Question

The AM of two numbers exceeds their GM by 10 and HM by 16. Find the numbers

Answer 1

Let the two numbers be a and b.

Their A.M. = `("a" + "b")/2`

G.M. = `sqrt("ab")` and H.M. = `(2"ab")/("a" + "b")`

We are given A.M. = G.M. + 10 = H.M. + 16

(i.e) `("a" + "b")/2 = sqrt("ab") + 10`  .....(1)

and `("a" + "b")/2 = (2"ab")/("a" + "b") + 16`  ......(2)

From (2) `("a" + "b")/2 - (2"ab")/("a" + "b")` = 16

⇒ (a + b)² – 4ab = 16(2)(a + b)

(i.e) (a – b)² = 32(a + b)  .....(3)

(1) ⇒ `("a" + "b")/2 = sqrt('ab") + 10`

⇒ a + b = `2sqrt("ab") + 20`

⇒ a + b – 20  `2sqrt("ab")`

So, (a + b – 20)² = 4ab

(i.e.) (a + b)² + 400 – 40(a + b) = 4ab

(a + b)² – 4ab = 40(a + b) – 400

From (3) (a + b)² – 4ab = 32(a + b)

⇒ 32(a + b) = 40(a + b) – 400

(÷ by 8) 4(a + b) = 5(a + b) – 50

4a + 4b = 5a + 5b – 50

a + b = 50

a = 50 – b

Substituting a = 50 – b in (3) we get

(50 – b – b)² = 32(50)

(50 – 2b)² = 32 × 50

[2(25 – b)]² = 32 × 50

(i.e.) 4(25 – b)² = 32 × 50

⇒ (25 – b)² = `(32 xx 50)/4`

= 8 × 50 = 400

= 20²

⇒ 25 – b = `+-  20`

25 – b = 20

⇒ b = 25 – 20 = 5

25 – b = – 20

b = 25 + 20 = 45

When b = 5, a = 50 – 5 = 45

When b = 45, a = 50 – 45 = 5

So the two numbers are 5 and 45.