The sum of the LCM and HCF of two numbers is 176 and their difference is 160. Difference between the numbers is 32. Determine the sum of the numbers.

The sum of the LCM and HCF of two numbers is 176 and their difference

1.	The sum of the LCM and HCF of two numbers is 176 and their difference is 160. Difference between the numbers is 32. Determine the sum of the numbers.
Answer» GIVEN :The sum of the LCM and HCF of two NUMBERS is 176 and their DIFFERENCE is 160. Difference between the numbers is 32.To find :Determine the sum of the numbers.Solution :LCM + HCF = 176 (1)LCM - HCF = 160 (2)Now adding both equations :⇒ LCM + HCF + LCM - HCF = 176 + 160⇒ 2LCM = 336⇒ LCM = 336/2⇒ LCM = 168Now putting this VALUE in (1) :⇒ 168 + HCF = 176⇒ HCF = 176 - 168⇒ HCF = 8Now we know,HCF × LCM = Product of 2 numbers.Now, difference b/w numbers = 32Let the numbers be 'a' and 'b' [Where, a > b]⇒ a - b = 32⇒ a = b + 32Now, ⇒ HCF × LCM = ab ⇒ 168 × 8 = (b + 32)b⇒ b² + 32b = 1344⇒ b² + 32b - 1344 = 0⇒ b² + 56B - 24b - 1344 = 0⇒ b(b + 56) - 24(b + 56) = 0⇒ (b - 24)(b + 56) = 0⇒ b = 24 or, b = - 56∵ We will neglect negative values here.∴ b = 24Now putting value,⇒ a = 24 + 32⇒ a = 56∴ Sum of the numbers = a + b = 56 + 24 = 80∴ Sum of the numbers = 80 [Option B]