Test Index

Sequence and Series

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Question : 61 of 86

Marks: +1, -0

Find the Numbers whose GM is 5 and

A M

is 7.5

12 and 13
13.09 and 1.91
14 and 11
17 and 19

Solution:

Concept:

For two numbers

a

and

b

, arithmetic mean

\frac{a+b}{2}

and geometric mean

\sqrt{ab}

.

Explanation:

Let the two numbers be

a

and

b

.

Given

AM = 7.5

, so

\frac{a+b}{2} = 7.5 \implies a+b = 15

.

Given

GM = 5

, so

\sqrt{ab} = 5 \implies ab = 25

.

The numbers are roots of

x^2 - (a+b)x + ab = 0

, i.e.

x^2 - 15x + 25 = 0

.

Solve:

x = \frac{15 \pm \sqrt{225 - 100}}{2} = \frac{15 \pm \sqrt{125}}{2} = \frac{15 \pm 5\sqrt{5}}{2}

.

Since

\sqrt{5} \approx 2.236

,

x \approx \frac{15 \pm 11.18}{2}

gives

13.09

and

1.91

.

Answer:

The numbers are

13.09

and

1.91

.

Therefore, the correct option is B.

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