The difference of two numbers is 20 and their product is 56.25 times t

1.	The difference of two numbers is 20 and their product is 56.25 times their difference. Find the LCM of the numbers. (a) 70 (b) 1125 (c) 225 (d) 5
Answer» (c) 225 Let the numbers be a and b. Then, a – b = 20 .... (i) ab = 56.25 × 20 = 1125 ⇒ b = \(\frac{1125}{a}\) ∴ Putting the value of b in (i) we get a – \(\frac{1125}{a}\) = 20 ⇒ a² – 1125 = 20a ⇒ a² – 20a – 1125 = 0 ⇒ a² – 45a + 25a – 1125 = 0 ⇒ a (a – 45) + 25 (a – 45) = 0 ⇒ (a – 45) (a + 25) = 0 ⇒ a = 45 or –25 Neglecting the negative value of a, b = 45 – 20 = 25 Now, a = 45 = 3²× 5, b = 25 = 5² ∴ LCM of (45, 25) = 3² × 5² = 225

The difference of two numbers is 20 and their product is 56.25 times their difference. Find the LCM of the numbers. (a) 70 (b) 1125 (c) 225 (d) 5

Answer»

(c) 225

Let the numbers be a and b. Then,

a – b = 20 .... (i)

ab = 56.25 × 20 = 1125

⇒ b = \(\frac{1125}{a}\)

∴ Putting the value of b in (i) we get

a – \(\frac{1125}{a}\) = 20 ⇒ a² – 1125 = 20a

⇒ a² – 20a – 1125 = 0

⇒ a² – 45a + 25a – 1125 = 0

⇒ a (a – 45) + 25 (a – 45) = 0

⇒ (a – 45) (a + 25) = 0

⇒ a = 45 or –25

Neglecting the negative value of a, b = 45 – 20 = 25

Now, a = 45 = 3²× 5, b = 25 = 5²

∴ LCM of (45, 25) = 3² × 5² = 225