…
2024
If the sum of the two numbers is 27 and their HCF and LCM is 3 and 60 respectively, then what is the sum of the reciprocal of the two numbers?
- A.
1/10
- B.
1/5
- C.
3/10
- D.
3/20
Show answer & explanation
Correct answer: D
For two positive integers, the product of the numbers equals the product of their HCF and LCM. Also, the sum of the reciprocals of two numbers equals the sum of the numbers divided by their product: 1/x + 1/y = (x + y) / xy.
The product of the two numbers equals HCF × LCM = 3 × 60 = 180.
The sum of the two numbers is given as 27.
Forming the quadratic with sum 27 and product 180: t2 − 27t + 180 = 0.
The discriminant is 272 − 4 × 180 = 729 − 720 = 9, so t = (27 ± 3)/2, giving t = 15 or t = 12.
The sum of the reciprocals is (sum of numbers)/(product of numbers) = 27/180, which reduces to 3/20 after dividing numerator and denominator by 9.
Check: HCF(12, 15) = 3 and LCM(12, 15) = 60, confirming these are the correct two numbers. Also, 1/12 + 1/15 = 5/60 + 4/60 = 9/60 = 3/20, matching the result above.