# Find two numbers whose A.M. exceeds their G.M. by 12 and their H.M. by 2526. - Mathematics and Statistics

Sum

Find two numbers whose A.M. exceeds their G.M. by 1/2 and their H.M. by 25/26.

#### Solution

Let a, b be the two numbers.

A = "a + b"/2, "G" = sqrt("ab"), "H" = (2"ab")/"a + b"

According to the given conditions,

A = "G" + 1/2, "A" = "H" + 25/26

∴ G = "A" - 1/2, "H" = "A" - 25/26     ...(i)

Now, G2 = AH

("A" - 1/2)^2 = "A"("A" - 25/26)

∴ "A"^2 - "A" + 1/4 = "A"^2 - 25/26"A"

∴ "A"- 25/26 "A" = 1/4

∴ 1/26"A" = 1/4

∴ A = 13/2                 ...(ii)

∴ G = 6                     ...[From (i) and (ii)]

∴ "a + b"/2 = 13/2 and sqrt("ab") = 6

∴ a + b = 13,
∴ b = 13 – a              ...(iii)
and ab = 36
∴ a(13 – a) = 36        ...[From (iii)]
∴ a2 – 13a + 36 = 0
∴ (a – 4)(a – 9) = 0
∴ a = 4 or a = 9
When a = 4, b = 13 – 4 = 9
When a = 9, b = 13 – 9 = 4
∴ the two numbers are 4 and 9.

Is there an error in this question or solution?

#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board
Chapter 4 Sequences and Series
Exercise 4.4 | Q 8 | Page 60